THE SCREW PROPELLER. - Zenodo

THE SCREW PROPELLER.

By F. W. LANCHEBTER, M.INET.C.E. (XEMBEB OF COUNCIL).

INTRODUCTION. Q I. I t can hardly be said that in the past the theory of the screw

propeller has been established on an altogether satisfactory footing. The more general theory of propulsion aa laid down by Rankine and the late Mr. W. Froude, although admittedly incomplete, ia in effect one of the most successful of the applications of the Newtonian theory,* and may be regarded as practically suflicient.; when, however, we pass from the general theory to the implement o j propulsion, the screw propeller, the outcome of the early at- tempts cannot be regarded as conclusive in any respect. Mr. W. Froude introduced into his treatment of the subject the

notion of regarding the blade aa made up of a number of annular elements. So long as this idea is not pushed too far it is of the greatest utility, and may be said to have taken a permanent place in propeller theory ; certain restrictions, however, are necessary.

The author’s method of treatment, aa set forth in his “Aerial Flight,”? ir, founded on the theory of sustentation and of the aerofoil developed earlier in the same work. In it the Froude conception of the blade aa the sum of its annular elements is adopted, but the component elements are given, as to their pressure reaction values, etc., the benefit of the aspect ratio of the blade as a whole. The theory is developed in respect only of the condition of highest efficiency, or optimum condition, and is carried to the extent of being made the basis of an actual design. The investigation in question and the rules and procedure laid down by the

* The theory of the hypothetical medium of Newton. t 1‘01. I., Chap. IX.

at UNIV OF CINCINNATI on June 5, 2016pau.sagepub.comDownloaded from

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264 THE IKSTITUTION OF AUTOMOBILE ENGINEERS.

author have been followed and worked to at the Royal Aircraft Factory with a very fair measure of success, the utility of the method being fully admitted. I n the present paper the author goes far beyond anything attempted in the original investigation and deals with the propeller more generally, under other than the optimum condition; also as a special case (in Part I .) a theoretical solution is given to the problem of the stationary screw or ‘‘ helicopter. ”

The results of a recent paper by the author, “A Contribution to the Theory of Propulsion and the Screw Propeller” (Inst. Naval Architects), are to some extent utilised-mainly in Part I. in the discussion of the “helicopter” problem. I n dealing broadly with the screw propeller, the author has adopted a result given in an earlier paper by Dr. R. E. Froude,* in which it is shown that, in the case of a propelling instrument of a purely hypothetical kind, one-half the acceleration must take place in front of the propeller and the other half behind it. This result undoubtedly applies in some degree in the case of a screw propeller, but how much before the medial plane of the propeller is reached and how much after we do not exactly know. I t being necessary in the investigation (Part 111. j to make some assumption, Dr. Froude’s result has been taken as i t stands. It will be noted, however, that the author’s treatment admits of any other or better ascertained result being adopted; a few f o r m of algebraic expression only will need modification, and the arith- metical results mill need correcting accordingly-the method is quite e h t i c . Q 2. The author takes the present opportunity of pointing out

a difference between his present and former treatment which is of some moment. I n his “Aerial Flight” (Vol. I.) the theory of sustentation and of the aerofoil in which the propeller theory 6 founded is based to a certain (and essential) degree on the experimental results at that time available, mainly the work of Dines and Langley; in the present investigation the whole of this i s discarded, and in its place (in spite of all the wind channel work which has since been done) the author employs as basis a purely theoretical conception of the nature of the fluid motion. This is

* “On the Part played in Propulsion by Differences of Fluid Pressure.” Proc. Inst. Naval Amhiteat#, 1889.



THE SCREW PROPELLER. 265

a~ set forth in the paper read last month before this Institu- tion;* it is a definite fact that the results given by experiment are, 80 far aa relevant to the present subject, mnsiderably lese reliable than those which may be deduced from the moat elementary, theoretical considerations. The form of treatment adopted h, however, such that when the state of affairs is reversed, as in due course

FIG. 1.

it may be, it will be merely necessary to substitute a curve plotted from experiment for one calculated from theory, in the graphio portion of the work.

I t may be briefly stated here that the author’s recent paper gives

“ The Aerofoil in the Light of Theory and Experiment.”



266 THE INSTI’I’UTION OF AUTOblOHlLE ENQINEEHS.

an amount of a theory of dynamic support founded directly on vortex motion. It is shown that the aerofoil gives rise to a vortex pair or system in its wake, and that the supporting reaction, the resistance, and (approximately) the sectional form may all be correctly deduced as related to the said vortex system. Beyond this it is shown that the vortex system commonly generated by a& aerofoil of approximately uniform camber is very closely that which would be set up in a two-dimensional region by the movement of a plane (or line, if strictly in two dimensions) through a very short distance by an impulsive or nearly impulsive force; this happens luckily to be a form which has been studied and solved by the mathematician, and the streamline system is that given in Fig. 1. The two-dimensional region is presumed to lie ‘‘ athwart stream,” and the width of the plane in Fig. 1 corresponds to the effective span of the aerofoil; it has to be assumed that the plane be withdrawn immediately the impulsive force has set the fluid in motion. The two-dimensional treatment would not be legitimate were it not for the fact that the vortex is shown to comprise a cyclio component around the aerofoil itself. The subject at the best is not in theory the easiest to understand, and in any case some knowledge of the hydrodynamics of mathematical theory must be presumed, otherwise the facts must be taken on trust.*

In the present connection, the most important fact to be accepted (whether taken on trust or otherwise), is that the ultimate or residuary vortex motion may be represented by a dynamic substitute in &he form of ti cylindrical body of the fluid of circular section, whose diameter is equal to the span of the aerofoil, or, in the case of the propeller blade, equal to the effective length of the blades.

We may obtain a graphic idea of what this means by reference to Fig. 2; here we see a blade arranged diametrically acrom the mouth of a slightly bent pipe; the blade represents the aerofoil, the pipe represents the melwure of the column of air which it con- trols. Now we know that the question of skin-friction or surface

* Reference should be made to the paper cited for a fuller exposition ; aleo to the author’s “Aerial Flight,” Vol. I. For the study of the mathematical treatment of hydrodynamics and vortex motion, either Lamb or Basset may be recommended.




resistance is one of vital importance in Q h t ; the only real advantage in fact of a foil of high aspect ratio over one o f low is due to the reduction of surface; hence, we are in a position to appreciate the enormous importance of the cyolio component in the periptery, and the positive and ludioroua futility of the many

FIG. 2.

schemes proposed from time to time to utilise pipes or scoops in place of the simple aerofoil or propeller blade.

The circular form of the peripteral “cylinder ” is, .it is true, due to the particular character of the impulse distribution assumed as baais; at present there is not much choice in this respeot owing to the baokwardneas of the mathematician, but when attention is focussed on the point doubtless other solutions will be found; the author is at the present time attacking the problem by graphic methods. In any case, the particular solution in question dow undoubtedly fit the actual conditions very closely. Beyond this, i t is becoming evident that even when widely different types of impulee distribution are in question the difference in the peripteral mea equivalent is not 00 great aa might be thought, thus it is not to be supposed that the result of assuming the circular cylinder as the vortex equivalent will, in any actual case, be seriously in error.

Q 3. When we apply the same concept to the cam of the propeller we have each blade in ita spiral path forming the diameter of a circle, its peripteral area, whose trace in space defines the equivalent of the fluid coming in effect within its grasp; thus the whole region of tho propeller stream must be considered as resembling a rope,



268 TRE INSTLTUTION OF AUTONORILE ENGINEERS.

whose number of strands is equal to the number of blades. The author finds this conception of the propeller race or stream one of considerable utility; it should be kept constantly in mind.

