The efficiency of marine propellers and the drag coefficient

B

G. S. BAKER, O.B.E., D.Sc.,

Honorary Fellow

A Paper read before the North East Coast Institution of

Engineers and Shipbuilders in Newcastle upon Tyne

on the 23rd March, 1945, with the

discussion and

correspondence upon it, and the Author's reply thereto.

(Excerpt from the Institution Transactions, Vol. 61)

?WCATLE UPON TYN

PUBLISHED BY TEE NORTH EAST COAST INSTITUTION OP ENGINEERS AND SHIPBUILDERS, BOLBEC HALL

S

LOON

P. & P. N. SPON, LIMITED, 57, RAYMARRET, S.W.I 1945

A RCH;EF

Lab. v. Sdcvu e

Techc

THE EFFICIENCY OF M[NE

PROPELLERS AND THE DRAG

THE EFFICIENCY OF MARINE

PROPELLERS AND THE DRAG

COEFFICIENT

By

G. S. BAKER, O.B.E., D.Sc.,

Honorary Fellow

A Paper 'ead before the North East Coast InstitutiOn of

Engineers and Shipbuilders

in Newcastle. upon Tyie on th

231d March, 1945

zith

the discussib and

correspondence upon i, and the Author 's ,rely thereto.

(Excerpt from the Institution Transactions

Vol. 61)

-- NEWASTI UONTYN

PUBLISHED BY THE NORTH EAST COAST INSTITUTION. OF ENGINEERS AND SHIPBUILDERS, BOLBEC HALL

LONDON

E. & F. N.. SPON, LIMITED, 57, HAY(ARKET, S.WI

3945

THE INSTITUTION IS NOT RESPONSIBLEEOR THE,

STATEMENTS MADE NOR FOR THE OPINIONS EXPRESSED TmS'PAPER. DISCUSSION AND UTHORS REPLY

C

idADE AND PRINTED IN GREAT BRITAIN

THE EFFICIENCY OF MARINE SCREW

PROPELLERS AND THE DRAG COEFFICIENT

By G. S. Bxit, O.B.E., D.Sè., Honorary Fellow.

(Commmication from the National Physical Laboratory) 23rd March, 1945.

Summary

The first part of the paper enumerates- the geieral characteristics whjch all screw efficiency curves possess. A simple formula for efficiency is then developed covenng these characteristics, containing only two unknown constants. It is then shown that these constants bear definite relations to particular features of the curve of efficiency plotted on a base of effective slip, which enable their value to be obtained.

Using the artifice of representing a propeller by a single mean blade section at a defimte fraction of the diameter, and with the usual assumptions made in screw theory, it is shown that one of the above constants depends dirctly upon the frictional drag of the screw, and through this relation, the mean drag coefficient of any screw can be obtained from the, usual model test data.

Part 2. To obthin some criterion from which to judge such results, and to check the conclusiois formed and the character of the drag, the profile drag data for a very large number of airfoils in straight flight, at various. Reynolds numbers, have been examined. The results obtained for good airfoil sections are shown in Figs. I and 4 and by formula (18). A note on circular-back and sharp-nosed blades is given at the end of this section. New and reliable lift and drag coefficients over a wide range of Reynolds number, obtained in' the Compressed Air Tunnel, National Physical Labor-atory, through the co-operation of Mr. E. F. Relf and Dr. R. Jones of the Aeronautic Division are given in the Appendix.

Fig. 1 shows a very definite (and at first sight 'disturbing) amount of laminar flow, 'especially at low Reynolds numbersa point commented on later. These results are

compared with the calculated drag coefficients obtained by Squire and Young. With the use of these data the minimum drag coefficient for the mean blade section of a propeller can be obtained, but to apply this to a propeller, it must first be corrected for any interference of adjacent blades. This so called "cascade" effect is given approximately in Fig. 7 which is based on a rather limited set of experiment data due to Shimoyama, and may require elaboration.

Part 3 gives drag coefficients obtained as in Parts I and 2 and deals with the assump-tions made and the possible accuracy of the results. These lead to'the suggestion that screw efficiency in certain cases might be improved by the retention of laminar flow over the blades. Eddy-making in the screw is considered and a number of examples are given showing the loss of efficiency. Other -changes' which are sure signs of this defect are given. The effect of roughness is shown by a series of tests (Table 6).

-Pars 4. Analysis on the. above lines enables one to deal with scale effect in pro-pellers. The drag coefficient for ship corresponding to its high Reynolds number is substituted for the model drag coefficient in formula (14) and a new a constant is obtained for use in the efficiency formula (1). Two typical examples are given and the factors which may affect the application of the results to a ship are considered. ' The scheme gives a very definite indication of the size of screw and speed oftest below which model data will have a doubtful, application to full size.

The work described has been carried Out as part Ofthe research programme ofthe National Physical Laboratory, and this paper is published by permission of the Director of the Laboratory. The Author desires to acknowledge, the assistance and some sbggestions received from Mr. L. T. G. Clarke, until recently of the staff of the Ship Division of the Laboratory, who has also been responsible for the special experiments

made for the paper.

INTRODUCTION.

IN a

lecture given by the AUthór* last year on the "Fundamentals of the

Screw Propeller," by making certain broad assumptions, the general effect of skin friction and drag of the propeller blades. upon the screw efficiency was explored. It was shown that the loss due to this cause depended not only upon the frictional coefficient, but also upon the pitch ratio of the screw and' its thrust

In5. Mech. E., Jaia., 1944.

loading. Three of the conälusions then formed may be quoted (with a little expansion) as they have a general bearing upon the subject of this paper

Loss of efficiency due to friction occurs largely from increase of torque, friction having only a quite small effect upon thrust, provided it is not serious enough to alter the effective pitch.

Skin-friction loss is thuôh more important with low-than with

high-pitch ratios. Hence it is specially important that with the former

there should be a clean and polished surface. This applies with

greater force to cast-iron than to brass screws, since the former

necessarily have thicker blade sections and a consequent increase in liability to eddy-making.

(3) Also, because. of this relatively small importance of skin friction with high-pitch ratiOs, blade area can be increased in such cases to meet high thrust loads with only a small loss in efficiency.

The methods adopted in that lecture are too general and do not lend them-selves to the analysis or study of efficiency for any, particular screw design ; but if, on the other hand, one turns to any detail, screw theory for this purpose, quite apart from the important assumptions involved, the iesult in any case is restricted to one set of conditions, and does not afford a measure which can be generally applied to any screw. To obtain such a measure one must, therefore, turn to experiment data, and use theory as a painter uses a canvasmerely' as a backing and something to hold 'his picture together. The efficiency curves for. over a hundred screws have been examined an4 the following results are based upon the data obtained in this examination.

PART 1. TuE CURvE OF EFFICIENCY

If the efficiency curve for a propeller is drawn on an effective-slip base, it always assumes one common form. It has zero value at zero and unity slip, and a maximum value between 02 and 04 slip.. If this is represented by a formula therefore sO s) must be prime factors in the formula. We know too on general grounds that loss of efficiency arises from two separate things, the

drag of the blades and the thrust loading. The latter varies directly with slip,

but at all moderate slips the frictional and.form drag will have one main com-ponent independent of slip, with, a secondary one increasing with slip, the induced drag also coming in this latter category. Both the drag and thrust effects will be affected by the pitch ratio, and a convenient formula to represent all these influences is to write

s(ls)

. .

a+bs

where ,a and b are constants for any screw, both liable to vary with pitch ratio, and s is the effective slip, i.e. calculated from the revolutions of no thrust.

The type of formula can be easily checked for any particular screw, when its efficiency curve is available, since the term S) when plotted to a base of i should give a straight line. This has been done for over 100 normal marine propellers of all types, over, a slip range from 0 to 40 per cent, and without a single exception, a straight line has been obtained. The effective pitch used in this analysis is that obtained from a straight line kT/J curve, or estimated in the manner* already given by the Author. The ordinate value of such a line at zero slip will give the a value of the screw, and the tangent or slope of the 'line will

give the b value. . .

THE EFFICIENCY OF MARINE SCREW PROPELLERS AND THE DRAG cor.rn mwr 281

The accuracy of a and b when determined in this way is considered on p. 288. Another and different approach to thefr values is possible from the following, equations. The first differential of equation (1) gives

d-ll.

a(1-2s)bs2

ds

(a--bs)2

- Hence when's 0

=

(3)

d- 1

whens=10

₌

(4)

provided equation (1) held good for slips from 0 to 10.

