Calculation of marine propeller performance characteristics

CALCULATION OF MARINE.

PROPELLER PERFORMANCE

CFIARACTERISTICS

L. C. BURRILL, B.Sc., Associate Member

A Paper read before tke North East Coast Instiiutibñ of

Engineers and Shipbuilders in Newcastle upon Tyne

on the 3rd March, 1944. .

(Excerpt from the Instil ition

.'

Transactions, Vol. 60)

t944 NEWCASTL'UFoN TTh

PUBLISHED BY TILE NORTh. EAST COAST INSTITUTION OF. ENGINEERS AND SHIPBUILDERS, BOLBEC HALL

-F LONDON

E. & F. N. SPON, LIMITED, 57, HAYMARKET, S..W.L

THE INSTITUTION IS NOT RESPONSIBLE FOR THE

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MADE AND PRINTED IN GREAT BRITAIN

CALCULATION OF MARINE PROPELLER

PERFORMANCE CHARACTERISTICS

By L. C. BuuuLL, B.Sc., Associate Member

SyNopsIs.The paper is concerned mainly with the development of a complete strip-theory method of calculation for marine propellers which includes the cor-rections necessary to take account of the differences between wide-bladedmarine.

propellers and normal air-screw types, in such a way that satisfactory agreement is obtained between theory and practice.

The usual momentum and thrust equations are modjfied to take account of slip-stream contraction consistent with the inflow velocities.

A serlésof standard curves are given which enable the lift and drag characterIstics

to be calculated for aerofoil and round back sections.

The effect of blade interference is presented in a manner convenient for

calcu-lation purposes.

- (1) Introduction

pRESENT-DAYmethods of designing marine propellers are based mainly on the results of systematic experiments with small model propellers,

such tests having, been carried out as a part of general research by most

of the leading experiment tanks

_{The main dimensions and proportions}

of a new propeller are usually determined by interpolating the results

of open-water tests with the aid of design diagrams expressed in

terms

of basic variables which embody the appropriate laws of comparison

and provide a very ready means of determining the optimum pitch-ratio

(or blade angle) for any given conditions of working. When sufficient

experience has been gained with the, uses (and limitations) of such design

diagrams by analysing the results of trial trip and

voyage data for a

large number of vessels and propellers of various types, this method

_of

approach proves very satisfactory in practice, and must be regarded

as the fundamental basis of general design work. At the same time,.

since the number of combinations of variables such as section shape,

blade shape, thickness, centre-line camber and pitch variations,

etc.,

which can be tested by means of models, is very limited, it is desirable

to have a more detailed method of calculating the performance

charac-teristics for given propellers, by means of which all the details of a new

design may be taken into account.

Such a thethod of calculation is provided by the modern vortex theoryOr

strip-theory, which has been developed mainly in connexion with airscrew work and is based essentially on wind-tunnel experiments with aerofOils; As

a result of the work of Prandtl, Glauert, Goldstein and Lock, to mention only a few of the most prominent workers in this field of enquiry, the basic theory has reached a very high standard of development and has been the subject of numerous technical papers and memoranda published in aeronautical circles. For airscrews having long narrow blades and low solidity (or disc-area ratio) the practical application of the theory presents little. difficulty, and very good agreement between calculated and experimental results has been established for some time. For marine propellers, on the other hand, the application of the theory as preseated by Glauert. or Lock, for example, leads to quite important differences between the calculated and experimental results, the discrepancies in power absorption being in some cases of the order of about 30% for propellers of quite. moderate disc-area ratio.

The general principles of the vortex theory and its application to marine propeller work were the subject of a paper entitled "The Vortex Theory of Propellers, and its Application to the Wake Conditions existing Behind a

Ship" read by Perring before the Institution ofNaval Architects in 1928, and have alsO been dealt with recently by Dr. I. Lockwood Taylor in his paper, "Screw Propeller Theory," read before this Institution in 1942. The present paper is F

270 ccUi.ATtoN OF MARThE PROPELLER 'PERPORMANCE CRABACrERISTICE concerned principally with the development of a complete practical method of calculation for marine propellers which includes all the correction factors necessary to take account of the differences betweenwide-bladed marine pro-pellers and normal airscrew types, in such a way that satisfactory agreement is ábtained between theory and practice. To this end, the usual momentum and thrust equations have been modified to take account of slip-stream con-traction consistent with the inflow velocities, a series of standard curves and

correction factors have been introduced to enable the appropriate lift and

drag coefficients to be estimated from the section drawings, and the effect of blade interference has been presented in a manner convenient, for calculation purposes. The method of calculation is presented in a form which willenable the performance characteristics of any propeller to be calculated as a routine operation in a normal shipyard or engine-works technical office.

(2) DeflAition of Angles

Fig. 1 shows the various angles which will be referred to in the text. The blade angle 0 at radius r of the propeller corresponds to thegeometrical pitch angle of the blade and is given by Pitch/

_2r.

The propeller is assumed to be rotating with uniform angular velocity l and advancing with uniform velocity V relative to the undisturbed streamlines at some distance forward of the screw disc, so that the corresponding advance angle ' is given by tan

1 V/

_{Ot '/2'}

_nr where n = revolutions per unit of time.

The axial and rotational inflow velocities at the propeller blades are assumed to be functions of V and fir and are represented by a V and a1flr respectively, so that the hydrodynamic pitch angle is defined by the equation

V+àV

V l+a

1+0.

tancf,==

_{f1raflr -}

_{1_al}

tançp.

The geometric angle of incideftee', is given by

i = 0

p, and the angle

between the face pitch line and the no-lift line of the section is (oi,, + rJ) where = no-lift angle relative to the nose

tail line xx and a = nose tail slope, so

that the true angle of incidence o

is given by r = ; +

+

. The angle between v' and '/ is designated .

The angle Oo which is equal to the face pitch, angle 0 plus the no-lift angle (plus the nose tail slope if any) may be conveniently termed the effective pitch angle of the section under consideration.

The pitch angle of the helical vortices in the ultimate wake at infinity behind the propeller is designated c and is givenby the equation tan - .

as. will be seen later.

(3) Goldstein Correction Factors

For propellers, having an infinite number of blades theinflow velocity may be regarded as uniform round any elementary annulus between radius r, and r + dr, but when the number of blades is small the inflow velocity will be at a maximum in way of the actual propeller blades (or helical vortexsheets) and will fall off to a somewhat lower value at positions between theblades. This effect was first discussed by Prandtl who alcuIated the flow corresponding to a series of parallel finitelifting planes spaced equally apart, and introduced the well-known Prandtl correction.