In the author’s theory, vortex motion plays an important part in the determination of blade section. As in the case of the aerofoil the camber is dissected into two elements: the primary camber, which is related to the two-dimensional motion, that is the ultimate motion left in the h i d : and the secondary camber, which ia considered as superposed on the primary and which is concerned with the cyclic motion about the foil itself. In effect this ia as given in the author’s earlier work,* but there the distinction of primary and secondary camber was not made, neither was it realised what assistance to the theory of the subject would result from the separation of the two functions. In connection with the propeller theory as set forth in the present paper, the primary camber ia that with which me me mainly concerned; the role of the secondary camber only becomes of interest when we come to the discussion of b l d e form. This is to some degree discussed in connection with the helicopter and again in Part IV. of the paper, but it ia a subject which, to do it justice, would need a paper entirely devoted to its consideration.

The blade section dealt with in the present theory is thus, generally &peaking, that of the primary oamber. It is, however, well to bear it constantly in mind that the actual blade is something quite different, although a dynamical equivalent.

The necessity of the cyclic or vortex theory becomes the more evident the closer we we mquainted with the propeller, and on any other basis the idea of the earlier workers in the problem of propulsion would have been perfectly sound; thus Fig. 3a would have been a correct and logical design. On the other hand, how would it be possible to justify the highly e5cient propeller of the modern aeroplane (Fig. 3b), unless by an alternative assumption involving action-at-a-distance ? Again, it ie well known that the cores of the vortex pair left by each blade may become visible if a little air is introduced, as, for example, when the propeller of a steamship is not fully immersed.

Q 4. In the present paper the symbols employed are those adopted in the author’s previous work; notably his “Aerial Flight” and

* “ Aerial Flight,” Vol. I.




the recent papers read before the Institution of Naval Architects and this Institution.* In the latter paper, however, the symboI e was inadvertently used to denote the angle of trail of the primary camber of the aerofoil; as this symbol was used for the effective pitch angle of the propeller blade in the previous work, it is being retainedjn the present paper in its earlier usage, and the symbol rr is adopted for the trail of the primary camber. The paper on the Aerofoil is being revised accordingly. Again, objection has been urged (perhaps with justification) to the use of the skin-friction coefficient aa being expressed in terms of normal plane pressure, in lieu of being expressed as a constant dealing direct with the

quantities density and velocity squared, aa is usual in other cases. This objection is most easily met by combining the coefficient with the normal plane constant, thus, $17, it being understood that when these symbols occur thus in juxtaposition, the value of 0 ehdl be taken as 0'62.

As a matter of convenience the present paper is presented in four parts, as follows:-

Part 1.-The Stationary Propeller. The " Helicopter '' Problem. Part 11.-The Propeller of Highest Efficiency. Part 111.-The General Theory and Solution of the Sorew

Part 1V.-Conclusions and Points not otherwise Discussed. Propeller.

* '' The Aerofoil in the Light of 'l'heory and Experiment."



270 THE INSTlTUTlON OF AUTOMOBILE ENGINEERS.

PART I.

QHR STATIONARY SCREW PROPELLER.-THE PROBLEM OF THE

“ HELICOPTER.”

Q 5. In the theoretical discurnion of the scww proptller the problem of the stationary propeller, that k to say, the propeller working under the conditions of the screw ventilating fan, has received comparatively lilmited attention; there is, for instanb, no authority to whom the would-be designer of a direot lift machino of the “ Helicopter ” type can turn to tell him whethei his projeds have any chance of succass or to guide him as to the lines on which to experiment, or to develop his ideas.

At first sight it might be supposed that the problem of the screw ventilating fan and that of the stationary propeller (as involved in the helicopter problem) axe one and the same; in neither c m a n we speak of efficiency in the mnse of screw propeller efficiency, for the mounting is fixed, m d there is no useful work done in propulsion; thus reckoned fmm screw propeller standards the e&- ciency is mro. If the object were the same in both the sorew ventilating fan and the helicopter,* the problems themselves oould, with certain reservations, be regarded as idenlical, but according t o the best authorities there is an actual difference; in the helicopter the problem is to obtain the maximum of lift with the minimum power expenditure; in works on the screw fan, stress is usually laid on the volumetric efficiency, in other words, the quantity of air passed. Without questioning the commercial expe- diency of this latter view, it is worthy of -remark that it seems to be bmed on the assumption that there is a certain sized hole in a wall and that somehow or other the necessary volume of air has to be got through that hole, the best fan is the one whose de2iuerylh.p. is the greatest. It does occur to one that it may sometimes be the better expedient to enlarge the hole, and thereby reduce the h.p. required; however, the problem has presumably been worked out by hard experience, and eventually there must be same maximum size of hole permissible, and once this is admitted the best fan is dearly the one which passes the most

There is actually much that is in common in the two problems; thus to obtaiu the greatest possible lift from a screw involves imparting t o the air the maximum quantity of momentum per second.




wind for a given horse-power. Now in the cast3 of the helicopter, since the direct problem is to lift a certain weight, the indireot problem is to wmmunioate a definite quantity of momentum (downward) to the air per unit time, which is a different matter to the displacing a given volume, and it is this which makee the two problems essentially different.

There is one paxticulrtr O&BB of the smew ventilator which perhaps may be regarded as exempt from the ordinary limitations -the ceiling punka31 aa f*quently to be seen in Continental restau- rants: here the diameter limit no longer exists, and the objeot is otherwise to stir up the air rather than diaplace it, and it may be generally inferred that the effectiveness of the appliance will be proportional to the downward force continuously exerted by it on the air: thus the conditions appear to be identical with thoae in the 0&88 of the hlicopter.

Q 6. In a recent paper contributed by €he author to the spring meetings of the Institution of Naval Architects, the fundamentd dynamic basis of the helicopter in the s&ntathn of ita load ia discussed BA a matter apart fmm the meam or mechanism to be employed; that is to say, the problem dealt with ie that of the sustaining of a Load by direct reaction on th0 fluid (air in the aotual problem), distributed over a certain area or disk; the treatment adopted being on the lines originally laid down by Dr. R. E. Froude, abeady referred to in the introductorg section of the present paper. On the baeia in queation it is shown that the weight in pounds which may be sustained per h.p., under the best conceivable conditions, is given by the following fomiuIa,*

(1) w = s o / - 2Ap . . . . . . . . . . . . . . . . h.p. 32.2 W

where A is the m a of the propeller disk, W is the weight sup- tained, and p the density of the fluid, in the case of air approximately 1/13 or 0'078.

The author, in the paper in question, pointed out that the above may be expressed in very simple terms: for one ton gross

* When the whole power expended is represented by the kinetic energy of the downwardly impelled " wake " stream. Under these conditions the velooity (downward) in the plane of the propeller is half the ultimata velooity, there being a contraction in the stream similar to that which takes place in efaux phenomena during the period of aoceleration.



27 8 THE INSTITUTlON OF AUTOMOBlLE ENGINEERS.

weight the pounds sustained per h.p. will be numerically equd to the diameter of the downward current, that i s to say, approximately 0'7 of the diameter of the propeller disk in f e e t . Thus 8 gross load of one ton sustained by a propeller 40 ft. in diameter will require one h.p. for every 40 x 0'7 = 28 pounds weight. This is, of course, the absolute theoretical minimum for the oonditions given,a@ from all losses incidental to the employment of screw or other mechanism by which the downward current is maintained. In the w e of the swew these include skin-frictional or direct blade resistance losses, also those represented by the rotational component of the " make."