If the efficiency 'curve were well defined experimentally at the two ends, these equations would give a and b, since 1/a is the tangent to the efficiency curve 'at

zero slip, and 1/a + b the tangent at unity slip. But in actual practice it is very

difficult to. measure the efficiency accurately at small slips, and equation (3) can be better used to define the average efficiency slope near zero slip from a known a value, rather than in reverse to define the a constant from the slope at zero slip.

Consider now the a constant and its physical meaning. The efficiency 'of a screw is given by

:-Tv

',

2nQ

. ' '

where 1 is the velocity of advance along the screw axis, T and Q are thrust and torque on the screw, n is the numbr of revolutions per second, and when

T=0

d

pDdT

'

'2

''

ds27vQds

'

()

From the analysis* of the thrust of propellers we can write for slips ranging

from zero to 35/40 per cent.

:-kT=

= kpsx(no. of blades)

. ' . . (6)

Hence = p k p n2D4 x (no. of blades)

and from (5) when T= 0

' '

=

< x (no. of blades) (7)

To obtain an expression for the torque Q, we adopt the artifice used by the Author in the definition of the effective pitch, and in thrust analysis, of finding a representative section of the propeller blàdës' at which radius the whole area

of the propeller blade may be assumed as concentrated. Calculations for two

very different propellers have shown that, if the drag is assumed proportional to the blade width at any radius, the mean diameter at which the frictional drag acts is 068 the screw diameter. In the earlier paper referred .to above, it was shown that the mean diameter at which thrust could be concentrated varied from 072 to 074 D, i.e. slightly 'outside the mean radius for the drag

if L is the lift and Dg the thag on the mean section, °L and 0Dthe pitch angles

at 068 and 074D respectively, then ' '

Q =[034.L sin Ql + 037Dgcos ODID

(8)

andwhenT=0

'

L COS eL Dg Sin OD ' - (9)

The Thrust of a Marine Screw Propeller. l.Mech.E., 1944.

Since OL and 0D differ only by a small amount, cos (Or. Or)) can be writtn as unity, and for simplicity substituting 0:37 for 034 in equation (8) the above

reduces to :

.

Q = 037 D

_{cos Or.} I

If Cjji is the mean drag coefficient and the screw blade area is A

Dg = - pA (1)2CD

(11)

and A can be written

=

D2 (b. a. r.)

andsince,whenT=0,v=pnD

' (12)

037

p2n2D3

Q = p A _{Sin2 Or. cos Or. .} ,. . . (13)

Finally from equations (3) (7) and (13)

1 k (no. of blades) Sm2 QL cos OL

14

ds - a -

091 Cn (b. a, r.) ( ) This gives a direct relation between the a coefficient in the efficiency formula

and the mean drag coefficient of the blades; All the terms in (14) are known

from the design drawing and' normal experiment data, except CD. This CD

is the drag coefficient for the mean section All really good airfoils of inlinite'

aspect ratio attain their minimum drag coefficient at angles of attack only

slightly above the zero thrust point, and the coefficient remains fairly constant over a small slip angle range. This is quite true provided nothing occurs to cause the flow to break down or depart from the flow of an idearl fluid over the section. It is not true if there is material eddy-making, unless this is strictly localized, i.e. does not vary with angle of attack. Taking the case of the good propeller, therefore, it will be reasonably correct to assume that the drag coefficient to be used in equation (14) is that for minimum drag for the' typical blade section,. Methods of assessing this are given in the next section of the paper

Next, as regards the constant b, since the main purpose of the paper is to consider the effect of friction on efficiency, this has been explored so far, only to the extent required to determine whether friction plays any part in its value. Starting with the bold assumption of equation (4) that b is constant over the whole slip range, we can write, when s is near unity

drj 'i

Krp

ds 1sKQ 2*(ä+b)

When experiment data is available at 100 % slip, this enables one to determine the value of (a + b) at this end of the efficiency curve. This has:been done for four sets of Troost's- screw data for B. type blade sections, and the results

are shown in'Table 1. The variation is relatively very small, and'taking this

'to be generally, true, equation (15) helps to define the nature5 of the b term. Since at 100% slip, all the blades are stalled, friction plays only a small part in the development of Kr and Kq. Hence b is practically independent of friction, except in so far as friction ,or eddy-nking'may alter the efféctivé pitch.

(10)

THE EFFICIENCY OF MARINE SCREW PROPELLERS AND THE DRAG COEFFICIENT 283

TABLE 1

Ana1ysi. of Propeller Data for Efficiency Constants. Comparison of values for high and low slips.

Some light upon the b constant is obtained from consideration of th general thedry of propellers. For a frictionless propeller of a given pitch, on the basis of minimum loss of energy and Uniform intake at any annulus, a curve of efficiency can be obtained to a base Cr or

j1. If

a screw of the same pitch

be tested we can draw a curve of Cr to a base of slip. Reasonable changes

in friction do not affect this curve, 'e and it can be taken as correct for the pro-peller without friction. If this. relation between Cr and slip is assumed to hold for the above theoretical case, this will give an efficiency curve to 'base slip, and the screw being frictionless should have a value of zero and b value unity. For ten screws of different type tried in this way, the b value was around O83

and a varied up to 0015. But assuming in the theoretical case that the intake

velocity dropped between the blades (as in fact it does) a new relation btween Cr and efficiency is Obtained, and again assuming the Cr and slip are relted as

in the actual test data, a second theoretical efficiency curve to base slip is

obtained for a frictionless screw, in this case with variable intake. This was

tried on screw B.W.4 of Table 4 and gave a = 0 and b = 10 as required,

whep the variation between the blades was about 20 per cent. It would appear, therefore, that the b value in the efficiency equation dpends upon this variation of flow through the screw, which will depend to a considerable extent on the mean blade width ratio and blade shape.

Since this. term also covers tip losses and increase of profile drag with slip, there will be a small variation with pitch ratio, as this affects the relation of slip to the angle of attack on which Cr depends.

This statement is based upon the experiment data in Table16. Troost

peliers.

s 10 KT p Kr

A0 a+

b low slipa from slow slip.b from b from high K KQ B 4/40 0-69 0243 0-0214 113 1-24 0-808 0-13 071 0-678 0-91 0319 0-0358 892 129 0777 0-094 0-706 0-683 1-12 0-383 0-0521 735 1-31 0765 0-070 0-714 0-695 1-33 0-450 00716 6-29 133 0-753 064 0-712 0-689 155 0495 0-0923 5-37 1-33 0-753 0-055 0-720 0-698 B4/55 0-662 0-253 0-0250 101 1-06 0-945 0-147 O786 0-798 0-870 0-342 0-0423 0-809 1-12 0-893 0-097 0792 0796 1-08 0-43

00613 70l

1-20 0-834 0-074 .0-770 0760 1-29 0-502 0-0839 5-98 1-22 0-820 0-060 0-760 0-770 148 0-55 0-1065 516 1215 0-823 0047 0-786 0776 B 3/35 0-705 0-223 0-0197 10-8 1-21 0-828 013 0-67 0-698 0-915 0-294 0-0327 9-0 1-31 0-765 0-086 0-71 0-68 1-14 0-365 0-0503 7-25 1-32 0-760 0064 0-73 0-696 1-34 0-42 0-0686 6-12 - 1-30 0-77 0054 0-74 0-716 1-55 0-461 0-089 5-18

l28

0783 0-045 0-75 0738 B 3/50 0-666 0-236 0-0219 10-77 1-14 0-88 0-120 0-75 0-760 0-876 0-329 0-0382 8-63 1-205 0-831 0-078 0-79 0-735 1-095 0-424 0-0604 7-02 1-224 0818 0-063 0786 0-755 1-296 0-49 0-0834 5-88 1-213 0-825 0-054 0-776 0-770 1-508 0-545 0-1076 506 1-21.5 0-824 0-045 0-788 0-78

PART 2. Tim DRAG OF AN AlaFom

There is an immense amount of data on the subject scattered in various papers,

reports and text books. A good deal of these data have been obtained for

special purposes, and are more useful to check rather than tO help formulate

a general law for drag. For. thin airfoils it is fairly clear on theoretical and

experimental grounds that the drag consists of two parts called the profile drag

and the induced drag. These two components vary in quite independent ways

with the blade dimensions and working conditions. The profile drag consists

in the main Of pure skin friction resistance, including the effect of form upon

this rdsistance, with a further addition due to the normal pressures on the

airfoil having a fore-and-aft component, due to the viscosity effect On the flow. If the blade were infinitely long there would be no other resistance, but when it has a finite aspect ratio, a certain amount of flow spills oVer the ends from the positive pressure face to the suction face forming fore-and-aft vortices which trail away down stream at each end of the blade, and these absorb a constant supply of energy, which constitutes the induced drag. The proffle drag co-efficient will vary mainly with the Reynolds number, and with variations arising

from the blade-thickness ratio and centre-line camber.