2 B/2 1 x . B = blades

f =; cos ' e

x. sin where _{= r/.R}

which appears to be generally adopted by Continental and American writers. The flow past a. series Of helicoidal surfaces 'of infinite length, which approxi-mates more closely to the actual propeller conditions, was later investigated by Goldstein (Ref. 1) who obtained an expression for the ratio between the mean circulatioti and the circulation at the helicOidal surfaces in terms of the. pitch angle 'of the helicoidal surface at radius r, the outside radius of the

CALCULATION OF MARINE PROPELLER PERPORMANCE cHARAcrEIUsrics 271

surface R, and the pitch or spacing of the surfaces, which is a function of the number of blades.

The Goldstein correction tà the usual vortex theory for infinite number of blades takes the formofa correctionfactor 14 (Or KE) which represents the relationship between the mean inflow velocity taken round any Snniilüs and the corresponding inflow velocity in way of the helical vortex sheets. The appropriate values of this inflow cOrrection factor for three- and four-bladed

propellers may b taken from Figs. 2 and 3in terms of

(or e) andx= r/R

foE the radius under consideration, the values shown in these diagrams being those calculated by Lock on the basis of Goldstein's equations and published in R. & M. 1,674. It may be noted that these values differ somewhat from those shown in Table 2 of Dr. J. Lockwood Taylor's recent paper, which are thosepublished by Schimamoto in Werfi Reederei Hafen, 1933.

In applying the Goldstein corrections to the design of airscrews iii R. & M. No. 1,377, Lok makes the assumption that the pitch angle of the helical vortices in the ultimate wake is equal to the pitch angle in way of the blades, but since in

the present paper the effect of slip stream contraction is introduced in the

equations, two values of the Goldstein correction, have been used, namely 14. corresponding to the hydrodynamic pitch angle tf' in way ofthe propeller blades and k0 corresponding to the pitch angle of the helical vortices in the ultimate wake.

(4) COntraction of the Slipstream.

Fig. 4 shows the conditions of flow through the propeller disc, corresponding

to radius r1 of the propeller. The axial velocity at the blades is V (1 + a)

where a is the usual inflow factor and in accordance with the Froude momentum theory the corresponding axial velocity at the helical surfaces in the wake at infinity behind the propeller is V (1 + 2a).

Considering the quantity of water passing through an elementary annuhis between r1 and r1 + 8r1 at the propeller disc during one revolution and the similar quantity passing through the corresponding annulus between r, and r0 + 6r forward of the propeller where the streamlines are undisturbed, we have for continuity of flow

-V V(l+KqSa)

-x 2rr -x,&r, = p X

x.2icr18r1 . . (I)

where k# = Goldstein factor corresponding to the flow conditiOns at the pro-peller disc, n revolutions per second, 1!n = time for one complete revolution

-Further assuming that the inflow is constant at eveEy radius (i.e. a constant) for a lightly loaded propeller, which is consistent with the usual theory then

p -

. 2rr 8r0 = p

J

--

(I +kja) 2Tvr1dr1. . (2)

from which

'= (1 + k.a)z

or approx. (1 + +k#a.) . . (3)

substituting this in (I) we have

-8r0 x r, x

= 6r1 x r1 x (1 + ka) x 27r

. _. (4)

8r0

-(5)

Now determining 81k, in terms of 8/0 we have

8r0 x2rr0x8/0=8r1x2r1x8/1

. (6)

or = 1 + k4sa. (which is correct) . .

Further, if r2 is the corresponding mean annular radius at infinity behind the propeller and k is the Goldstein factor corresponding to the flow conditions

272 CALCULATION OF MAJNE PROPELLE&PE1ffORMANCE CHARACrERISTICS at inflnity behind the propeller (i.e. pitch and siope of the vortexsheets at co') then by similar reasoning we obtain

(1 ± 2k,.a) ,vr22 =. (1 ± k#.a) r12 -. . . (8)

or r2/,.

(1 ±

4 ₍₉₎

1±2ka1

Summarizing we now have

ro=.r1x(l±k#.a)

rnX(1±2k:.a)

. .

(5) Momentum and Thrust Equations

Using the above result, the quantity of water Q Sand the elementary thrust dT, expressed in terms of V, a and r1, may be obtained as follows for positions

at infinity ahead of the propeller, at the propeller disc, and in the ultimate

wake.

6..j

rfl

(a) Infinity ahead

= V.2Trr, x

r0.p = V.2irr1(1 + ka)* x 8r1(l + kaYp.

= V.27rr1(1 ± ka)r.p.

While the momentum equation for thrust bebmes

dT0 = V x V.2irr1(1 + k.a)dr1.p. (12)

(b) At the.propeller disc (mean)

= V(l + ka x 2ir1.r1.p.

-. . . (13)

and dT1 = V(l + ké,a) [V(1 + k&a)2iw18r,] p. . (14). from which the mean thrüt at the propeller disc

dT dT0

kaV2(1 + ka)2ir drjp.

. . . (15)

(c) Ultimate Wake (at infinity behind)

V(1 + 2ka)2r,r2.p = V(l + ka)2r18r1p.

. . (16) and dT2 = V(1 ± 2ka) [V(l + kã)27rr1 dr].p. . (17)

So that the net mean thiUst in the ultimate wake

dT3 - dT0 = 4r1 kaV2[1+ kä] fr1.p.

(18)

which may be expressed in the form dT

= 4r1 k.a. V2 [1 + k./.a].p . . (19) This expression is comparable with equation (4) in Perring's paper when

= 1 O (for infinite number of blades). i.e. dT = 4vprV2(1 ± a) a.dr.--1erring

CALCULATION OF MARINE PROPELLER PERPRMANE CHARACTERISTICs ₂₇₃

but differs from equation (22) in R. & M 1,377, which is

- cos .

Lock (22)

where

u = aV,u = V(l + a) and

= lJk, and iS thus equivalent o

= 4irr, kaV2(l ± a). p.

. _{.. (20)}

It will be seen that the use of equation (19) in place of equation (20) leads to an increase in the calculated value of-the inflow velocity a,- which has the effect of reducing the final thrust and torque obtained from the _alculations. This effect is greatest in way of the tip sections, typical values of K and IC being 63 and 56 respectively for x = 9, and may therefore be considered

as a tip

correction

Equation (19), which has been obtained from momentum cOnsiderations, may also be derived in terms of the vortex theory method of approach used by Lock by comparing the thrust at the blades to the thrust in terms of the mean circulatiOn in the ultimate wake as. shown in Appendix I.

(6) Thrust Equation in Terms of Lift and Drag of the Blade Sections-.