Admitting the-deficiencies of the Froude hypothesis on whioh the abovo computation is based, the power required, apart from the final means of propulsion, must actually be in excess of that stated. The author has introduced a modification into the Froude treatment involving a, factor to represent the losses of energy proper to the modified regime (aa distinguished from instrumental losses), this being denoted by the symbol Q, a quantity less than 2 and greater than unity; the expression becomes,

h.p. w = """J3*w.. . . . . . . . . . , . . . . (2)

Thus, the weight sustained per h.p. is lw than under the prece- dent conditions in the mlation of 1 : 4 There is at present very little information on which to base estimates of the value of Q ; for the main purposes of the pmeent pqper,, Dr. Froude's lluw will be taken as baais, i .e . , Q will be taken as equal to unity; thus our resulk must be regarded w on the optimistic side of the truth. The efficiencies me shall deduce will be rather better than in practice w0 can hope to obtain; it is not, however, in the present state of theory always certain that t b methods dealing with the means of propulsion-the instrument of propulsiondo not auto- matically include also the losses of rhgime.

Q 7. Let- V , = velocity of the blade element in its circular path. c2 and w = respectively the axial and circumferential com-

= axial wake velocity component in the hypothetical ponents of the impressed wake velocity.

'' plane of propeller."




rr = maximum angle of a hypothetical impelling surface; the maximum angle of the primary camber ; (compare preceding Paper.”)

A = area of the foil (or its containing rectangle), such that

A n is the square of the span and 4 A n is the

peripteral area.

Ic

m Et = energy per second in downward component of wake

= mass of air dealt with per sec.

on the Froude basis, 21 = 4 2 (otherwise Q = 1).

‘ \ FIQ. 4.

6 C = the direct resistance, or skin-frictional co-efficient in

n = aspect ratio. Q = constant as defined.1- F = density of h i d .

absolute units.

* “The Aerofoil in the Light oE Theory and Experiment,” p. 171, m t e .

’ ‘=Energy in wake stream‘ Energ-y expended Comp. ‘‘ A Contribution to the Theory of

PmpuMon.” Roc. Inst. Naval Architects, 1915.

LANCHESTER. S



274 THE INSTITUTION OF AUTOMOBILE ENGINEERS.

Referring to Fig. 4, we may write at once-

and

Now,

1’0

0 9

I

0.8

w

(3) 9’ 0

J’ - 4 .................... ’ - s1n n n v = u tan2 - .................. (4) 2

Tr - s A ~ P V ~ m - - - A n p V , = z - sin n

1 - 4

( 5 ) illt = ~~ - - ~ .............. mtvrrZ n A n p v r r 3 - 8 sin n

l o d e . n 20 deg.

O E D I X A ~ = EFFICIENCY.

FIG. 5.

90 deg.

Energy/ t lost in direct blade resistance

............ (6) t C A P V , ~

= Y, X fCApl.’,‘= - sin3 n

................... (7) BZC

~ r n sinZ n or in terms of Bt =-

and aerodynamic loss (in rotational wake) is,

.................... (8) E t tan? 2




(9) or in t e r m s of E , =tanZ. ...................... ?1

2 :. total energylt loss in terms of Et is,

r) 85C + tan2 ~ ................ (10) rn sin2 TJ

1 ~ :. EfEciency is, .............. (11)

8'c + t anz i rn sin2 r) 1 +

Graphs of efficiency calculated Aom this expression are given in

1 0

0 ,

0 8

0'7

10deg. r ) 23 deg.

. .

OBDINATES = EFFICIBNCY.

fie. 6.

Fig. 5; for values of aaped ratio n = 2-5 and n = 5.0. ZC" is taken as 0.01, a value which under ordinary circumstances is not f a r from the truth.

The basis of the foregoing is the assumption that Q = 1. Further graphs are given in Fig. 6, on the basis of Q = 2 ; Figs. 5 and 6 thus represent the probable extreme limits.

Q 8. We next require to investigate the question of the number of blades, that iS to say , we have to determine the minimum number neoessary to deal with the whole of the column of fluid-air-

F 2



27 6 THE INSTITUTION OF AUTOMOBILE ENGINEERS.

oontemplated in theory as passing throgh the propeller disk area. The foregoing investigation refers specifically to a single blade, but it is clearly applicable to any number of blades, provided tshat there is no interference, in other words, on oondition that their peripterd are= do not overlap. As a matter of fact, the question of interference is one of degree rather than one of actual definition; thus it is one in connection with which we need to establish some sort of a convention. In the rather similar case of the spacing of the two members of a ’biplane aerofoil tho same question crops up; hore the matter has been settled experimentally; it is shown that when separated by a distance somewhat in excess of the chord dimension, the ‘‘ biplane ” disposition gives between 80 per cent and 90 per cent of the lifting power of the two members separately, the resistaneellift ratio being lowered in almost exactly the sa,me proportion. The interference still exists therefore, and results in a loss of about 15 per cent of the effective area of the supporting surface; but in spite of this it is generally agreed to accept a spacing of 1 to 1.5 times the chord as being the practical limit of seriow interference.

Now, when we a,re concerned with the problem of superpomd wmfoils or planes, the conditions (as pointed out in the author’s “Aerodynamics,” 210) are not the same as where the blades of a propeller are in question. In the former case it is the quantity termed by the author the “sweep ” of the foil which is important, and the practical separation distanoe bf two aerofoils is in fact the nearest approach to a positive measure of this quantity; in the latter case it is the peripteral area which is decisive. The point of viex of the author, as hmmonising the otherwise rather anhbiguoous porsition, is, in brief, that i f experiments were con- ductcd with numbers of superpmed YQ& the individual spacing would require to be increased as the numbers become greater until in the limit the s M n g would require to be that corresponding to the peripteral area, that is to say, in the ordinary way the spacing, when the number becomes sdicient, would approximate to from 2 to 4/5 of the span. I t is difficult to say that interference would cease even at this distance, but it can be definitely stated that it mwt beoonie sensible and even serioius i f the distanoe is appreciably less; the propeller blade conditions are analogous to those of an aerofoil system of superposed naembers when W e iaurnber becomes indefinitely great.

The basis, therefore, on which we shall conipute the number




of blades permissible will be as follows. An expression will be found for the volume swept per second by the peripteral area of a single blade; complementary to this an expression will be given for the volume paasing per second through the propeller or rotor disk area; the latter in .berms of the former will give the appropriate number of blades. Owing to the f a d that the above is not of necessity a whole number, and that fraotional blades are inadmissible, the designer will usually have the option of adopting a number of blades which will reault in interference in some degree, or of adopting one less blade, in whioh c&s8 the whole 05 the supporting reaction as computed from bhe propeller disk area will not be realised. In 'thb farmer ca& the mechanical efficiency will s d e r to some (usually small) degree; in the latter m e a rather larger propeller or rotor 'diameter will have to be adopted than otherwise necessary.

Let " span," i.e., effective blade length, = $ diameter of rotor, take periptery as defined by a circle whose diameter = span, thus,

peripteral area - - - 0.14 a.