The induced drag

coefficient will vary with the relative extent of blade tip to blade area, i.e. to the mean aspect ratio or the blades, and to the pressures involved, i.e. to the

lift coefficient. For thin airfoils of elliptic plan and the same section throughout, of aspect ratio X, theory gives the relation for small lifts

:-CD=CDo+

(16)

where C is the drag coefficient for a lift coefficient CL with aspect ratio X. CDO is the profile drag coefficient for infinite aspect ratio, and includes thern skin friction resistance with any increase due to the fullness of the blade, and any drag due to the pressures around the form. With departures from the elliptic distribution of lift, there are small increases of the last term, but these do not

concern us. When this formula. is applied to blades of fair thickness, and

what might be called stream-line form, over a quite fair range of CL from zero upwards, the profile drag remains almost constant at a minimum value CDO increasing slightly as CL increases, with a quite rapid and serious increase as the stall is approached. Any feature of the blade which increases the difference between the flow around the blade and the theoretical flow without friction

or breaiaway, adds to the drag. In this category comes steep curvature of the

sections at the trailing edge, producing a general rise in profile drag at all slip angles, and sharp leading edges which produce a rapid rise in drag with any

departure from the slip giving minimum drag. One of the principal sufferers

in these respects is a circular-back-section screw .and some special notes on this type are given later.

'The drag coefficient required f9r equation (14) is the average value corresiond ing to the thrust curve as the slip approaches and reaches zero, and for all good forms f section this can be taken a,s slightly above the minimum value of the profile drag coefficient CDO. Jacobs, Ward & Pinkerton (N.A.C.A. 460/1933) have given this coefficient in terms of the thickness ratio and the centre-line

camber of the blade section. Their formula is givei for Reynolds number

3 x 10 and is as follows

:-CDO =00056 + 00l t + 01 t2 + k2

. (17)

thickness of airfoil, where t is the ratio.

. chord

and k2 vanes with the centre-line camber

ratio. This formula has recently been repeated by Burrillt who quotes the

adopted k2 values. But for our purpose we require a more genera] formula applicable at ALL Reynolds numbers.

THE EFFICIENCY OF MARINE SCREW PROPELLERS AN]) THE DRAG COEFFICIENT 285

Most of the drag data contained in the undermentioned reports* have been

examined for this purpose. In any existing set of experiments with varying

thickness of blade, the thickness ratio does not usually extend below a value of 005 or OO8, and when exterpolating such results to zero thickness, there is room for variation, according to whether one tries to make the drag curve

"fit in" with flat plate data for turbulent, transition, or laminar flow.

To avoid any bias of this kind, the scattered data on the effect of thickness, in these various reports; have been analysed on the following assumptions

:-That since thickness effect is almost entirely due to stream flow, there can be no sudden change in the effect as thickness is reduced, and the exterpolation of a drag curve 'for smaller thickness, beyond the known part of the curve, must retain 'the same characteristics as the known part of the curye.

Thatfor the same reason the increase of resistance for any thickness is largely a relevant quantity, influenced by Reynolds number change in the same way as is the zero thickness resistance.

These assumptions have been closely 'supported by the experiment data, and a common law for resistance applicable to all Reynolds numbers, and for all symmetrical sections has been obtained as follows

:-CDO = (flat plate value) (1 + 25 t) + 01)6 t2 (18)

The flat-plate value to be used in the formula is given in Fig. 1, and for general

-information the similar curves for flat plates in turbulent and laminar flow are also shown in the figure. It would appear from the figure that, even in a turbu-lent wind tunnel, and to really high Reynolds numbers, some part of an airfoil blade is in laminar flow.

This law can be used for any fair symmetrical section with maximum thick-, ness anywhere from 03 to 05 chord from the leading edge, and for thickness ratios up to at least 02. The mean "no thickness" line in Fig. 1. is reasonably definite down to a Reynolds number of 2 x l0 but below l5 x l0 increasing

variations were found rising to 30 per cent. at I 1) x 105probably due to

variations in tunnel turbulence, made of experiment, nature of blade surface,

etc.

The effect of this uncertainty upon propeller model tests is given in

Part 4.

So, far we have dealt with purely experimental data, but a check upon the analysis is to be obtained from R & M. 1838/1938. "The Calculation of Profile

Drag of Aerofoils" by H. B. Squire and A. D. Young. In this the drag is

obtained for a blade with various points of transition from laminar to turbulent flow, and for various thicknessei. Figs. 2 and 3 give the profile dragt. co-efficients based upon the above report for three transition points, and on each diagram the curve of CDO from formula'(18) is shown, On Fig. 3 a fifth curve is shown in which zero thickness value has been taken as for a fully turbulent plate. Taken 'broadly there is a general confirmation of the variation of CDO with thickness, but the results suggest that the mean transition point at R = lO is well back from the leading edge, whereas at R = I 0 it is' close to the ledjng edge, and at both Reynolds numbers it moves nearer the leading edge as the blade

thickness is increased.

It should be borne in mind that, in the report it is

assumed that the transition takes place suddenly, and the wake is supposed to contain no' energy other than that due to the distortion of the pressure dis-tribution around the airfoil due to the boundary layer, i.e. there is no eddy-making and' no breaking away. The comparison is regarded as justifying. the use of formula (18),, leaves the possibility that the line for zero thickness in

Fig. 1 is' a little low, and emphasizes the caution about the uSe of low Reynolds nnmbers in model-screw tests.

Nat. Adv. Corn. Aeronaptics (U.S.A.) Reports 460/1933, 628/A38, 586/1937, 352/1930 6489/1943 R. & M. 72 & 248/1916, 362 & 562/1917, 1147/1927, 1708/1936, 1SO4 & 1826/1927. Also Bairstow Applied Aerodynamics.

t In the re1,ort these profile drags are calculated for sections having a 20 per cent, centre-line camber. The

Most marine propeller blade sections are not symmetrical, and an addition must be.made to the drag 'coefficientto take account of the increased camber of the suction face. These allowances take the form of an addition toCDO

calcu-lated from equation 18. They are based on the centre-line camber ratio,, and

have been obtained from methodical series of blade tests in the various reports, by calculating theCDOfor 'thickness alone from equation 18 and plotting the difference between the actual CDO_{and this calculated value to a base of centre} line camber. Fig., 4 shows the mean curve.

It is quite deflnfte up to 4 per

cent, camber, reasonably so at 6 per cent., but above this, the additional

allow-ance is greater the nearer the maximum camber is to the trailing edge; in

fact, when eddy-making begins it can multiply greatly and indefinitely, and lead to very high drags.

Addition to CDO for centre-line camber

:-Centre-line camber )

from nose tail line _J Additional drag coefficient upwards.

As "a general check upon this analysis and to show the limits of ccuracy, results

fOr eight.typical airfoils' from N.ACA. Report 460 are given in Table 2. The

departures between calculated and actual vary up to 10 per cent. These results

apply only to sections of reasona6lë stream-line form with rounded leading

edge, and withoutany droop of the trailing edge (i.e. the thickness ratio t must be at least twice the centre line camber ratio).

I

'TABLE2

'Comparison

of

Calculated and Actual CDmiii. atHigh Reynolds Numbers

Circular-back and Flat-faced Sections

In the opening paragraph of Part 2, it is assumed that drag increased steadily 'with slip. When a blade section has a flat face, and particularly when the leading

edge is sharp, the minimum profile drag occurs when the flow is directed

approx-imately along this face, whereas zero slip occurs along some such line asFBC

in- Fig. 5. Flow along this line involves a type of hairpin bend around the sharp edge A with the inevitable formation of an eddy on the 'positive face at A, and the movement of the transition point on this face right. forivard to A. Such sections, therefore, at zero thrust have an abnormally high drag which;

with increase in slip, first rapidly drops to a more normal value and then

increases more or less as usual. If in Fig. 5 the edge A were washed back to

A1, the zero lift line FBC would, not be materially affected, but the minimum drag would occur with flow along A5C, and with some curvature at A1, the

These three results are obtained from NACA REP. G28 in which actual results have been corrected down by

roughly 0001 on the supposition that the estra turbulence in the tunnel warrants a correction based on and derived from the" turbulent" plate curvea correction not supported by this work.