To evaluate the elementary thrust in terms of the lift and drag of theactual blade sections, the Kutta-Joukowsky circulatiOn .or "vortex" theorem,which

states that the lifting force produced by an aerofoil is directly proportional to the product of the strength of the circulation flow and the velocity of advance, is applied and we have .

L=p.W,r.

where I' = circulation strength, L = lift force,

Vt', = local stream velocity, from which

-= B.c. p. WL2(CL.COS #. - CD.Sin qS)compare Lock (26) . (21)

where B number of blades, C chord, CL = lift Qoefficient

-Drag CD = drag coefficient

so that = B.c.p.(1 + a)2. V2 (CL. cos - CD sin ) _Sin2 . (22)

(7) DeterminatiOn of inflow Factors "a" and ' a'" in 7'nns of_c6. From paragraphs (4) and (5) we now have two expressions (19) and (22) for dT/dr,.

Equating these, and writing tan y = CD/bL and e = solidity where x = r/R, we obtain

a

(1 + /cç6a _(cL.&

_{Cos ( +}

23

-1 +a'\ -1 ±a I

2kcI

2sin2çS.cosy

..

(

In a similar manner two equations for dQ/d,, can be obtained, as follows, i.e.

4irr2 ka' ). V(1 + ka)p.r,.

. TOrque

(24.

and

=

W,2(CD. cos ± Ci:.. sin g). (25)

These two equations may be equated as before, and we obtain

a'

(.1 + k.a \

(CL.a \

sin ( ± r).

274 CALCULATION OF MARINE PROPELLER PERFORMANCECHARACTERLSTICS

Ii will be seen thattheterm( 1 is common to equatiOns (23) and (26), and

is shown in appendia LI to be equal to i p . (27)

where is the inflow angle.# - ip.

(8) Relationship between Angle of Incidence o and Ljft Coefficient" CL."

Fig. 5 shows the relationship between liftcoefficient CL and angles of mci-dence for a typical aerofoil.

In this diagram, the line AA represents the

theoretical relationship between CL and where , (theory), is the value of the nolift angle relative to the nose tail line calculated by means of the Glauert theory of thin aerofoils (R. & M. 910), the slopeof this line being 2 per radian, or 1096 perdegree.

The line BB rOpresents 'the relationship between lift coefficient and angle of incidence as Obtained from actual wind-tunnel experiments (corrected for infinite aspect ratiO and tunnel interference, etc.). For the range of angle of inidence th *hich we are concerned in marine propeller design,this relation= ship is also lineax for' aerofoil-type sections (round-back sections will be -dealt with later in section (10)) and may be represented by a straight line passing through (expt.)

(expt)

where

(theory) -

.

'.'

. . (28)

and having a slopO Ic3 x 2 where K and K3 areftinctions dependent on thick-ness ratio, centre-line camber and position of maximum camber, as will be seen later.

The line cc corresponds to ihe results obtained

from experiments with

cascades of aerofoils, the experimental no-lift angle corresponding to single aerofoil results being diminished by the angle ixg where _(theoiy) kgz, and the slope being reduced to kgj x k x 2iv

where K is a function of gap ratio

and blade angle. - ' '

The following expression

CL (in cascade) = IC X k3 X 27C X (ct°si

tg°) x'3 .-

. (29) is therefore Obtained as the final relatiotiship between lift coefficient and angle

of incidence. A method of determining the

appropriate, correction' factors.

Kgs and K5x will be dealt within section (11).

(9) Expreàlon for Angle of Incidence (including Cascade corrections) From the foregoing paragraphs we now have

the following equations

:-(+ka\

I'CLG\

cs(+y)

1 +a

_{I +a )}

k2K,)

2siii2.cosi

a'

(1+Ka'

(c.a'.sin(+i

' _{. (26)}

1a1 \ 1±a /

-

\ 2K/ .sm2#.cosi

1-CL

K, x K x 2ir x (rx0, - oc) X

. (29)

so that, as shown in Appendix ifi,

the following expression for

° can be

obtained.

(+go)

(j'5)xKe.siñ#.tanP)< [1(1=K)] (30)

(23)

çALCULAIIONOF MARINE PROPELLER PERFORMANCE CHARACrERLS'TICS 275

This expression will enable the value of a to be calculated for any assumed value of (3 and since the value of + (3 is equal to O - p, the correct value of &' can readily be obtained by a process of progressive approximation, as will be seen later.

The appropriate value of CL can then be calculated from (29), and the

elenfents of thrut and torque in terms of

and can be ascertained froxn the expressions

K1Q=. (1a').(l +tan2).CLsmS

(31)

2

2

K'r

_{- d.i}

_dx

x.tan (# + y) . .

(3 )

the value of a1 being most conveniently calculated from thCs expression

(tan-tan).tan(+y)

=

_{l+tan.tan(±y)}

. . . .

. 3

(10) Calculation of Ljft and Drag

To obtain a systematic method of calculatidn which will take proper account of changes in design, it is essential to have a systematic method of determining the lift and drag characteristics of the blade sections. The method suggested for this purpose isthat of relating the results of actual tests to the cakulated results of. aerofoil theory by means of coefficients plotted in relation to the

;

significánt variables, (i.e.. thickness ratio, centre-line camber, position of maxi-mum camber, etc.).

As will be seen from Fig; 5, the determination of the no-lift angle a, is of primary importance. The most satisfactory method of determining the theo-retical no-lift angle is that developed by (llauert in R. & M. 910, the value

of a0beinggiven by z, = J'°y./ixdx.

where

f1(x) = l/i.(l -. x)Vr(i - x)

. . . (34)

The integration of y.f.(x) may be carried out in a simple manner by applying the following formula, which has been derived by adopting a division of the chordal length into 20 parts and applying an adaptation of Simpson's multi pliers in conjunction with the f1(x) coefficients given in R. & M. 910.

ao(meory),(fyj+f2y9+f3y3...f19y1

. .

(3)

where the values of f1, f2, f3, etc., are given in Table 1.

From this table, it will be seen that the shape of the tail part of the centre-line camber curvC is very important in determining the yalüe of a,, and this is confirmed by actual results with marine propellers. Various simple forrnul have been suggested for the determination of a,, butthe Author prefers to draw out the actual centre-line camber in any case, and the above method can then be applied very readily.

To determine the actual value of a, in terms of a,

(theory) the values of Kao given in Fig-6 to a base of thickness ratio £/, are applied to the results of the above calculation.

It will be seen that this correction. factor,, which represents the mean of a large number of aerofoil tests, has a mean value of about 92 up to thickness ratios of '10 and then diminishes to about80 for a thickness ratio of '20.