Now we will call the path of the centre of the blade the peripteral azis-it is a spiral-and the velocity along the peripteral axis is clearly the v1 of Fig. 4, henca the volume included by the periptery per sscond

9 a ' - 6 4 -

=0*14 aV1 Now the effective area of the rotor or propeller disk is its total

area less that central portion not swept by the blades; that is to say, a circle 2 of the diamehr of the disk requires to be deducted to correspond with the assumption that the effective blade length is 3 of the diameter; in any case this central portion cannot in theory be u t i l i d so 'long as the instrument is a screw of any

kind. n u s , effective area of disk = - a or = 0.937 a. 15 16

the value of Q ; in the limiting values, And the velocity u of the air through the disk depends upon

Q = unity u = v2/2 Q = 2 U = V ~

In the first case, denoting thle number of blades by the symbol N,

or, N = 3 * 3 3 sin r,



27 8 T H E INSTITUTION O F AUTOMOHILE ENQINEERB.

In the second case tlue number is twice as great, or N=6*66 sin n

We will amume--as it is oertainly fair to do-that on t b peripteral axis the angle n is given a value in or about that OP high&. e5ciency. If, in any case, this assumption doas not represent the fad, it is open for the. &signer to give the angle in question whatever value he please, and to revise the present calculation to suit his own particular case.

Thus, i n the first case, referring to Figs. 5 and 6, n =2'5 rr =25 degrees

sin n = 0'422 N = 1'4

n = 5 ' 0 n =21& degrecs

sin =0'366 N = 1'2

In the second case, n = 2.5 r, =21 degrees

n = 5.0 TI = 18 degrees

sin n = 0'358 N = 2'36

sin n= 0.309 N = 2'06

From the foregoing we are justified in stating that two blades, in practice the minimum which can be adopted, are more than sufficient; any greater number will have no advantage, and can only result in a lowered efficiency. The particular condition in which the number comes out materially above 2, i.e., the maximum possible value of Q in combination with a low aspect ratio, is an extreme which can be reasonably ignored: even i n this case the provision of three blades would mult in considerable interference. Q 9. Our next consideration will be that of the camber and

sectional form of the rotor blade ; we have also to consider th8 question of the variation to be assigned to n a t different poinb along the radius.

I n oonnection with the latter point i t is to be remarked at the outset that the rotor of a helicopter is not bound down by the ordinary considerations which control the angle variations in the ordinary screw propeller. The whole angle n, in tho case af



THE SCREW PROPELLER, 279

the helicopter, is amlogow to a part only of the gliding angle .y, as used in the author’s “Aerodynamics,” the angle e representing the affective pitch angle in the surew propeller theory (mmp. Part II.), is zero in the blimptar, since the latter is presumed to sustain without axial a h n c e through the air. In practice, it is true, the maahine requires to lift, it must be capable of a certain vartiaal velocity, but this GJ not the essential feature of the problem; it is rather inc6cleIln;al. Given that the requirements of sustentation are properly met by the design, it is only necessary to supply the needed additional horse-power and drive the rotor above its normal speed to obtain a definite upward velocity. The detail of the regime under these conditions need not trouble us. I n the extreme, if it be considered poasible a d desirable to obtain a high speed rate of ascent, then the problem, ceases to be in fact a special case, and may %e treated as f ~ a

ordinary matter of screw propulsion, and the mom general theory of Part 111. w i l l then apply; at pre-t, however, the h.p./wt$ght problem becomes far too severe if a high rate of asoent be contemplated; we must remember that no machine of the type under consideration has yet been even made to lift sucaess~ly , let alone, to soar like a rocket.

Thus, the angle @ being zero, the angle of the blade in the helicopter may be graded from point to point along the blade length according to the will of the designer, in the same way as the addendum or slip-angle in the sorew propeller, and in deciding this point the designer requires to take into mmunt, on the one hand, the distribution of the load over the disk area required bJr theory, and, on the other hand, the falling off in effi- &ency when the optimum angle-m shopPn by Figs. 5 and 6-45 departed from. I f the angle be made constant from root to tip, the resulting prassure per unit area on the blade will manifedy include a multiplier varying as the square of the distance from’ the axis, thus impasting an unduly high velocity to the outer concentria elements of the stream. If , alternatively, the angle m be made vayiable along the blade length, in such way as to be represented by a true helical surface, then this multiplier bewrnw praprtioual to the radius itself, which is that required by t3leory to give a correct distrilbut.ion to the momentum in the wake stream, but at the same time results in the employment of angles of somewhat low efficiency. To some extent the conditions may be recon- ciled by a tapering off of the blades tomnrds their outer cstrrmi-



2&0 THE IXSTITUTION OF AUTOMOBILE ENGIXEERS.

ties, by which means the pressure per unit m a at the different pa& of the blade may be kepi? proportional to v2 (which corresponds to a constant angle) without thiwwing too great a proportion of tho total load on the outer conuentiic elements; this ia closely analogous to the tieatment of the same diffimlty in the theory of the screw propeller (romp. “ Aerodynamics," Ch. IX.) . , The author believes that in practical design it mill be found‘ desirable i n both the rotor of the helicopter,* and in the screw propeller, to accept a compromise, namely, a tapering blade, but not to the degree corresponding to ri = constant, in other words a reduction of angle and partial ‘‘ wash out ” of the primary camber towards the blade tips, and a slight sacrifice of efficiency consequent upon the extent of angle and camber variation adopted.

The treatment of the question of sectional form in accordance

FIQ. 7.

with the author’s recently published methodt is a matter of com- parative simplicity. The primary camber is first assigned (Fig. 7) to give the correct value to the angle r i , precisely as i f the problem were one of two-dimensional motion, and the secondary or addendum camber is then assigned to accord with an angle a calculated from the cspression,

as given in 12 of the paper cited; the camber so obtained (Fig. 7) represents the forin of mid-section. The secondary camber

(13) tali a = 0.39 n tan ri . . . . . . . . . . . . . . . .

* Assuming foi. the purpose of discussion that the helicopter is to be regarded

t “The Aerofoil in the Light of Theory and Experiment,” p. 171 , w t p .

as a thing of potential utility. It is by no mean8 certain.




should, strictly speaking, be graded to ordinates set off proportional to those of a semicircle whose base h the span, i.e., blade length. Probably there is little real difference if the camber is carried without change to the extremities; in theory this should, give a very slightly increased totalyressure reaction with a rather- more-than proportionally increased resistance: exact data as to the difference are at present lacking, either as computed in theory or as determined by experiment.

9 10. Whilst discussing the question of sectional form, it is worth while to add a few remarks of general interest on the point. The actual sectional form of an aerofoil, except in the erne. of small models of paper or mica, cannot be the lamina represented by the curved line shown in the figure, since, for structural reasons alone, the section requires to possess solidity. Again, it is by no means necessaxy that the actual foil should' have its anterior and posterior margins coincident with those of the theoretical surface; the extent of the foil forw,ad may be curtailed, as shown in the figure by the shaded section, without any loss of sustaining reaction, and the pressure reaction will, near the leading edge, be greater than the mean pressure to make up for the loss of area; indeed this, as an experimental feature, is quite familiar. On the other hand, there is no disadvantage in allowing the posterior margin to extend, con- formably, of course, to the lines of flow, some little distance abaft the theoretical limit; by this means it aan be ensured that the pressure difference between the upper and lower surfaces dies away before the parted layers of the stream re-unite, and 80 the production of eddies at the trailing edge is minimised. The reason for the unsymmetrical character of the treatment fore and aft as hero given is due to the f a d that the pradioal requirements of stream line form will not tolerate a blunt after edge, whereas a blunt leading edge is not detrimental: in some cases of experimental determinations of the resistance of spar sections it has been reported a9 advantageous. It is evident that a blunt nose to the section renders it adaptive to considerable variations in the incidence of the lines of flow where the fluid i8 entered. To some degree the extension of the after edge minimises the detrimental results of disconformity, such aa will arise when the attitude of the foil is inappropriate to its loading and camber-otherwise, when its camber is not oorredly adapted to its pressure constant, the C of the equation P = C p V 2 . In the wings of birds the adaptability



282 THE INSTITUTION OF AUTOMOBILE ENQINEERS.

of the after edge is greatly inm& by its delicately graded flexibility, and so without doubt variations of flight veloeity are provided for; this is a feature whi& the author believm might well be introduced into the design of the flying machine, especially in cases where the greatest possible variation of velocity is mn- sidered important.