Section. Centre line camber ratio. Thick-ness ratio., CD0 Actual C0 rein. -For ' plate, -Plate with thickness, With thickness and camber. NACA 2506

..

0-02 0-06 00053 00062' 00069 0-0073 2212 - - 0-02 012 ,, 0-0078 ' 0-0085 0-0087 '4312 -. 0-04 0-12 ,, 0-0078 0-0097 0-0095 4315 - -. 0-04 0-15 ,, ' 0-0085 00104 0-0107 2518

..

0-02 0-18 ,, 0-0095 0-0102 0-0112 Gott 436 - - 0036 0-11 ,, ' 0-0073 00089 ' 0.0082* Clarke Y6 -. Y14 - -0-017 0-04 006014 ',,,, 0006200082 000680-0101 00059* 0.0091* 2% 4% 6% 8% 0-0039 0'0007 0-0019 _0-0043 0.01J70

THE EFFICIENCY OF MARINE SCREW PROPELLERS AND THE DRAG COEFFICIENT 287

abnormal. rise of drag approaching zero slip would be eliminated, i.e. we are

approaching good aerófoil shape. if the above rapid change in drag were

properly shown in the efficiency curve, this woUld be drawn with a hollowed and curved finish as it reached zero slip; but this is rarely done, and the straight finish usually adopted gives an averagevalue to the drag in this region. Hence thea value, obtained by this analysis from such efficiency curves is never quite as high as that corresponding to the actual * drag coefficient at zero slip but is higher than that due to the minimum profile drag of the section.

Another factor increasing the drag of circular-back sections is the relatively high mean camber ratio for any thickness, and the steepness of the back face near the trailing edge. The increase in drag with camber is shown in Fig. 4, and when the back slope reaches20degrees t low to 10 degrees at high speed,

high and uncertain drag due to eddy formation is experienced. These slopes

for circular-back sections are given in Table 3.

TABLE 3

Circularback Sections, Sharp Edges -Slope of back face at trailing edge

A few words- might be said here about the sharp leading edge f circular back sections. H. L. Dryden in his work on flat platest has shown that a sharp leading edge allows of a high degree of laminar flow, and hence a low drag Quite a number of blade sections have had their resistance lOwered by making the leading edge sharp, but the advantage has been restricted to quite a small angle range, and outside this, the resistance has been definitely worse. On a circular back section the advantage is usually cancelled by the loss due to high

camber and eddy-making. But this suggests a line of advance which might

be followed up with more normal sectiOns, provided ship blades surfaces can be made smooth enough to allow of laminar flow.

To provide some reliable experiment data for circular back sectiOns, by the co-operation of Mr. Relf, Superintendent of the Aerodynamics Division, National Physical'Laboratory, some special tests have been made in the

com-pressed air tunnel.. A brief summary and three tables of some results are

given in the Appendix. A full account will be published shortly as a R. & M.

of the Aeronautical Research Committee. Data are given for three sections

of different thickness ratio, each being tried with three different radii to the leading edge, so as to determine the effect of such curvature.

* The correct value of CD at zero thrust for any sctew which has been tested in the region of no thrust, is given by the formula

:-688 sui2O COS U

CD = KQO x _p2(b.a.r.)

where KQO is the international torque constant at zero thrust and U is the effective pitch angle atO-73D.

t N.A.C.A. 562.

The foUowin examples are taken from N.A.C.A. 460 and 628.

Symmetrical blades I = 006, 012, 018. All showed lowest minimum drag with sharp leading edge. Clarke Y t 0115 Sharp edge gave lower drag for small range about CL 03

N.A.C.A. 18, 19, 20. Sharp edge best around CL 02

Gott. 398 A.B. Sharp edge beet around Ct . 04 S

L'hickness ratio t ..' .. .0-043 0-065 0-088 0-111 0134 0182

centre-line camber ..

..

0-021 0-032 0-044 0-055 0-067 009I .ngle of tangent at trailing

CaFcade 'Effect on Drag

When an airfoil is one of a number in a row as in Fig. 6, moving through the water in the direction OK, there is mutual interference between the blades

and the pressures developed differ from those for an isolated airfoil. These

differences cause a change in the effective pitch and in the growth of thrust

with slip angle, but they also affect the drag. Due to 'the restricted passage

between the blades, the pressure distribution over the blade will be modified. This will be most noticeable in the region AB in Fig 6, the presures on the driving fàce'at B, and on thesuction face at A, will be reduced, and these vari-ations will increase the ' form " drag. The best experimental data on the subject 'is that of Shimoyama. * His tests were made on one set of airfoils (t = O'145 centre-line camber ratio 0{)47, Reynolds No. 21 x 10°) at a series of pitch

angles 0 and with i/c varying from 2'O to 075. The drag was determined by

pressure measurements around the airfoil. The broad effect Of varying i/c

is supported by Gutsche, t but there are fairly' wide discrepancies between the

two sets of data as regards pitch angle effect. As Shimoyama's methods were

more accurate and his results more detailed and consistent, his data have been

followed. It will be understood that data on one airfoil in various cascades are not sufficient for generalization, and Fig. 7, gi,ving the effect of cascade gap on the drag, is put forward as a stop-gap until something with sounder foundations has been obtained. As the increase in drag in the cascade is due to the form or body Of the airfdil, it has been plotted as a fractional increase of the form drag of the, isolated airfoil. There is a strong possibility that the ordinate of' Fig. 7 will be proportional to the thickness ratio or centre-line camber, but' this awaits experimental justification. The final minimum drag coefficient of an airfoil in a cascade is therefore made up as follows

:-Although not involved in thi analysis it may be 'mentioned that the increase in profile drag due to cascade becomes greater as the blades develop thrust, and this would have a very appreciable effect on efficiency calculations in some cases.

PART: 3. T DETERMINATION OF SCREW PROPELLER DRAG

When the thrust and efficiency curves of a model screw become available, with the methods of Part I and the data of Part 2 we should be in a position to calculate the separate values of the constants a, b, and k, and so obtain the mean CDO for the screw (equation 14) to compare with the CDO calculated for the mean blade section (equation 18). The process is quite sound, but before taking this step, a few words 'need to be said in regard to inherent difficulties which limit the accuracy of the work, and two assumptions which'have to be made.

The first .difficulty arises from the relatively small value, of the constant a compared with b in formula 1, for all usual pitch ratios (see Tables 4 and 5).

The a value is obtained as already stated by drawing a curve ofS 1 .s') to

base S. This gives a straight line up to at least 40 per cent, slip, the ordinate of which at zero slip is the a contant. Generally the efficiency curve does not exist

at slips below about 10-15 per cent ; hence the S (1 s) line must be

exter-polated to zero slip, and any small 'change in the'line is liable to make a material fractiOnal difference in the relatively small a value at zero slip. Unless special care has. been taken in this respect, the constant a cannot be defined nearer

Memoirs Foe, of Eug., Kyuslzu Imp. 1Jth)ersity. Vol. 8, No. 4, 1938. SC8Iff. Gesei. 1938.

zero thick- \ raddition for

addition for i

rcascade

-CDO = ( ness value 1+ I thickness -I- centre-line 'I x I multiplier I . (19)

\from Fig. 1,

Lformula 18 camber Fig. 4J LFig. 7

J

THE EFFICIENCY OF MARINE SCREW PROPELLERS AND THE DRAG COEFFICIENT 289

than about ± 5 per cent. These figures also give about the limits* ofaccuracy

with which CDO can be determined by direct measurement in a wind tunnel. The second difficulty is the method ofobtaining equation (8). It has been assumed that the resistance is distributed as if it were only skin friction and 'form drag, so that if there should be localized drag; due to eddy-making near

the boss for example, or any form

of

breakdown of flow, some error will occur

here. This difficulty assumes importance only when there is an unexpected and improperly large drag in the screw, as disclosed by a high value of the a constant; its general effect being slightly to undervalue this already high ,a value.