The slope of the lift curve may now be determined by applying the factor actual slope

k, shown m Fig. 7, where k

_{- theoreti}

_slopeand is plotted to a base of '/, the effect of centre-line shape being negligible in this instance. Theabove curves have been based on tests carried out in the Variable Density Tunnel at Langley

276 CALCULATION OF MARINE PROPELLER PERFORMANCE CHARACTERISTICS Field, U.S.A., and published in N.A.C.A Repbrte,352, 460 and 628, but have been checked against other data published in Germany, and give very satisfactory agreement.

To determine the appropriate drag coefficient for any angle of incidence use is made of the fact that the drag cjirve is approximately parabolic in character, and rises equally on each side of the minimum 'alue CD tiu. The value of

CL opt. which corresponds toCD

mu. as obtained from an analysis of test

results is shown in Fig. 8 forvariôus values of and centre-line camber. These results have been plotted in Fig. 9 to a base of for various positions of maxi-mum centre..line camber, and for practical calculation purposes the empirical formula

CLopt. = (127 - 48

t/)

x centre-line camber . (36) can be applied up to thickness ratio of about 22, which represents a high value for marine propellers.

To determine the value of C for any given value ofCL itis thst of all neces-sary tO calculate the value ofCDmiii. using the formula

CDmiii. .0056.+ .01 L/

4 .l0(t/)2 + k

. . . . (37) where k2 is an empirical constant obtained by analysing the results of actual

tests for standard aerofoils, and then to add the small quantity i

CD

= Cv

-. Cv mun. which is given by the formula

Cv = K3

(CL -

CL opt.)2 . . . (38)

- The appropriate values of I'2 may be obtained from Fig. 10 in which this

constant is plotted for variOus values of the centre-line camber to a base of '/, while the values of 1(3 may be obtained from Fig. 11 for values of CL - CL opt.

less than 5O and from Fig. 12 for values of CL - CL opt. greater than 50.

For small values of CL CL opt. the approximate formula i CD = 0062(CL

-CL tj8 may be used;

(11) Round-back Sections

The methOds of determining CL and Cv outlined in the foregoing paragraph are based on streamline-shaped aerofoil sections, and are not applicable to round-back (or segmental) sections. In dealing with the latter type of section, advantage may however be taken of the fact that all round-back sectiOns are similar in shape, so that CL and CD may be standardized on a base of thick-ness ratio (or effective thicicthick-ness ratio where the edge thickthick-ness varies from a standard value of about 1)08c).

From an examinatiOn of all the available data for this type of section, which is reasonably consistent although not extensive; the curves shown in Figs. 13 and 14 have been obtained, from which the appropriate value of CL and Cv may be determined in terms of thickness ratio and angle of incidence relative to the flat underside or pitch face. In order to indicate the accuracy of the curves some of the actual results for given thickness ratios have been indicated in the diagrams. Comparing these round-back results. with the method out-lined above fOr aerofoil sections it is found that the values of Kcco agree very satisfactorily and that the values of K810 agree with the limiting slope of the lift curve at CL = 0. The lift-curve is however no longer linear, but curved and concave towards the base line. For this reason Gutsche uses two charac-teristic straight lines to represent the values of CL above and below zero angle

of incidence cc = 0, but from- the curves given in Fig. 13 it has been

found that the lift curve can be represented by means of an equation of the form

CL.= K. 27r.cx. X K,.. . . - . . . (39)

where Kr.b. = 1 - 02 (cc) so that the method of calculating CL outlined in inction (9) may be extended to cover round-back sections by introducing the further correction coefficient Kr.b. =1 - 02 (cc) which can be used in equations (29) and (30) in a manner which will be described later in connexiOn with the calculation of for any given section.

CALCULATION OP MARfl1! PROPELLER PERFORMANCE CHARACTERISTICS 277

To determine the drag coefficient for round-back sections use is made of Fig. 14, the appropriate value of CD- being lifted for the calculated lift-coefficient and- thickness ratio, as follows:

The angle of mcidence relative to the flat underface is obtained from the expression i (a x kgs)

+ ag and the corresponding value of Ci

is lifted from Fig. 14. The final value of CD is then given by

CD Cri + CL [ao - F4S.Oi°j X . ₍₄₀₎

this value of CD being referred to the hydrodynaniic pitch line corresponding

to the angle #.

-(12) Cascade Effect or Blade Interference.

The effect of blade interference may be estimited from aerofoil theory in a manner similar to that discussed by Weinig in his paper "Strömung Dirch Profilgitter" published in "Hydromechanische Problëme des Schiffsantriebs" 1932 but the Author prefers the experimental approach of F. Gutsche who

tested series or cascades of aerofoils and round-back sections at different

angles of pitch and various gap ratios.

The necessary corrections to the lift curve take the -form of a :correction to a,, and to the lift curve slope and Figs. 15 and 16 show the appropriate cor-rection factors Kga, and Kgs in terms of 1, and . These curves have been based on the resü1t published by Gutsche in 1938, the only difference being that Kgcx has been expressed in terms of the theoretical a,, in place of thickness ratio. These results differ somewhat from the earlier figures published by

-Gutsche.in 1933 and are believed to be the most satisfactory published figures obtainable on this subject, but it is suggested that these results should be amplified by fUrther systematic experiments, and a series of tests on this subject caEried out in this country would be Welcomed by propeller designers.

-(13) Standard Calculation Form*

An example of the staedard form of calculation for any radius which it has been- found convenient to adopt is shown in Fig. 17. This calculation applies

to 7 radius of the Dutch Tank B.4.40 screw having a pitch ratio of 10, the

working condition being that corresponding to J = V/nd = P70, or

(Taylor)

= 1447.

-In this instance, the values of l and 0,, are calculated from the propeller drawing to be 1766° and 27.74° respectively, and Kgc,,

= 87 while Kg = 874.

As a first approximation, the value of a may be obtained from the empirical formula

a= (0,, ')x (52. 2x)= 1008 x 38 = 384°

.- . . (42)

.Thisgives -

#=:0o_.23.9O°

and.

_=#çp=624°

. .

-The correct value of e is then given by the expression

4

tane

_{1+2tani'.tan(±y)tañ#.tan(#±y) .}

.

(3)

but for practical purposes this may be taken á.s

- (44)

The value of Ke frOm Fig. 3 is 8l2 and the value of C2 _in equation (30) is given by

2 x573

. 1146

C2

₌

_{T 2965 X 3905 X 874 -}

1133 - also sin

= 4051and tah f3

1093

so that a2 becomes ll33.x 812 x4051 x 1093 =408°.