On referenoe to Fig. 7 it will be at once evident that the SO-

d e d chord angle, as measured from an actual aerofoil (or pmpeller blade), is a quantity entirely without meaning; the leading edge may be ourtailed to, a greater or lees degree, or the trailing edge may be pushed out further or less faz along the lines of 00w in the wake, without affecting t h e dynamic value of the urnfoil in any measurable degree, but with as many different s o - d e d chord angles aa the variations made. The chord angle of hypothesis, which is equal to d 2 , has no real relation to the so-called chord angle, and is a quantity which must either be given by the designer, or calculated from the dynamic properties of the itctual foil on the lines laid down in the author’s “ Aerodynazliicsi,” or in accordance with the newer regime discussed in his recent paper.

Q 11. We now revert to the main subject. We may allow that in view of all the conditions, the rotor effioiency cannot be aa high as theory indicates as its maximum; probably, i f we take it that efficiencies of the order of 85 per cent are possible, we shall not be far from the mark. It must be &called that the term efficiencj in the present connection is an altogether different matter from that with which we have to ded in the general theory of the screw propeller; it does not represent any ultimate work done, except in the form 03 the kinetic energy represented in the downward wake, so that there is nolthing contrary to established experience, even though the full theoretical efficiency of over 90 per cent should be attainable.

I n the w e of the aewfoil of a flying machine, we know thaC after all other considerations have been talien into acooqnt there remains the factor olf aerofoil weight as tending to curtail the area which should in practice be found most advantageous; now in the case of the helicopter it is this self-same question of the aemfoil weight, or rather the rotor weight, which almost entirely determines the best diameter to u p ; mre it no& for this factor the power required could be reduced indefinitely by progressively increasing the diameter; this is clear from equation (1).




If, for the purpose of illustration, we suppose the total weight it is required to l i f t be one ton, there should be no great diflicultg. in klesigning a rotor of 50 or 60 ft. diameter within the permissible weight which could be assigned to that part of the machine. The conditions would indimte a “monoplane” structure of, say, 56 f t . length: about 6 ft. wide in the central portion, tapering to about 3 ft. at the extremities, the two blades thus bsing embodied in a single structural member. Calculating on the basis of equation (l), the weight sustained per h.p. expended i n the downward motion imparted to the air is approximately 40 pounds, and if we allow for an energy loss as due to a value of Q = 1.5, a reasonable allowance to make, the weight per h.p. will be 40 divided by or approximately 32 pounds, or 70 h.p. for the one ton weight total. If , now, we take the rotor efficiency, to be on the safe side, as 75 per cent, we find the total required to be between 90 and 100 b.h.p. The author is firmly of the belief that a machine designed to the dimensions given, and in accordance with the requirementa of theory as herein laid down, would lift satisfactorily i f driven by reasonably efficient gearing and a power expenditure of 100 b.h.p.

There axe, we know, several problems i n connection with the direct lift machine beyond that of sustontation which need to be solved before any such machine can be deemed even an engineering success, but these fall rather outside the scope of the present paper. There is, for example, the question of rotational anchorage: evidently some provision is needed in any actual design to prevent the car rotating instead of the rotor; obviously the arrangement of two rotors having reverse rotation offers a simple solution, and here there appears to be a possibility of aa actual improvement. Thus, i f two rotors of opposite hand be arranged on concentric shafts a large portion of the energy otheraisc lost in rotational wake may be recovered. The author has not pursued the matter further; an investigation to cover the condition in question would be an affair of some length.

8 12. 0 1 1 looking into the various schemes for direct lift which h w e been proposed from time to time, it is quite clear that owing to a want of appreciation of the principles involved, the conditions of least power expenditure have not been complied with. Thus, n-e are familiar with designs in which are embodied members fashioned after the manner of an American windmill, with an incredible multiplicity of blades, also with machines with fundamentally inadequate rotor area, and others with other defects of an equally detrimental kind.



284 THE INSTITUTJOK OF AUTOMOBILE EXGLNEEKS.

One of the standard methods of miscalculating a helicopter is to figure the lifting value of the blades on the assumption that. they are fully analogous to the aerofoil of a flying machine, using the established aerofoil data for the purpose; the fact is entirely ignored that( ultimately the lifting reaction is tied down by the Newtonian principle, that is to say, as due to tho downmrcB momentum of the air passing through the c.ircular area swept by the rotor blades, in accordance with the teaching of Ranlrine and Froude. All question of blade interference is thus ignored.

Summarising the position, me may take it that there is n o present prospect of making a direct lift machine without a considerably greater expenditure of power than that required for a flying machine of ordinary type of equal weight, or in the alternative, should this be achieved, it can only be done by the adoption of a diameter fa r greater than the span of a flying machine of equal weight, the diameter of rotor required being in the region of twice the span of the machine of ordinary type. At the same time, unless there are unaccounted losses of efficiency of unsuspected magnitude, it is equally clear that the direct lift m h i n e is, as a problem in engineering, capable of present day solution; whether the problem, when solved, will result in a mmhine of any possible value, military or otherwise, is quite another and in itself a very debateable question.

PART 11.

THE SCREW PROPELLER UNDER THE CONDITIONS OF MAXIMUM

EFFICIENCY.

Q 13. The screw propeller under the conditions of maximum efficiency may be coqsidered and treated as a special case. In the general treatment it is necessary to take account separately of the losa due to the axial or direct (rearward) velocity component and that due to the circumferential or rotational component, i n addition to the skin-frictional or direct blade loss. So long as me confine ourselves to the condition of maximum efficiency, the whole problem is vwtly simplified; it is demonstrated in the author’s “Aerodynamics ” that for this particular condition the whole of the losses may be lumped into one as expressed by the least gliding angle, or least resistancellif t coefficient of the blade, considered as an aerofoil adapted to move in th0 helical path.

The method of treating the blade of a propeller as a nqmber of annular elements is originally due to the late Mr. W. Froude,




whose paper of 1878 is summarised in White’s Naval Architec- ture,” p. 606 (3rd Ed.). The present author, working on a similar basis,* has obtained values for the pressurelvelocityz relation of least resistance for the different annular elements of the blade and has worked out a rational method of blade design in accordance with this relation; curves of efficiency also are obtained based on the proved constancy of the minimum gliding angle, the whole of the real complexity of the problem being by these means evaded; it has, however, to be frankly acknowledged that the treatment aa such is that of the special case, and can only be applied with a certain amount of ‘‘ interpretation ” when the essential conditions of tnaximum efficiency are departed fromior are rendered impossible by the limitations imposed.