The first assumption made is that the drag data in Figs. 1, 4, and 7 derived

from wind-tunnel tests can be used for marine-screw calculations. This

implies that differences in the degree of turbulence in a wind tunnel and for an open-water tank are not sufficient, to modify the transition from laminar to

turbulent flow at any Reynolds number.

In actual fact the turbulence in a

wind tunnel is definitely higher than in a ship-model testing tank, but the condi-tions of a screw test, particularly the minute high-frequency vibration often set up, will offset this, and so far as we know there should be no serious error in using airfoil data.

For full-scale estimates we are noi so certain,' and there are two possibilities. On a single-screw ship there is a considerable amount of turbulence in the wake,

particularly within the limits of th

ship - friction'belt.

Hence it would be'

expected that the " zero thickness" drag of Fig. I would rise to the." turbulent

flow" line. Luckily the screw on such a ship is working at a Reynolds number

about or above lO, and such a change has only a, minor effect on the drag

coefficient. The second possibility is one of the future rather than the present.

if one could control the flow to the propeller (say in a twin-screw ship) so that the turbulence was definitely low, then with great care in the preparation of the surface of the blades and their design, a high percentage of laminar flow could be developed, with quite considerable advantage to the efficiency.

The second assumptiOn is in rógard to the mean Reynolds number for a model propeller. To be consistent with the earlier assumption as regards the mean radius Of the frictional drag, this has again been taken as that for a sec-tion' at 0'73 D at the revolutions for no thrust.

with v = speed Of advanée in the test (ft./sec.).

c =.chOrd width at073D

8 pitch angle at 073 D:

Reynolds number in fresh water

₌

l'n

_e X 10

If 'the tests are made at constant revolutions per second = n (19)

Reynolds number in fresh water

O5's 0.x l0

In salt water 123 and O536 change to 129 and 0557 respectively.

The thrust and efficiency of a large number of screws, have been analysed by the methods detailed in Part I and values of the constants a, b, and k for each screw found. From equation 14 the mean drag coefficient Cno for each screw

has then been determined. From Part 2 the CDO for the.mean blade section

has been' computed, to compare with the experimental value for the ,screw A fairly large number of examples are given in Table 4 for screws having

reason-ably' good blade sections to which the method should apply. In Table 5 are

given similar data for some eight circular-back screws.

Variations between the drag coefficient found in these two different ways occasionally rise as high as 30 per cent, but the average discrepancy for about 100 screws ana1jsed in this way is about hail' this amount for reasonably good blade sections. it is not easy to explain these departures without more exact

data (or rather it is too easy). Possibilities lie in extra smooth or rough blades, in our uncertainty as to the correct measure for cascade effect on drag, in the presence of slight, vibration during the model tests, and in the drag at no thrust being a little above the minimum. All of these things produce varying amounts of laminar or turbulent flow oyer the blades. These discrepancies do not liç all in the same direction, and there is a definite suggestion that in certain sets of tests, the transition point from laminar to turbulent flow is well back on the blade. The analysis in Part 2 has shown that such. a condition is possible with airfoils even under fairly turbulent conditionsa conclusion, supported by recenta aeronautic development on full-scale machines. It is worth drawing attention to the fact that, given a not too turbulent ship wake, this condition might be produced on actual ships. Fig. 3 shows that the drag coefficient would be reduced roughly 50 per cent by holding the transition point at 04 chord from the leading edge, with a handsome increase in efficiency at all moderate slips.

t should be realized that such a condition would not be .affected by small transient changes in slip angle, but depends upon the type of streamline flow, and the surface being sufficiently smooth to avoid the formation of minute eddies which might cause local separation of the laminar layer. Such a result is therefore quite' within the bounds of possibilities on certain classes of ship. More complete and accurate data is required before any certain explanation of the above discrepancies can be given, arid with this in view, two small sets of tests have been made to.explore the effect of eddy-making and varying rough-ness of surface.

Eddy-making. All of the drag coefficients for the circular back screws in Table 5 and for Adams' propeller (normal surface) in Table 6, are considerably

higher than those estimated for a good airfoil under similar conditions. It

was reasonably certain that this was due to two possible causeseddy-making at the sharp leading edge, or at the trailing edge near the boss, due to breakdown of flow mainly on the suction face , To explore this latter possibility, tests were made on the last three screws in the table, modified at and near the blade roots.

The alteration was much the same in each case, and is shown in, Fig. 8. The

chord at the blade root was increased as much' as the boss would allow, using

all the increase to ease the angle of the suction face at the trailing edge. The

alteration died out at about 0;7 diameter, so that all the blade outside this

diameter and on the fore side of the maximum-thickness line remained un-changed.

This did not entirely remove the eddies at the trailing

dge, but

was sufficient to show their effect. These screws are marked B in Table 5,

which gives the results obtained. Comparing the original and modffied screws, it is clear that eddy-making on such sections not only increases the drag, and hence reduces the efficiency, but also reduces the effective pitch, involving more revolutions for a given thrust and a consequent further reduction of efficiency. The maximum efficiencies for these screws are given in the table. The improve-.

ment with the modified root sections is surprisingly largemuch too large

to be attributed to the mere change in drag. The removal of the eddies and consequent increase in pitch both tend to improve the fore-and-aft flow through the centre of the screw, with a. resultant reduction of rotary velocity 'and it is probable that these things have played a part in the improvement.. An en-deavour was made to' test the effect of eddy-making drag pure and simple,

without any pitch change. Screw T.2 in Table 6 was, made for this purpose. It had no rake, a sythmetrical blade outline, and symmetrical blade sections of

the same thickness ratio throughout. The leading edge was almost circular in

section ; the trailing 'edge was very tapered. This was tested with the blunt edge leading and then reversed, i.e. with the tapered edge leading. The results show a 50 per cent increase in drag coefficient, but despite the fine blade section,

* R. & M. 1826/1937. Full-scale tests on machine with wing section RAF. 2S, thickness ratio 14, centre-Iine camber 019. This showed a minimum profile drag coefficient 0064 for the wing. With transition point from leading edge chord x 0 1 2 4

The estimated C00 was 0082 0082 0073. 0054

THE EFFICIENCY OF MARINE SCREW PROPELLERS AND THE DRAG COEFFICIENT 291

TAIBLE4

Analysis of Propeller data for Drag Coefficient Redsonably good Blade Sections

Blades of sanie bickness atall radii, with sharp edpes. Screw. Blades.No.

Blade Area Ratio. Efiec-tive p. Average Section. kr $ p Efficiency data.

-Mean per a b From Flat For c. r. blade screw, plate. ai±foil.

BW 1 4 --40 684 065 -026 110 140 7O 0129 0082 'W07 2 '55 655 -05 020 1I5 -142 -77 0098 -0070 0089 3 4 7O -635 -035- -013 115 -159 -78 '0077 0064 0081 4 40 99 O65 -026 -119 084 76 '0157 -0096 '0126 5 .55 -97 '-05 020 122 '085 -76 0115 -0084 0104 6 -70 -946 035 013 -126 -082 82 '0087 0073 -0085 7 4Ø 130 '065 026 110 056 -755 'O142 -0100 '0126 8 -55 1-29' '05 -020 120 060 -79 '0121 -0085 -0104 9 - '70 125 '035 013 '125 '060 -80 -0094 '0077-0089 -705 -132 130 '-67 '0132 -0088 '0114 915 -130 '086 -71 '-0134 '0086 -0112 B 335 3 -35 1-14 : 065 '026 - 127 '064 -73 -013 -0084 -0109 1-34 '128 '054 '74 '0142 '0083 -0108 1-55 129 '045 75 0142 -0081 -0105 666 144 120 755 -0084 -0085 0102 876 145 078 -790 '0087 '0084' '0100 B 3'5C 3

5

1095 045 -018 --147 -063 '786 0102 -0083 -0100 1296 - '143 '054 -776 '0106

0082 -00'

151 145 '045 -788 0107 '0082 '0099 69 ..11 13 0125 '0094 -0122 91

'lii

094 '706 -0122 -0092 -0119 B 4'4C 4 -40 112 065 -026 -11 072 '744 -0149 -0091 -0118 133

-108 4 -712

-0164 '0090 -0117 1:55 -110 -'055 :720 -0170 0089 -0116 66' - 116 147 -786 -0101 '00890106 :870 fl9 '097 -792 0108 -0088 B 455 4 -55 1-08 -047 '019 -119 074 -770 0115 -0088 0105 1-29 119 - -06O -770 .0117 '0086 -0103 148 -119 -047 -786 -0116 -0085 -0102 15 , 378 -608 '098 -030 102 -160 -70 -0119 '0098, -0140 LP 32 4 '422 '594 -078 -025 104 -158 -70 -0103 -0096 -0124 16 608 '546 -'047 -014 115 '17-5 --77 '0076 -0075 -0088 10 -647 542 -042 -013 -116 -190 80 -0077 -0074 -0094 22 -422 -638 -078 '020 -107 -170 -70 -0127 '0096 '0122 24 ., -505 -608 -062 0. 110 -15 '706 -0089 -0085 0100 LP 33 * 505 -475 -026 0 -109 -135 -93 -0057 -0082 _-0082 26 - '505 50 -062 0 -105 -160 -93 -0065 -0080 -0094

TABLE 5

Analysis of Propeller Data for Drag Coefficient.