Note :DeaI isigonometrical tables suitable for use in connexián with these caiciilatláns, published by the Civil Sesvice Coinmlonth, are obtainable fresu H.M..Statlon.ry Othce price id.

278 ALCLAT1ON OP MARJNE.PROPELLER PERPORMANCE CRABACIERISTICS The small deduction represented by a x (1 in equation (30) may then be calculated, and is found to be

119 x 1093

408x

₄₄₃₁

12

so that a3

ecoines 4O8 - 42° = 396°.

The next approximation toais then given by + &'

where 8a=(a3a)x3+'12X

i.e.

.= '3'84° + '07°= 39l°.

This is placed at. the. head of the next column of the table, and the calcula-tion is repeated This gives a new value of a = 39l° and it' will be found that

agreement between a and a can usually be established in two

applications of this process, to the second place of decimi1s

The remainder of the calculation is stiulghtforward, and the values of the elementary thrust, torque and efficiency are found to be

K1Q.7 O595 .

36l5

= '678

As a check on the arithmetic of the calculation, the value of may then be calculated from the expression '

_{- tan (q}

and this should agree to three

plces of decimals. ..

In dealing with round-back sections, the process of determining ais slightly more complicated since the value K.,.o. = (1 - O2a) must be introduced in the expression for C2 according to the assumed value, of a and the value of CL). must be determined from Fig. 14, in the method describedin section (11).

For the case under consideration, assuming the aerofoil section to be replaced by a segmental section of similar thickness-ratio

= 0728 the following

modified results are obtained

:-4.17 x 90 = 3'75°;

Lccg = .150. Oo = 2804°.

K,.b..= 1 02x

a°=

.92sothatC,becomes23

l231

a is then found to be 4'23 and' = 2382,

while CL =

(45)

and since it is usually found that the results for '7R are representative o the vvhole propeller it will be seen that the efficiency of 'the equivalentsegmental propeller would. be about 5% less than that for the aerofoil propeller.

In order to standardize the method of calculation, it hs been found con-venient to calculate the section characteristics such as centre-line camber, position of maximum camber, nose tail slope, calculated at,, etc., from the. actual propeller drawing and to, plot these to a base of x = r/R when a series

of 'fair curves should be obtained as shown in.Fig. 19.

-Calculations similar to ihat shown in, Fig. 17 are then made for x '2,

4, '6, '7,

8, 9 and '95 from which the thrust and torque grading curves may be plotted as shown in Fig.. 19.

A suitable method of. integrating the thrust and torque grading curves is

shown in Fig. 18, Simpson's multipliers being used in conjunction with a close division of ordinates, particularly near the tip since the final integral is sensitive

to changes in this region.

-CD =,Ci -i-

x3529 = '0110 + 0034 = 0l44.

from which tan y = '0414

?°. = 237°

CALCULATION OF MARINE PROPELLER PERFORMANCE ca&cmnisiics 279

From Fig. 18 it will, be seen that the final result of the calculation for the

B440 screw compares with the test results as follows

:-Kq = 03000 by calculation 1,

' 2.

°'

C {)2940 from curves erence

fo

Kr = 18337 by calcUlatiOn

'

difference + 24°'

C1 = 1790 from curves

)

= 682

by calculation .

"

= '680

from curves WUerence V ' /0

The final integral figures for the same propeller but with. round-back sections

are as follows

:-KQ =''03247

Kr = 18811

-7

=645

which may?. be compared with. the Standard Taylor data for 4( disc-area ratio

as below..

= 031O(J (i.e. difference + 47%)

= .634

(incL boss effect).

(14) DIscussion of Practical Applications

The method, of calculation described in the foregoil3g sections has been applied to a wide range of. propeller .types over a period of several years and has been found to give very satisfactory results. For example, test calculations

have been made for standard screws of B.440, A.440,. E.455, B.335, and

B.35O type covering a range of J values from 50 to 90 (or = 100 to 200) and' pitch ratios of 80, 90 and 10, and it'has been found that the calculated thrust and torque coefficients agree- within ,5% and efficiency within ± I %, the calculated torque being usually about 3 to 5% in' excess of the test results.

Satisfactory agreement, in respect of optimum pitch ratio has also' been

obtained. , .

Other calculations have been made for three

and four-bladed ãerofoil and round-back propellers .up to 90 disc-area ratio and also for high slip values

up to 73%, with similar atreement, so that the method may be 'used with

confidence for design work. . . .,

For practical design purposes the procedure 'usually adopted is to prepare a preliminary propeller drawing based on routine design methods and then to make a complete calculation for the assumed mean wake conditions. This should give satisfactory agreement with the. designS conditions and may then be used as a basis for further investigations. For example, calculatiotis may be made to check the optimum diameter and pitch-ratio corresponding to the proposed propeller particulars, by operating with the 7 section only; since the results for this section may usually be taken as characteristic for the whole propeller. Various pitch variations from root to tip may then be examined

and also the effect of typical radial wake variations based on model tests. Typical sections, such as those at the top of fillets, at mid-blade, and near the tip, may then be investigated to determine the optimum centre-line camber, and a complete calculation for the final propeller will servO as a final check and provide valuáblO information regarding the radial distribution of loading for strength and cavitation calculations, etc.

Since the appropriate angles. of incidence, lift coefficients, inflow velocities, etc., are known for each section of the propeller, pressure distribution .calcu-lations may be carried out by means .of the well-known Garrick-Theodorsen method of calculation, published in several Reports of the. NatiOnal Advisory Committee for Aeronautics or by the methods proposed by Dr. Lockwood Taylor in his recent paper, on this subject read before the Institution during the current session, and although the effect of blade interference (or cascade effect) requires further investigation and comparison with experimental results, the information which can be derived from the results Of such calculations has already proved to be of considerable practical value in design work.

28) CALCULATION OF MARINE PROPELLER PERFORMANCE CHARACTERISTICS

BIBLIOGRAPHY

S. Göldstem, The Vortex Theory of Screw Propellers, Proc. Roy. Soc. 1929.

W. G. A. Perring, The Vortex Theory of Propellers, and its application to the

wake conditions existing behind a ship, I.N.A. 1928.

J. Lockwood Taylor, Sciew Propeller Theory. N.E.C. 1942.

C. N. H. Lock,. M.A., F.R.Ae.S., An Application of the Prandtl Theory to an

Airscrew. R. & M. 1521..

C.. N. H. Lock, M.A., F.R.Ae.S., Tables for Use in an Improved Method of

Airscrew Strip Theory Calculation. R. & M., 1,674.

C. N. H. Lock, MA., F.R.Ae.S., Application of Goldstein's Airscrew Theory to

Design. R. & M., 1,377.