One of the most interesting of the actual results reached by Mr. Froude is that the condition of maximum eaciency, that is to say, the most efficient annular element of a blade whose efficiency is everywhere maximum, has an angle of 45 degrees. The author in his investigation obtained a very similar result, namely, that the most efficient element of the best possible blade will have an eflective pitch of 45 degrees less half the least gliding angle. Now these two results in any case are in close accord, for the least gliding angle is probably between 0’05 and 0.10 (radians), or, my, between 3 degrees and 6 degrees; hence the author’s result is in fact that the best effective pitch angle is betwleen 42 degrees and 13$ degrees. I n order that the matter shall be quite clear, plottings of the curve of efficiency are given in Figs. 8 to 13 for values of 7 the gliding angle being (in radians) from 0.05 to 0.10 inclusive; in the figures the radius of the propeller in terms of pitch is given by abscissae and the efficiency corresponding to these radius values is given by ordinates; the angles corresponding to the abscissae values of pitch ratio are given as an irregular scale. It is to be pointed out that both the radiwlpitch values and the correqonding 0 values relate to the eflective pitch and the effective @oh angle, in other words they represent the motion of the propeller Z-I,S if the fluid were a solid and the gliding angle a certain constant angle of friction. This analogy is, for the purpose of quantitative deduction, complete, and the diagrams may be applied (and have been so applied by the author) to the representation of the efficiency of screw gear or worm gear; the analogy

* “Aerial Flight,” Vol. I., chap. IX. (Constable, London).



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is duc to the fact (which the author believes he was the first to demonstrate) that the least gliding angle as a function of velocity is approximately constant.

Now tJm probability is that Mr. Froude mas not dealing witla the effective pitch angle but rather with what is sometimes termed: the true pitch angle, so that there is no reaaon to suppose that the difference between his result and that of the author represents any real discrepancy; the difficulty is to satisfactorily defino the true pitch angle. Tho conception is, we may say, founded on the idea of a screw which is a true helix, and when pterygoid sections are employed in tho design of the blades it is difficult to fix any definite physical feature of the blade as giving a measure of the pitch angle. The difficulty is exactly comparabre to that of the chord in wing form discussed in Part I. of the present paper.

I f we take the case of an aerofoil designed to the condition of least gliding angle, then we know that the gliding angle must approximate closely to the trail angle of the primary camber n, for the dynamic reaction may be represented by the angle of tlie hypothetical chord, which is nl2, and under the condition of least resistance this is half the total.* Two proofs of this may be offered; firstly, there is the obvious fact that the curvature of the primary camber represents in reality the deflection of a current -that defined by the peripteral area-and implies a uniformly applied acceleration and a uniform pressure distribution; hence, the mean normal whose inclination to the vertical is nJ2 gives the direction of the resultant reaction. Secondly (and really the better proof) a dynamic demonstration may be given.? Taking, then, n as equal to tlie gliding angle, and assuming Mr, Froude’s 45 degrees aa being the angle of the h.ypotheticn1 chord, we find that the author’s result and the previous result of Mr- Froude’s analysis axe identical, for half the gliding angle may be expremd as n/2, and, adding this to the author’s value, w e come h k to the 45 degrees of Mr. Froude, Fig. 14. With reference to Fig. 14, it is well to point out that the equality of r and w only apply provided the condition of least resistance be assumed (direct resistance = dynamic resistance), also that t h e camber curve shown is definitely the primnry camber and bas not of ncccssity physical existence in the final blade form.

\

* “Aerial Flight,” Vo!. I., 6 164. t Part IV. ; 4 so.

IANCHESTEK. T



290 THE INSTIl'C'1'ION OF .4UTOMOHlLE ENGlhTEERP.

Q 14. The bmi4 on ~vliicli the author derived the curves given in Figs. 8 to 13 is wortli inontion tw bearing on the mope complete investigation and solution given in Part 111. of the present paper. In Fig. 13 the gliding angle nOh = r is Yhown as superposed on the angle of effective pitch rlOb=e; now i f we, for the tiine being, regard the problem as that of a flying machine climbing steeply,* then an applied force o f propul- .&n sufficient to climb the gradient On if no resistance of any

FIQ. 14.

kind were experienced (the work done being represented by tho change of altitude do), will, under the real conditions, result in the w h i n e climbing the gradient Ob, the actual gain of altitude being db; thus the efficiency is bd,'ad. On the basis of the independent treatment of the annular elements of a propeller,

* A rigid proof is given in the author's '' Aerodynamics ') (1. c. Rntej. The analogy between the smew blade angle and the climbing angle as here employed only holds if in the latter the thrust be suppomd to remain horizontal.



each element is supposed represented by &.diagram similar to that given in the figure, and thus each element will have an efficiency proper to its own values of y and 8.

shown that the efficiency thus given by the expression (2 +-+J

is maximuui when e = Now the angle e is definitely

related to the iadius/pitch, whereas the angle y ir, within the €imitation imposed by the condition of minimum value, under the control of the designer; hence the author hm from the

The author tan 0

-- '. 2

Fra. 15.

outset given tho preference to the angle e as a datum, that is to say, the angle of egective pitch rather than the blade pitch, whicli depends upon some arbitrary definition and in any case includes a function of the variable angle 3. A propeller designed on the basis of the 0 angle includes its slip factor in the design. and in any case the quantity which the nayal architect terms "slip" requires very careful definition before it can be regarded as one of scientific application.* If the hypothetical chord of the blade were something tangible which could be measured, the blade pitch could be appropriately defined as the. helical pitch of the chord; but we have already seen, vith reference to Fig. 7, that the real or

* Comp. 4 29. T 2



292 THE INSTlTU’FION OF AUTOYOHILE ENGINEERS.

“so-oalled” chord has no neceasary relation to the hypothetical chord, and is a measurement of no dynamic import. The difficulty of basing propeller theory of the blade pitch is again com- plioated by the fact that the designer may elect to vary hie Y angle according to some arbitrary scheme that may be found advantageous in practice, and so the blade pitch will vary f& point to point in a manner that will defy systematic treatment., For the foregoing reasons the author regards any attempt to rationalise propeller theory on any basis other than that of effective pitch a8 tending to chaos and foredoomed to failure.

Referring once inore to Fig. 15, it is clear that for any given value of 0 the efficiency is greatest when the line ab is the shortest, and it requires no demonstration to show that this corresponds b the angle y being the least possible; therefore, the curve of maximum efficiency of Figs. 8 to 13 must be definitely founded on the minimum value of y; by no jugglery can it be otherwise. Thus we have the remarkable fact that the conditions of best efficiency for the screw propeller can be laid down without giving a single thought to the question of the rotational component of the wake: as already stated, this is so in the special case of nnaximuru efficiency only.

Q 15. I n all propeller design the question sooncr or later arises as to how much of the efficiency curve to include in the blade length selected. If we take, say, the Y value 0’07, the graph for which is given in Fig. 10, it i s quite clear that we have the option of utilising as much or as little of the curve as we please or may deem expedient. For example, i f we select just the very best portion of the curve with a view to obtaining the highest possible e5ciency, we shall have a propeller with a pitch diameter ratio round about 3 to 1, and the arnis to carry the blades mill require to be considerably longer than the blades themselves; clearly the losses in the resistance of the arms will be too serious, and the proportion of the disk area usefully employed mill be so small that any such design is out of the question. This question has been discussed very fully in the author’s previous work; i t is a matter i n which there must always be some latitude and discretion left to the designer; it is equally a matter in which some convention is necessary as representing normal practice.