Comparison of CD0 for Circular-Back Screws will; cDO derived

from streamline sections o[same dimensions

The value for airfoil Is or a gbod airfoil with no eddy-making, i.e. neglecting any bad effect from the circular back shape of the sections.

f

Both edges of blades rounded off.

Approx. radius at average section

008 S. -Possible variation ± 00005.

-§ These screws were derived from those immediately above by extending the blade surfacesat and near the roots (see text).

No. blades. Blade Area -Effective . Average Section. kr jer . Efficiency data. Maxiinnin Efficiency. Centre-Root -. -atm. . . f line Camber. Section i. blade, . a b Froiii _screw. Flat _plate. For airfoil.. J I . 3 . 0629 1418 0037 0-018 019 0160 0:070

0'75

00143 . 00068 00O81 0720 2 3. 0-719 . 1404 0-031 .0-015 0-19 0164 0074 0-77 0-0128 00065 0:0075 0708 3. 3 0872 1392 0025 0012 0-19 0176' 0076 079 00114 00060 00068 0692 4 3 0955 137 0022 0-011 ' 0-19 0177 . 0077 082 00105 00058 00066

0665-LP .8t

4 0505 0-575 0062 0031 023 ?.1n13 0222 070 0-0125 00084 0:0120 049 F 55 3 . 0-245 128 0-107 0053 040 0150 0142

- 062

00591 .. 0011: 00175 064 F55B* -3 0285 148 -0107 0053 0245 0J25 0112. 062 0040 0011: 00175 0-70

A A

2 0-178 -0875 .0084 0042 036 0150 -0142 061. 0031 00085 0013. -'-0645

A AB

2 0200 0-888 0084' 0042 025. 0147 011.2 063 -0022 00085 0013 0685 'B 13 3 0205 115 0111 0055 0405 0126 0132 065 0:048 00105 .0018 0613 B 13B 3 ' - 0230 121 0111 0055 0-24 0-116 0114 056 0036 00105 0018 -0-675

THE EFFICIENCY OF MARINE SCREW PROPELLERS AND THE DRAG COEFFICIENT 293 the eddy-making behind the full trailing edge when reversed has spread forward

and Md a small effect upon both the effective pitch and the

just as with

the circular back sections.

Roughness of Surface. Experiments have been made on several occasions with badly roughened model screws, all showing a very considerable drop in

efficiency as a result; bu

in practice, the roughness one, has to consider is relatively small, and to determine its effect one of the wide-bladed methodical series screws (BW. 6) was tested with four surfaces, in addition to the normal smooth surface. Particulars and results are given in Table 6. For the first test (b), a milling cutter was rolled over the back or suction faces,with a little pressure

on the tool.

This roughening could not be taken quite to the edges of the blades. For the secOnd test (c) the driving faces were treated in the same way.

For the third and fourth variations (d and e) the coarsest sandpaper* was

rubbed heavily and diagonally across the blades in two directions at right

angles; on the suction faces alone for variation 1d), on both faces for ce).

'The roughness was ." taken off" by the Metrology Division of the Laboratory at several sections, and average values of the depth of grooves cut (top to bottom) are given in the Table. It would appear from these results that on a screw

1-0 foot in diameter, a fairly open spaced roughness of one-thousandth inch depth reduces the efficiency some 3 per cent, at the maximum, the effect dying out at higher slips. Roughness of this order will not affect the thrust of the propeller, since the pitch and kr/sp both remain practically constant; hence the revolu-tibns to propel the ship will not be altered, but' more power will be required due

to the increase in drag. But closer spacing of the roughness, or increase in its

depth, leads to loss of efficiency by pitch loss as well as by drag loss, and rapidly becomes serious.

In 1929 results were given by Adams for a screw with normal surface, and with a stippled paint surface. Thesehave been analysed and the results are included

in Table 6.

The effects of the rough paint on both pitch and drag are

much more serioUs than in the preceding tests.

The roughness factor

(depth of roughness)

(average chord width) was approximately 0-0002 and 00004 m the Enghsh and American tests respectively. The latter figure is much greater than would be experienced in normal practice, and the results would only apply to a badly

fouled propeller. Taking a broad view of all these results, certain general

conclusiOns can be formed

:-(I) The constant a derived from the efficiency curve of a screw gives a good approximation-to the "drag" of any propeller.

Small changes in the condition of the blade surface affect only the .a valve.

When any screw test shows a Jo lower than would be expected from its mean blade section, and this is associated with a too high value of both kr/sp and a term, the screw is definitely suffering from eddy-makingmainly near the blade roots.

A highly frictional blade surface has the same effect upon Jo and a

as eddy-making, only to a smaller extent; but the kr/sp,'instead of being above the normal value, is a little below it.

Circuia±-back sections with thickness ratio above -09 will suffer from

eddy-making and therefore have a high drag coefficient

Stream-lining the tail of such sections, and/or keeping the maximum ordinate nearer the leading edge, and lifting the leading edge will avoid a good

deal of this defect. Durex Abrasive (l - 40D).

TABLE 6

Analysis of Propeller data for. Drag coefficient.

Effect of surface roughness and eddy-making.

All four blàded screws

* Depth of cut gener'illy 0001 ii ch, occasionally 0 0015 inch, spaced 025 inch apart. t Cuts closer together, more uniform, maximum dep h 00013-inch, spaced 012 to 026 inch apart.

Cuts along two diagonals at right angles roughly.

(Amer.) Soc. N A. & M. N. 1929.

Plate 22.

The C00 from

airfoil is approximate only, as speed of test is not known.

§ A brass propeller, with symmetrical blade section qthte full at leading edge, very tapered trailing edge; same thickness ratio tlroughout.

Average Section Efficiency data C00 Screw Effective t Mean Section a 1) F ypo of Roughness B.W. 6-a 946 035 013 '19

l2

082 820 '0087 0085

1'4ormal smooth metal surface

..

652

-airfoil type, blade

. b ' 946 ,, ,, . . ,, 087 . 816 '0091

Short sharp cuts* on suction face only

... ... 640 area _{ratio 7} c 938 . ,, ,, ,, 124 086 836 , '0087 .

Short sharp Cuts* on both laces.

... . . ' '635. d '904 ,, ;, / ,, '123 100 '845 '0097

Additional long sharp diagonal cutst on suction face.only

. .. .. .. ... ... '602 e -913 ,, -,, ,, 120 117 '835 '0113

Cuts as above but on bothfaces

.. ' .. ' .. .575 Adarns '93 ,'065 032 '225

'1W 14

64 '0233 0125

Normal surface formodóls .

. .. . . 35 No.1037 '38 87 -' ,, ,, ,, ' '100 '236 '68 032,

Stippled paint on both faces, roughness008.inch

'

.

'

'48

circular back sections

. . . . ,' T 2-995 088 0 088 '111 06 '875 0091 '0105

Normal smooth brass surface

-.. .. .. '680 T 2 reversed 985, ' 114 085 '90 '0138

-Screw reversed, i.e. with b!unt trailing edge

..

.

.

THE EFFICIENCY OF MARJNE SCREW PROPELLERS AND THE DRAG COEFFICIENT 295

PT 4.