H. Glauert, A Theory of Thin Aerofoils. R & M., 910. Aerofoil Data, Reports 352, 460, and 628, etc. N.A.C.A.

L. Troost, Open Water Test Series with Modern Propeller Forms (Results for

B.4.40). N.E.C. 1931.

Fritz Wëinig, Stramung dUrch Profilgitter. Hydromechan1sche Pröbleme des

Schfsanfrlebs. 1932. .

-F. Gutsche, Einfluss der Gitterstellung auf die eigenschaften der im

Schiff-schrauben entwurf benutzten Blattschnitte. Schiff. Techn. GeseIl. 1938..

BooKs." The Elements of Aerofoll and Airscrew Theory." H. Glauert.

(2 At

CALCULATION OF MARE PROffLLER PERFORMANCE CBARACrERISTICS 281

APPD

I.

EQUATiON - FOR d3 IN TERMS OF THE MEAN cIRCULATION IN THE ULTIMATE WAKE.

(See Lock R,M. No. 1371 for notation)

(i) At propeUer

disc-=fW.L.(LOCkeqn.l7). 3= B.COS+.(LOCkI8).

= B.p.r.

.pv.(l+a).

whereW

infinity

behind.-= B.p..W2.cosE

= K

2'rc.r2TanE'

_{d w}

v+xu

V(+2lc a

2' _B en _{2vnean Sin E'}

_{Sin '}

k 2itr2ian

'

V+

drá

& e"z

B Tan B

-.

=

f.ke.Uz 2ur2.(v+ku)

but r2 =

r1 (:::Y

-- _and

fl+k

\V2

di = dii

or

=

411.ri.kiaV(I + ka)p (ie. eqn,19)

APPENDIX II.

EQUATION FOR

I+ka

P +a Tän'ili

-

l+Tan.Tan(+)

p- -Tánr

_I

(I

_l+Tan.Tan(+}

Tan '4r+ Tan +.TanlPTan (+ ) + Kian - KTan.

Tan.Tarn}Táñ(+W) +Tan4.

1. -

(HK4(TancTan)

I Tan.(l+Tarr4r.Tan 1i+j)

When Tan S' = is small compared with Tan, suth as is usually

correct, then neglecting .

=

I (IK.).Tanp

P _Tan+ wherep

282 CALCULATION OP MARINE PROPELLERPERPORMANCE CHARACTERISTICS

By efirition_.. Tan jb -. Ten

i....

Cos(+')

\.2KJ

2 S1n20.CosV.

±

Sin.

\ 2i< 1 Sin 2.Cos V

from which

(.2t'

Ii±

I Cos(Ø+)

\2K)

in2.Cós Tan

L \2K)2Sin2Ø.CosJ

but

Krf. _o.Kgs where o

o4)- inEqn.

57.3

F. where

F -

Ks.1i.o.Ks

573 K:

o°

t

Sih20.CosY r1

F.oc. _TA p.Cos(0+V)

_j

- .F p.SinØ ) Lien /r.

Tanr 2 Sin.Cos

Tan 0. Sin 2 0 .Cos? Sii 2 $.Cos V cC°

p.Tan.Sin(Ø+) F.p.5in(Ø+y) Tan&.Tan(Ø+y)

1 Tãn.Sin 2$.CosY.Sèc(+V) Tan %lr. Sin 20.CosV.5ecO+y)

\. Tan.Tan(Ø.')J F. p.Tan'ô'.Tan(Ø-i-)

oe(I

s Tan .Tan(OY))

APPENDIX rn

EQ.VATION.FOR ANGLE OF INCIDENCE IN TERMSOF

NGLE5 AND \6'. 2 Sin.ø F.p. p Tan --.From p

Sin .20.CosY.

_{(TanO_Tan)}

F.p. Cos(O+').

Sin2O.CosV.(TanTaná).x

Cc5($+ Tan'&. Sin(O i- J) F.p.

_ 2 Sin o.coso.Cosy. (Tan 0

tan)

F.p. Cos 0 . CdsT'' Si 0.Sin ' sTan. Sin Ø.Cos'.+ianfr.CosSTn

Tan 0Tang

1+ Tonfr.Tàn 0 Tan0.TanV +Teh.TanV

(CL.

or (CL.

APPENDIX I CORRECT EXPRESSION FOR lanE. definition Tan & = Tan

Tan Tan

where a

Tar[r+Tan+.1an(++J

jTan Tan gr).'Tanf+b!

I + Ian .Tan ( +

and, a' =

so that TanC

CALCULATION OF MARINE PROPELLER PERFORMANCE CHARACTERISTICS 283

APPENDIX Ill (contd.).

o _2.Sinct,.. -I Ep. I+Tanc.Tan'4r TanY Tan Taniiy 2Sin.Tan Fp. ITanp.Tan cx? -2 Sini.Tán (1+ _-)

neglecting the small quantity £...

=

('

3)

_{K.Sin +.Tan x}

_[i

_{- f (}

and the 'small addition to ci.. represented by &ci.. may be conyeniently dealt with, after the drag coeff. C0 has been determined, by adding to CL the correspondin9 small quantity SC. where

C,. Tan p. K9s. Since -= K95x d..Tanp.Tan' 2TanTan* +TaniTan4,Tan (c+W) I + 2 Ta n

n(i.Tan *Tan

fl,r

This expression may be used for very low slip coriditions,when Tan is high,but for

normal design slip conditions the value of E may be obtined. with sufficient accuracy

284 CALCULATION OF MARE PROPELLER PEBPORMANCE CHARACTERISTIS TABLE 1 STATION POSITION - i i iris MULTIPIJERSL FUNCTIONS. FROM

LEADINQEDC.E f1,12 etc. (f1Ky1)etc

0Q5C

-

5'04

Y2

Ol

OC

338

015C

301

y4

020C

2-87

025C

281

y6

030C

-

284

-

035C.

292

Ye

040C

3.09

- -y

045C

332

050C

364

Yii

055C

407

060C

464

Y13

065C

5..44 Y14

O7.00

665

Yis

075C-

85.9

-Yue

080C

l-140

YI7

085C

1705 Yia

o.oç

3540

Yu9

095C.

18620

o(, (Theory)

_(f,yi

"2Y2 + f3y3+ :f19

Theoretical

x

Chord

.FIG.I. Didgrarn of Ariqies.

FIgs. 2 & 3 ... Plates VIII & IX.

PROPELLER DISC ULTIMATE Sc WAKE

---/

H

CALCUITIONVFMARiNEPROPELLEIUPER1ORMANcECHARAcTERTSTI 285 Sr. UNDISTUR8ED FLOW.