For the purpose of the investigations forming Part 111. of the present paper, the convention taken is that the diameter of the propeller is in every oase twice that at which i ts efficiency is



g r a t e s t ; the meaning of this, as to the amount of the efficiency curve included, will be clear on reference to Figs. 8 to 13. A further convention has been adopted to the effect that the active portion of the blade is three-quarters of its total radial length; thus, the central portion of the propeller, of one-quarter the diameter of the disk, is regarded as “blind,” the blades within this circle am treated as spar sections, or alternatively the central region may be supposed occupied by a boss as aictually the case in the Gifford propeller.

The oonvention as to the type of vortex motion in the blade peripteq m d n s as before, the periptery being taken as repre- eented dynamically by the content of a cylinder whose diameter ia equal to the effective blade length. The justification for this will be found in the author’s preceding contribution ‘‘ The Aero- foil in the Light of Theory and Experiment,” and in the Intro- duction to the present paper.

PART 111.

THE GENERAL THEORY AND SOLUTION OF THE SCREW PROPELLER.

Q 16. We lmve seen that for any value of the gliding angle 7, the

efficiency of propulsion is given by the expression -- - where 0 k the effective pitch angle for any given element of the blade, and that the maximum efficiency is that corresponding

to a value of e = ; the mean over the whole blade

tan e tan ( 0 + r>

90 aeg. - Y -

2 being leas than this value by an amount depending upon the portion of the curve of efficiency utilised by the blade length selected. Now the value of y may be its minimum value, that is to say,

it may be the least gliding angle, but this is not essential; the equation applies equally whatever the angle happens to be, it applies equally in fact to the case of worm gear or the wmmon screw prem-the angle Y representing the effective angle of friction. In the c u e of the propeller, as in the w e of the aerofoil, the condition of least gliding angle come- eponds definitely to a certain P/Va value, expreseed aa we know in absolute unite thus, P = C p V , the value of the constant C being (for least resistance) approximately equal to Js and oommonly lying between 0’25 and 0’35 under ordinary condi-



294 THE INST11 IJTION OP AIll'OMOHILE ENGIKEEHS.

tions of design.* Evidently when conditions prn i i t we shall make usc of the most advantageous P/V2 value and design on the lines of the preceding section for optimum efficiency, choosing our effective blade length and corresponding pitchldiameter ratio appropriately; it is when there are requirements to be considered other than efficiency (and in some degree there are always other requirements), that we have to go more deeply into the problem and eEect something in the direction of a compromise.

8 17. There are two particular limitations more commonly niet with as imposed by engineering conditions (as apart from purely scientific considerations), both of \vhich as it chances tend in the same direction; these me a diameter limit, and a pitch limit: the first of these is usually due to questions of ground clea,mnce, or in the case of the marine propeller, limitations imposed by dpaught or proximity of hull, etc.; the second is due to the incom- patibility between the engine or prime mover and the propeller as to revolution speed-more frequently than not the engine or motor constructor aslrs for a higher revolution speed (in order to save weight) than that required for the propeller of optimum efficiency. Under the conditions of restricted diameter, the thrust reaction has to be sustained by a lesser total area representing B

lesser ni<ass of air than that corresponding to highest efficiency, and i f any attempt be made to design to the same constant C a.9

before, the blade area will be found to represent a greater proportion of the total disk area than under the unrestricted conditions. If, as we may assume to be the case, the whole column of air or fluid be fully occupied by the periptery of the blades in the initial design, it is clear that the maintenance of the p m - . sure constant at its theoretically best value will result in interference, and ultimately it becomes neoessary to adopt a higher pressure constant and put up with a higher value of r with a corre-

tan 0 sponding drop in the value of efficiency.

that is to say, a 108s of tan ( 0 + Y)' It may not be considered self-evident that the restriction rsa

to pitch, i.e., the adoption of a pitch less than that of highest efficiency tends of necessity in the same direction, but inveetiga- tion shows this to be the (lase: it is a fact otheiwise well known tQ thosc who have practical experience of propeller design. As this point is made abundantly clear in the investigation which follows, i t may be dismissed from immediate consideration. The

* The Aerofoil in the Light of Theory and Experiment ; Appendix IV., p. 219, f I I S t C .



THE SCREW PKOPEI.I.EK. 2%

questiou of the weight of the propullur also evidently tends to act in the same direction as the diameter limitation; apart from other limiting factors, it is evident that if, firstly, a propeller be debigned for highest efficiency, a reduction of diameter could be made without sensible loss of power, whilst a definite and measurable gain would by this means be effected by the reduction in weight.

A t first glance i t might be presumed that to meet the artificial restrictions such as those under discussion, all that is necessaxy is to calculate a highor value of y to correspond with the new (higher) pressure constant, and to plot, or select from the plottings already given, the appropriate efficiency curve, and on this to bgse the design of a propeller as before: a little consideration, however, shows that the matter is by no means so simple.

9 18. It is a fact which has immediate bearing and otherwise one of considerable interest that whereas in the case of the fixed propeller the investigation for least power involves the quantity representing energy lost in the rotational wake, in the w e of the propeller of greatest efficiency this quantity is entirely ignored. We know that in the latter case there is power expended in the rotational wake, and under the conditions of highest efficiency, in the regien of 45 degrees, there is an approximately equal par- tition of energy between the sternwad wake and the rotation; the question arises as to why and under what circumstances we are justified in adopting tlie method employed where tho conditions are varied. When tho author originally gave the result of his earlier investigation,* though fully alive to the fact that the condition of best efficiency constitutes a special case, and tliat the method was not generally applicable, he was not in a position to give a more complete sulution. I t was quite apparent that the blade, considered as an aerofoil, could not.glide a t anything less than its least gliding angle, and the conditions governing this had been fully defined by the author in the preceding portion of his work, in which the angle itself was shown to be independent of the velocity. Thus, in Figs. 8 to 13, for any particular value of 7, we have already seen that the curve gives the maxiinum posddt) efficiency for every point aloiig the blade, that is to say, for its each radial element. There is evidently no going beltiud the reasoning, once granted the hypothesis of radial elements of blade. iMoreover, there does not appear to be anything in the

* Aerial Flight Vol. I., oh. ix.



296 THE ISSTI’ICTION OF AUTOWOHILE ESGIXEEKS.

hypothe& itself which could render the results opan to suspicion. On tho other hand, in the extreme case of the value of ‘Y being vanishingly small, the angle 0 beconies 45 degrees, and it would appear superficially that something must be wrong, since, of the energy demonstrably lost, half only is expended in giving sternwad motion to the fluid and the other half is thrown away in giving useless rotary motion.

Tho truth is, that this result, although paradoxical from some points of view, is perfectly sound in fact. Thus, in the extreme m e , where 7 is vanishingly small, the propeller diameter for best e5ciency is virtually infinite. More generally, whatever the value of 7 may be, i.e., however small, the condition of least gliding re- quiras that the energy lost dynamically-in generating the Blip stream--shall be equal to that lost in direct resistance-in the augmented skin-friction of the blades; hence a low y value is only compatible with a low value of 5 and corresponds to a corrc- apondingly low pressure constant C and the area of propeller disk required to fill the conditions becomes greater in like ratio. Thus, however small y may be, even if we consider i t a vanishing quantity, we cannot ignore the direct loss of energy, for the propeller diameter is increased-in the limit to infinity-if the condition of bast efficiency is maintained, and the angle approximates more and more closely to 45 degrees. The extreme case only appears as a paradox so long as we imagine the finite diameter with an in- finitely small ”/, then the whole energy expenditure is distributed. between the direct (marward) wake and the rotational component; evidently, if this were the condition, a clear gain mould result if the angle Owere diminished so that for a given rearward streani a less loss in the rotational component would result; once the falsity of this conception is realised, all difficulty vanishes. The foregoing exposition, while clearing up any difficulty that

may be felt as to the results given by the late Mr. u’. Froude, and more fully i n the author’s “Aerodynamics,” serves also to explain the essential differenm in the conditions mhcn the broader genemlised problem is substituted for the particular cme of maximum efficiency; it is quite evident that if we are dealing with a propeller of restricted diameter so that the direct resistance losaes become comparatively small or negligible, the question of the rotational component loss becomes a dominant factor in the economics of propulsion; and further, it is definitely only in the special case of the condition of highest efficiency that considera-




tions relating to the rotational component cease to affeot the solution and may be ‘ignored.