SCALE EFFECT

Starting from the conclusions in the last section, when the usual. model test data is available for any screw, we are in a position to examine the changes

which occui in the efficiency curve passing from model to full size. FOr this

purpose it is asfmed that the change will not affect the b value for the screw, but there will be changes in the a value due to Reynolds number as shown in Fig. 1, plus any effect on the drag due to roughness of surface or variation of

transition point. Before accepting' this statement completely, however, a word

must be said about the test methods adopted to obtain the screw data. Mddel tests are made under two conditions : with a ficed speed of advance, or with fixed revolutions per second. The Reynolds number for the mean chord at 074 D is given by

(velocity through water) (chord at 074 D) kinematic viscosity

For fixed speed of advance v, this equalsv (1+02 at) chord in foot sec. units.

it x 074D

chord For fixed revolutions per sec., this equals _{cos 02} XflX

I 23

X l0

in foot sec. units,

where 02 + effective slip angle = affective pitch angle 0, and at is the mean

transitional intake velocity at this slip angle. The term cos 02 increases only

slowly with slip; hence the Reynolds number with fixed revolutions is reason-ably constant for ordinary slips. But with fixed speed of advance, the (1 + a) factor increases quite materially with slip, and sin 02 dimimshes. Hence the

Reynolds number increases steadily in such teitsthe increase over the slip

range 0/40 per cent. may rise to 20 or 25 per cent.' This change in Reynolds number during the test will always cause a small difference between the efficiency curves obtained in the two different modes of test.

The best way to illustrate scale effect in general is to take two widely different examples. First, a model of a mercantile-ship screw, diameter 11) ft., similar in general to BW4 of Table 4. This was tested at 42 ft; per second advance speed,' and at no slip (when at is nil and 02 equals the effective pitch angle) the Reynolds number is 1 85 x l0, and the calculated CDO for the mean section at the Reynolds number is *0.0126. The full,size screw at no slip would have

a Reynolds. number above l0 for which, from Figs. 1 and 4, the meansection

would have a CDO valuC of 01)064 + 01)01 = 01)074, ie. a reductmin of 01)052. .At 40 per cent. slip and the same advance speed, this reduction would be approx

irnately 00045, giving a mean of 01)048. The a value for ship would be that found for the model screw (01)84) corrected for a reduction of 00048 on the model mean section CDO (01)157)

ship CDO = 01)84

(01)157_01)048)

= 00585

Provided the ship screw had smooth blade surfaces, the efficiency for, model

"and ship would be given by:

.

model

_{- O1)84+076s}

and

ship -.

_{± 076s}

All Sgures in tiis 'example taken from 3w4, Table 4.

TABLE 7

Comparison of Model and Ship Screw Efficiency Mercantile Screw similar BW4, Table 4, at l.'85 x 1O

For the second example, at the other end of normal pracfice, wide bladed

screw J4 in Table 5 has been chosen. This had a diameter 0'615 ft., mean

chord 051 ft. and was tested at 10 ft. per second, giving a Reynolds number

of 8'2 x l0 and a cDo for its mean sectiOn if a goOd airfoil of 0'0066. The

Reynolds number, for ship at no slip would vary from 3 x l0 upwards according to service, with a calculated mean. section CDO for ship screw of good airfoil llades of 01)061, a drop of 0-0005. The a values in the efficiency equation

would therefore be :,

For mOdel (Table 5) = 01)77

For ship 01)77

(O.0105_O.0005)

o'0735

TABLE, 8

Comparison of Model and Ship Screw Efficiency Wide bladed destroyerscrew jested at 8 x l0

The mercantile screw would normally work at an effective' slip between 03 and 04 where the scale effect for a smooth screw is some 8 per cent, average, the wide bladed would work at about 0-25 slip where the' scale effect is lZ per cent.

These results depend upon two assumptions in regard to the ship screw. First that the extent of any laminar flow on the ship screw corresponds with the difference between the fully turbulent line and the zero thickness line in Fig. 1

at its Reynolds number. This would. have no effect on the wide bladed ship

screw as the two lines meet before 3 ,x I 07, _{But for the mercantile screw, the}

extra turbulence behind the ship would' (at least in a single screw ship) carzy

the transition point to the leading edge of the blades.

The effect however would not be very material, andat mOst would increase the ship CDO by 011007 and reduce the above 8 per cent average to 68 per cent.

The-second -assumptiOn made in these comparisons -is that the blade surface of the ship screw is" smooth,"-in the hydrodynamic sense, as it is inour models.

Effective slip ratio

---

0-15 0'2 0'25 0-3

04

Model efficiency 0'64 0-678 0-682 0675 0-62,

Ship Efficie'ncy, with quite smooth

blade surface -. - - - 0735 0'760 0'753 0733 ,O-633

Ship efficiency, fully turbulent flow,

roughness 211 thousandths inch

..

0-682 0713 0-713 0'70 0:637

Slip ratio - - -. 0'! 0-15

02

0'25 0-3 0-4

Model efficiency

-Ship efficiency with smooth

0-629 - 0-635 -0-665 ' 0-664 ,. 0650 0'592 bladesurface - 0-643 0-645 0674 0-672 0657 0597

THE. EFFICIENCY OF MARINE SCREW PROPELLERS AND THE DRAG COEFFICIENT. 297:

It has been laid down by Prandtl that the permissible roughness of anr airfoil,

in order that its drag shall be that for a smooth surface, is

given by

At the ship Reynolds number, this gives a roughness of a. little less than half

a thousandth of an inch(taken on a 3'6 ft. drord).

Itis clear from this that

unless quite consideiab1e care has been taken with the. surface, there will be

some addition to the "smOoth" drag, due to roughness. It need hardly be

pointed out, that roughness on sections of fairly high thickness ratio tends

to promote eddy-making towards the trailing edge. It is difficult to obtain

satisfactory data as regards the effect of small roughness on drag. Experiments in the compressed-air tunnel with airfoils having athickness ratio of 012 have shown that close spaced grain roughness 1/20,000 and 1/8,000 of the chord had practically.no effect at Reynolds number 108, added approximately 0'002 and

0004 to the minimum drag at l0 and the effect was still growing. On the

mercantile screw chosen, these correspond td roughnesses of 20 and 5'O thou

sandths of an inch. Assuming the smaller exists on the ship screw, this would

raise .the minimum drag from 0'0074 to 0'0094 to which must be added the effect

of moving the tra'nsition point to the leading edge already mentioned, namely,

00007. 'It might fairly be assumed from Table 6 that for this roughness the b term in the efficiency equation still held constant, although it would begin to vary with the greater roughness. Taking this; therefore, as a typical screw, the total

effect would be represented by an a value of 00725. The efficiencies for this

condition are given in the last line of Table 7. On. a mercantile screw similar

to BW4 the scale effect will, therefore be about six. sevenths, that shown for

smàOth blades in the Table, but some of this will be cancelled by. the roughness

ofthe blade surface of the normal commercial screw, as shown by the.last line

ofthe Table. With the wide-bladed screw, usually more carein the preparationt

ofthe surface is taken but it is doubtful whether this is sufficient to remo\e

entirely all effects from roughness. The probability therefore is that h1 this case the ship screw will have anefficiericy slightly less than in the model.

These results have been obtained with two

particular screws, but it is a

simple matter to work out for any sërew its mean Reynolds number for model

and ship and so get the change in drag. When the test results are available the model a can be worked out, and a new value for ship obtained,'from which the

efficiency curve for ship can be obtained, it is perhaps desirable to say here

that the straightness of the kT curve on a J base only lolds up to 35/40 per cent, slip; hence this method of getting scale effect is of use only up to this maximum slip.

Quoted from Jones & Williams. R. & M. 708/1936.

t Measurements made by Dr. Berndt, Tech. High School, Dresden, on small pieces of metal give the roughness of propeller bronze as normally worked (in Gerrnany}aTs 1 '1 thousandths inch, but withregrinding and polishing

this was reduced to 05 thousandth. Unworked cast-iron bad a roughness 70 thousandths inch.

-Reynolds number in millions

..

1 2 5 10 20 50

Permissible excrescnees in

S

NOMENCLATLJR.E

T = thrust of screw in lb.

Q = torque of input of 'screw in foot lb.

D = dianiete of screw in feet.

A = blade area of screw in square feet, = D2 b.a'.r.)

b.a.r = blade area ratio.

n revolutions -per sec.

v = velocity of advance of screw in feet per second.

p = density of fluid = I 988 (salt) = 1 P938 fresh water.