/

//

/

1/'

//

j

/

Meet Velocity 4j Meat Velocity P2

286' CALCULATION OF PROPELLER PERFORMANCE CHARACrERJSTICS

expt

.. theory

SLOPE OF AA 2tr per RADiAN

-SLOPE OF B-B K5 a 2i

SLOPE OF CC K.a K. 2n

C..cn cascade K.. * i (a /57.3

95 b SC 80

i!!!PIIIIII

"rh1

V

19111h

bli

Ilhb!u.IuuI!I.Ih

. k

IIIHIU!iiiii!!

- IS

u.I.

05

o6y

-IS 17 z-z #ILS OF

MAX CAM8ER 3O'. FROM L

CALCULATION OP MARINE PROPELLER PERPORMANCE ARACtE1US11cs 287

CAI.E OF T)CNE5S RA11O YAU OF SLOPE CORRECTION FAC1

FIG. 6.

SCALE OF THCIaES5 RATIO

VALUES OF NO-LIFT ANGLE C0ii0N rACiER FIG. 7. 050 030 020 21 4%

g 0

SCALE OF CAMBER

MAX CAMBER 40t PROM LE

4

I CAMBER

OLAGRAMS SHOWING ç FOR VARIOUS 'LUES OF 4t CAMBER AND POSITIONS OF MAX: tCAMBER

. . .3

CALCULATION OF MARINE PROPELLER pwo1u.cE cHARcrER1sTics 289

0070

030

Iues of,3

=

₌

LCI.

I

.l

1

4

SCALE OF CEPifRE-UNE CAMBER

F1G.IL VAL.UES.OF Ka FOR (CC-CLOP).LESS THAN 50..

I-290 CALCULATION OP MARINE PROPELLER PERFORMANCE CHARACTERISTICS

THICKNESS RATiO

= 08

0140

i.oMO

0120 '0100 0080 0060 0040 '6 '7 8 '9 tO SCALE OF Cc -Ct,,.) THICKNESS RA11O 16 .7

8

'9 I'O SCALE OF (c-c.) CDCD- CDM - K, (c,-c)' .5 '6 '7 '6 .9 SC.AiE OF

(cL-c)

THICKNESS RATIO - '20

4lllUU

i.I

.5 '6 .7 '8 '9 SCALEOF (cL-cL,,J

FIG.12 \MJJES OF K,FOR (C-C10,) ATER THAN .50.

0160 40 0120 0100 '0080 '0060 00040 0160 0140 '0120 0100 '0080 '0060

2

o

SCALE OF Ufl COEFflCIENT C1

8 o S.'

U.

5' i.. -C 3 9 U, p 0 z (a, w

B

.'

R

lIuuulIuIu...h:luu.uI.k'..uu..I..i.li...

...:'.

i'i...

5' SI '. '?

...5

'

I.u.iiu.0

UU 1U

IIUU

_UiiIURi

_l.U,I

uuIu.u...u..uu...i'...

I

u

.I

.iiiu.

I'...

iu.. USULuI

.

U

U.

U..'

U..

U...

U U U

XI

2 0 o0 -4

I

C, 1 m 40.

Jo

Th It SCALE OF DRAG COEFflCID1T o

g

llllii,iII,gIu'iuIIUtlIIIIillIP!

IUllIII,lITlU4UUllUlIIIPllIUI

.iiriLul,lIUIIUII!IIuuIPlIlIllBII

ll.IIltIIfIIIV1UIUP4UlUlUIIU11111U

,uiiiuIiIrIIIUuIupmUuI

UIl.XIULII1IlHUIIWáIIIIU

miulu

iii

imuuiixr

uuiruuus

.u.liwrn'UI'uIIIIUIIUIUIUI

IlIIIL1IiIULIlL'uuIIIU

RiiflIUlLIIuNI

UiiuUIIIIuI

,giii

rim.

iiauci'uui'ui

u.uu..u

0

2

$CALE OF DRAG C0EFF1CIaT C 0 2 (4t 'It' C,

(

(

N

S3LISDIaL3YUYH3

2NThlOFdd

3TI3tOid 1YP IO MOLLYIflD1V3 Z6Z

CALCULATIONOF MARINE PROPELLER PERFORMANCE 293

FIG. 17 ..CALCULATIO

DUTCH TANK

B 4 40

SCREWS

-'R-'7O

OF-THRUST, TORQUE &EFEICIENCYAT

I X'>

.7Q

j

a 7Q

_{J6= 1447}

DIAM 788

PITCH : 788.

t 3 75rn/n t,,

_{c -}

_'0728

c5k5Om/m

PITCH RATIO lOmax). _{3Q°X.9Q 279°}

Va 414 f.p.s. _{(nose-tail slope) 62°} N

-

C. p.S. _{Tan6° .V+Tr4545 .; 2444°} - BXC

4x5I53905

_{(0 + 0Q+ oc)}

-.o- Q37x3IO

27 85° 2irr K

K:Q37

:2774°

Tanljf/rnc

3l83 .'.$ _17:66:.

K5 - 943

Kg . 874. e0- J1 10 08 OG = . IO'08f52- 2x) = 38 Assumed0, '384° 391°

c

2 = . x

=e-o, 2390 2383

. 1133 943xux '3905 X'874 624 617 .. .-. . .

'$+$. 30'14 30-00

_{CLOP, 267} K1(fig3) O'8I7 0813 . C1

I

SInG 4051 .4040 _{CL- CLp 087} Tàn

lO.IO8I

. . Oo.CKgS,$'r 408° 4.030

-- Ch

'00760 Tans 443l 4417 _{4-0872x '0062}

_'00005

- K(fig3) '881 '881 _IIx354 '00068

- 12

- 12

00833 cx 396° 3.910 .

Tan==.O23S

Ct,,'943x21rx391x874

_{- 354Q}

57.3 . 1-35°

l'2x &24

_-'07

SC1 -'a X '00833 X-fO8I x874

0004

2383°.

=..:

C1 -3544

(+)- 25'IB

..

Cos' 9997 Tan 4701

Sin(-)

4255

C3=-=

3635

as=(Ta_T8)Ta

1234x4701 0481

l+Tenfanc+)

I+'44I7x470I .

(I-a')

- '.952

i4 =C3 x (I_a.)2 (I+Tan24)Cj. Sn(4H-') = .3635x.9063 X ll951X3544 x ₌

_.o590

..

.9997 . -2K'o 2X'0595 K 361 xTari(+i) 7X 4701

X

--2i'r

j(

= 6283- X '0595

-

678 CHECK7= Tan($t') 678

Fig. 18

0

0.c 0

JTCH TA4< B 4.40 SCREW

0.1WE$

ItGRONT16LET-ND TORQUE G4ADING

JÔ70.