L and DE lift and drag forces on an airfoil blade, at right ankles and along the limo of motion rOspOctively:

p. = effective pitch ratio or- Jo = when T = 0.

o and OL = effective pitch angle at O73D. -. - .

-On - = effective pitch angle for dragtaken at O68D.

-= screw efficiency.

-s "= effective slip ratioi.e., slip calculated from the revolutions

-. for nO. thrust.,

-- T'

kT = thrust constant =

- k.s.p. x no. blades..

= torque constant _pn2D

- drag

-.Ci, drag coefficient _{p (vel.) (blade area, One side)} Cno - minimum proffle drag coefficient.

-.

.:

- maximum thickness

t thickness ratio of airfoil

-. chord

- centre-line camber. ratio.- If.

two lines are drawn on any blade section,

one straight from centre of lOading to centre of trailing edge,, the other cambered so that it everywhere bisects the thickness, the maximum distance between these two lines, expressed as a fraction of the chord is the centre-line camber 'ratio.

Transitionpoint. The point on a blade section, forward of which the flow- is laminar, aft of which it is turbulent.

Reynolds number

=

where c is the chord or width of, blade at its mean

section, i.e. at 073D, and v is the velocity of this section through the fluid. 'TSifl 0 _chOrd ratio

taken on a screw at 073D, the gap is measured

-'

t '(O73D) \

perpendicular to the blade and is given by

TILE EFFICIENCY OF MARINE SCREW PROPELLERS AND THE DRAG COUFICIENT 299

APPENDIX 1

Test Results of Circular Back Blade Sections

These tests were carried out in the compressed air tunnel of the Aerodynamics Division, National Physical Laboratory, by R. Jones, D.Sc., and D. H. Williams, B.Sc. Three "parent" blades 40 ft. by 067 ft, were made in metal, with a good sniooth surface, fiat faces. circular backs and sharp edges. These were tested at, a series of angles and over a wide range of Reynolds number On the completion of these tests, the leading edge was cut back 04 inch and rounded, and the tests repeated. A further cut by 04 inch and rounding of'the leading edge was then made and the tests again repeated. A full report. of the tests will shortly be published as a R. & M. of the Aeronautical Research Committee but the data relevant to this paper are given in Tables 9, 10, 11.

The points mentioned in the body of the paper must be carefully borne in mind when examining or using these data, or they become inexplicable in places. Referring to Fig. 5, the effect of eddy-making on the face at A is shown in the Tables, principally by the rise of profile drag as the lift diminished, a rise always more' noticeable (for a given thickness of blade) the smaller the edge curvature. The laminar flow condition near A is apparent in the drag with all three sharp edged sections at low Reynolds numbers but is disappearing as this number increases. The range over which it exists is difficult to define owing to its effects being mixed with the varying and more im-portant eddy-making.

The eddy-making at the trailing edge has two or three effects according to the angle of attack. It reduces the increase of the lift coefficient CL per degree, and this reductioh is much emphasised on the thick sections when a double eddy forms on the back (angle of, attack above about 20 deg.). One peculiar effect of this eddy-making shows at negative angles of attack, particularly on the thickest sections. At about inijius 60 or 70 degrees on these blades, the eddies formed along D E in Fig. 5. are becoming smaller, and at about minus 100 deg. have been completely washed away. This produces a temporary increase in the effective blade area, and as a result the lift co-efficient instead of diminishing with angle, remains nearly constant over this range at a small positive value, and even at minus eleven degrees there is still a small positive lift. This phenomenon shifts with Reynolds number (probably also with degree of smoothness of surface) and a little with roundness of leading edge, and makes it very difficult to determine a "zero lift" angle, except by ignoring these changes and pro-ducing a mean CL line to zero lift.

It is perhaps desirable to repeat here that the effect of a sharp leading edge in reducing profile drag over a limited lift range (Ct. = 01 to 03 for t = 005 and a good smooth blade surface) is common to most sections and can be obtained with good airfoils free from eddy-making as well as 'from circular back sections, given the 'necessary steady conditions, and the sharp edge will have the same defect outside this range and particularly at smaller angles, as it shows with all these sections.

TABLE 9

Circular Back Sections, Infinite Aspect Ratio Smooth Surface, Sharp Trailing

Edge - LEADING 03X10' EDDE SHARP

074

RE'(NOLOS 124NUMBER. &

45

= 05

$.5x106 0( _{CL C CO3 0}

_{CL C}

_{(. CL C}

₍ C1.

C,

39 164 3-49 4-9 236 4 44 41' 2-50 441 5-I 25' 485 5-ö 2s 493

30 O3S 2-40 3-! 02.9 2.oS 30 0.53 203 3:2 062. 237 32 o'53 2-44

Ia 040 1-76 2-a o-44

37 22 o40

39 22 034 .55 22 034 :54. 1-3 1-47. 119 2 -58 oq2.

Ii

'I 37 0-99 13 52 I'o4 3 l'3 ₁₀₅

04 264 29!

04. 251

075 03 229 07

0-1 227

083 04 22'!

07q

0.55 3 4 o-84 or 3-34.

-

os 324 0-73 0.6 3. 076 0-5 323

16 410 .1-IS 4-14 104 .5 4 a S :5 4'I o-8s

2.5 4.92.' 1-5: 2.5 503 -3! 25

5o.

3.5 25 502 '24 24 505 120

3-S 5-71 1-91.

35 55!

io 3,5

536 1-83 34 582 iS! 4 5-38 -is 5.5 736 37: 5-5. 7-5! 370 55 758 3-7!

55 744 375 '54

752. 3-65 7.5 8% 74! 7-5 5.55 72.3 75 I 337 7.33 7.5 3.73 7.35 74 5.9k 7-35

jOE

5-5 'oi

io-7

85 97 o'4

34 9.23 .4 .5 97 .

4 35

914.

LEAD NO.

O.29v.,OG

EDGE WITH RADIUS

074

REYNoLDSJ-2aNUMBER.

II THICKNESS . 053 -.

43

$ 4xIo'

CL C00 0( _{CL C0}

CL 'D0 c&

CL C00 C,. C00 4-8 ZSo 443 42 :14 224 5.0

232 320

---35 0.55 229 ---35

015 . Iqi 044 92 31

054 55 23 051

5.03

§ I !

5'O° 65

II

SoS 4-So

6-5 805 £31 4-5 8,8 5 17 65 815 523 &5 8-06

75 372 7-qq 75 8r74 7-53 7-5 S7z 75!'

-

'5

867

8-7. 886 11.1 qoe '07 3-7 q-oo 10.7 86 9-13 9(9 5-5 c15

-LEADING 0.27210'

EDGE WITH RADIUS 0-67 REYNOLDS NuMBe. ' 1-14 20 TI.1ICKESS 4-0 056

'50

I5 CL C00 C( C1.. C00 '( CL C00 C, _{C00 tX}

0L C00

32 049 236 i 64

S4 3.5 ooq 1.83 , j 2-3 o8a 1-70 23 0-97. -24 2-3 577 35 2-3 0-54 1-31 2-3 0-53 i-od

1-4 178 l3S j:3 1-62. l.Iq 4 I58 I-iS 1-4 44 015

i.

1-41 o-37 0-4 2.4o 1-35 2-4.7

l8

5 248' iae ' p42. o86 5 242. 0-82.

06 3.44 -36

06 335

1.1$

os

-4I I'll 0-5 _{3-54 o44} 0-5 3-36 o-7.

14 4-34- 1-39 i-s 4-.4 119 4-32 I-iS 1-5 4-30 a-3d. .1-4 4-2.8 0-79

24 520 146 3.17 1-35 2-6 52, 1-35 2-4. 5-il 0-92. 2-4. 5-14 085

A-s 6-84 3-13 45 6-82. 2.Sq 45 4-94 27 ₄₃ 7.05 aa 7-08

oq.

6-5- 8-35 6-24 6-4 s-3s 5-85 4-4 838 586 6-1 ags 1-2.2. Gf - 8-90 i-IS

74 S74 9-37 7-4. 8-80 7.4 3-39 8-it

73 qg 4.3Q

7-0 975 1-43 3-9 869 2-7 8-S S I-al 3-3 q.o8 24 8-5 _{9-44 8-47} 8-2 :0-2

-IS OEG.AGI.E OF ATTACK OF FACE NUMBERS WIT,! - AB0E A&E 'NEGATIVE.