PITcH RAIIO -I-Ô(ma447

X *

Ko SM. f(Ko) K, SM !00

-4

-975 00298 I 00296 01860 I 01660 660 0-0422 0-0211 02475 01237 665 0-0600 00500 02965 I 0-2955 oo 0-0644 34 00408 03233 ,% 02425 650 00592 2 0184 03550 2 0-7100 600 0-0612 I 00512 0-3693 I 02993 750 00613 2 01226 0-3700 2 0-7400 -700 00595 I 0-0595 0-3615 I 0-3615 650 0-0650. 2 0- 1100 -0-3360 2 06720 -600 0-0486-I 0-0488 0-3000 I 03000 -550 0-0409 2 0-0916 02560 2 06120 5O 00333 I 00333 02090 I 0-2090 -460 0-0260 2 00520 01635 2 0-3270 400 00195 0-0195 01249 I 01249 -350 00140 2 00280 00935 2 01870 -300 -0-0095 I 00095 00645 I 00645 -250 00057 2 0-0114 00395 2 00790 200-00096 1'2 - .00013 0-0173 V. 00087 24 08988 24 54926 069680-BQ

.0033 KQ+00IOO

, 14 - 0-02996 0-00004 -0-1830940-00026 (00300 Dy c.1cIolm) (018337 sy cfcIoon) (00294 C0 Iro.j c.rv.) (0-1790 C from q.re) F5cE' 2 D7FERENCE - +24t n . i .!3

-2Q 0-18337 0 6283.0 00300 s,

V (o,cato.)

_!.

("a.' Ce.1

0660 DFEREE+o-3 I

)P

I

,,--

.-.

,,._

r

.0

!i1I

0 -C2 0 00 0-4 Fig. 19 080 0 60 50 C. 30

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DISCUSSION ON CALCULATION .OF ::.MARINE

PROPELLER PERFORMANCE CHARACTERIScIICS*

Dr. E. V. TELFER, Associate Member:

This is a very useful paper and places at

our disposal a large amount of detailed

work already tested by successful practical

application. The paper, therefore, merits

our consideration and respect, since only

by such care for detail can better propellers

be produced. The following comments are also directed to detail.

Mr. Burrill in his Fig. 6 shows that the theoretical zero-lift angle always exceeds

the actual. While this is so for tests carried

Out at low Reynolds numbers, I doubt

whether it is sufficiently true for full-scale propellers to -justify the correction work

in-volved. The correctiOn admittedly.

in-creases with thickness ratio, but so also does the effect of Reynolds number upon

the blade resistance.

That the actual slope of the lift curve is less than the theoretical value of 2sr is, however,

fairly certain ;' and it does reduce with

increase in thickness ratio. The data which

I-. examined in -this connexion showed that

the decrease was linear with increase in

thickness ratio.; and extrapolated to zero

thickness ratiO the actual results endorse

the theoretical value of 21T. These data all

refer to segmental sections. Not all sec-tions show this same influence of the thickness ratio, but as the segmental type is at present the most important and most

practical, it is useful to note that by reduced thickness obtainable by the use of stronger

materials, theoretically attainable values

can be approached. The neglect of this changing slope value may have serious

consequences in the design of fast-running propellers of the destroyer type. The use of

too low a slope underestimates the thrust

carrier of the blade element and also

exaggerates the angle of incidence; the

accuracy of correlation with blade-cavi-tation tests may thus be destroyed.

The slide shown by Mr. Burrill of the suction contours on the back of a blade to

illustrate cavitation- zones is intriguing.

From a cavitation standpoint such a

dis-tribution is obviously xce1lent, but it has serious drawbacks from a blade-resistance

point of view. This follows from the fact

that disposing the rhaximum suctions

further abaft the maximum thickness than

usual increases the pressure drag very

considerably. This reduces the

blade-element efficiency and possibly by a greater

amount than would be gained by freedom

from cavitation. In uther words, an

efficient blade still- efficient in cavitation may be better than an inefficient section

free from cavitation.

There is one point in connexion with the treatment of cascade effect which can

use-fully be emphasized. it is essential, to-, Paper by L. C. Burrifl, B.Sc., Associaie Member,

Sec p. 269 ante.

...

/

wards the -boss particularly,. to allow for

the thickness of the root sections in ca1cu

lating the annular momentum. The effect

is

to reduce the thrust and the blade

efficiency at the root sections quite

con-siderably. It is clear that Mr. BUrrill does

not make this

cOrrection. In Fig.- 19 giving the radial 'ariation of elemental

efficiency he shows a maximum-efficiency at

the boss for aerofoil screws. This, I

sug-gest, is quite impossible, and shows that

theory requires overhaul in- its treatment of

the boss problem. I would strongly advise

against the feeling of complacency which a designer may acquire by his contemplatiOn

of a result which he knows to have an overall accuracy of the order of -97% and yet may have a lOcal accuracy of a much

lower order. The danger of this state of

affairs is seen when a designer imagines that because of the apparently high root

efficiency he can design to carry more thrust

at the root. Most experience shows -that it

pays to unload the -root rather than to load

it.

The unusual distribution of efficiency shown by Mr.. Burrill is accompanied by,

to me, an unusual distribution of incidence

angle. I generally find, with a constant

face pitch, that the incidence angle increases

towards the tip and does not decrease as

is shown by Mr. Burrill. Can the Author

state that the variation he finds is. the usual

one? If this is the case, it points to some curious difference of treatment which is worth further investigation.

Mr. Burrill's development of the

correc-tion for slipstream contraccorrec-tion is a useful addition to the subject. I would suggest,

however, that he should enlarge on the

explanation of the method and particularly show the effect of the correction on incidence

angle. -.

The numerical method used by Mr. Burrill for the calculation of zero-lift angle is one

which has been suggested by Glauert,

Munk and others. It gives good results, but I prefer the method given in my 1940 papert to this Institution, since it deals with slopes

of mean camber line which are generally more tangible than the ordinates, and the integratiOn process avoids wide variation

of multipliers. The method also enables

simple formuin to be derived foi the more

usual sections. Both methods

clearly-en-dorse, however, the importance of

trailing-edge pitch; and I have previously

em-phasized the practical advantage of checking the trailing-edge pitch of each blade -only

at the 07 radius to obtain a very close approximation to the true mean- - for all normally continuous pitch variations.

t "The Mean Pitch Determination of Variable

Pitch Propellers," by Dr. E. V. Telfer. N. E. C. Inst.,.

Yol. 56 (1939-40.) -

-H9