5 19. To cut the matter short, in the expression for efficiency, tan 8 the quantity y is not a constant in relation to e a8 tan (0 + y)’

in the special casc, but is a variable, and before we can give a solution we require to find a form in which y may be expressed us Q function of 8. Beyond this, the relation of y to 8 is clearly connected in some manner with the particular nature of the restriction, whether it be diameter, pitch, or other limiting factor; this limitation, therefore, must first be defined.

Therc is a possible alternative to the above procedure, namely, by a more complete and generalised analysis of the problem it should be possible to obtain a more comprehensive equation directly applicable to all reasonable variations in the conditions. There are several objections to this alternative, apart from whioh the mathematid difficulties seem to be far from light; the chief of these is that it does not lend itself so readily to the incorpora- tion of experimental data, and circumstances may sometimes occur to Tender this necessary, or at least desirable. The former plan of operations in the present paper has therefore been adopted.

The first question arisiug is, in what manner is it desirable to initially specify the terms of restriction? The author has given preference to a form of expression involving the relative wakc velooity, thus the limiting condition is taken as defined by a constant K representing the rearward wake velocity in term of the velocity of flight: in the case of the marine propeller the velocity of the vessel. If the flight velocity be represented by the symbol v, and the rearward velocity imparted by the propeller by vz, we thus have K = vz/vl.

The vdue of K so defined does in fact determine the diametral reijtxiction, at least this is so if we assume the whole of th0 fluid passing through the effective disk area as operated upon; this is evidently the correct basis. The relation by which the constant K and the diameter are connected is to be sought in the Newtonian theory of propulsion in amordance with the teaching of Prof. Rankine and Mr. W. Froude, but it is also controlled by the stream contraction, allowance for which may be made on the lines 8%-

gested by the theorem of Dr. R. E. Froude to whioh referenoe baa already been made, i.e., that the velooity u at the point of



298 THE INSTlTUTION OP AUTOMOBILE XNGINEERS.

pamage through the propeller is equal to e , + :. ALterna-

tively, tho author's extension of the Froude investigation might

I s

I I I I I I I I

L \ "'I

II 23 * o b

be made the baeis of the allowance.* In the present paper this- has not been thought desirable.

Spring Meeting, Inst. Naval ArchitrotN, 1915, Paper Nu. 6.



THE SCREW PHOPE1,LER. 299

8 20. The mathematical work resolves ihel€ into two very simple steps, supplemented by certain graphic transformations; thus tke &st step is to find an expression for E in terms of 8 and w ; from thia expression we make a plotting in which ordinates repreeent valuea of 8 and abscissae r ) , graphs being given repreeenting K=conetant, Fig. 16; we may call these graphs &o-K b e s . Next, following the method of the author's previoue paper

0'07 Y o'(l8 009 0'10 011

FIG. 18.

(Appendix IV.), the constant relation between r ) and C is established, and Y values are calculated or plotted, Fig. 17 (as in Fig. 18 of the paper aforesaid), with C as abscissae, the soale chosen for C being preferably such as to correspond with the v d e of Fig. 16. From Figs. 16 and 17 corresponding value8 of 8 and r are read off, and these form the basis of a further plotting, Fig. 18, in which ordinates represent 8 and abscissae



Y values; the graphs once again represent ko-K lines traneformed now by the substitution of Y for rr aa abscissae.

The final step is the calculation of e ie ienq from the expiweion tan e - employing conjugate values of e and y for each tan (e f 7 '

particulax value of K as read from the appropriate bo-K line given in Fig. 18. In the plotting, Fig. 19, 0 values are given by abscissae, ordinates = efficiency. In Fig. 20 the efficiency is replotted against radiua in terms of

pitah, aa in Figs. 8 to 13; the corresponding 8 values are given as an irregular male.

1'0

0 9

0'8

0 7

6 0 6 2, it! 0 6

04 H

9'3

0 2

01

The two brief nzathematical step in the proceas as leading to

(1) Referring to Fig. 21, we have

the nesult given finally in Fig. 20 are as follows:- K in terms of e and r ) .

................ c.2 = COB (0 + ;) 2' (1)

(2) ..................... i.1 = V sin 0

and from the construction _ - ;, - rr in circular measure (approximately)




or GO8 + 1) ................ (4) h- = v sin e

(2) C/* relation. By previous investigation (" The Aerofoil, etc.," Appendix IV.),

(5) rn v c=-q- 7 ....................

but

or

In the graphic work, Figs. 16 to 20, the value of n (the aspect ratio) hits been taken ELS 6; this must be regarded as rendering the rasults 86 plotted lhore exactly applicable to the aeronautical propeller, since it is rare that in the marine propeller the figure n = 3 is exceeded. The values of R which form the basis of the plotting have been arbitrarily taken as 0.15, O . Z , 0'3, 0'5, and 1.0 in addition to the calculated value of highest e5ciency 0'07.

8 2l. It is of interest to note that in Fig. 20, p. 302, as we follow the curve of highest efficiency from its highest point outward from the axis, the value of K progressively increws; the reason of this is that y can never have less than its minimum value asaigned to it, whereas e diminishes the further we go from the axis. Thus we pick up in turn the curves of efficiency corresponding to the higher values of K , so that the curves denoting these values may be regarded as branches from the maximum curve, the points of junction being indicated clearly in the figure. It would, perhaps, be more accurate to consider the maximum curve to be the envelope of the others, that is to say, to consider the curvm of higher K value to be continued on the right hand of the points of contact, the efficiency again falling below the maximum; euch continua- tions would represent portions of the blades carrying a lower preasure oonstant thsrn that of least 7 , a state of things which clearly is useless, unless, perhaps, under conditions in which cavitation is threatand.

It is probable that, in the other direction, were any attempt to be made to trace the curves close to the axis (Fig. 20), the method would sooner or later break down owing to certain of the approxi- mations on which it depends ceasing to apply with sufticient exactitude; they could, however, without doubt be extended if required somewhat beyond their terminations as drawn.



The present treatment has been developed on its existing lines with a view to retaining aa much elasticity as possible, and to this end mathematical work has been almost entirely eliminated in favour of graphic methods; thus, in place of the theoretical curve of resistance gradient (Y values), Fig. 17, any actual curve plotted from experiment may be utilised without modifying the procedure i n the smallest degree. I t has already been explainod

FIU. 21.

that this was not done in the example as presented owing to the fact that the basis of existing experimental values is so unsatis- factory: the theoretical curve is less likely to prove a source of error than any set of wind channel figures mitli which the author is acquainted. Q 22. Our next step is that of employing the infornintinii which

is rendered available by plotting the curves given in Fig. 20 to the determination of the best pitch/dismeter relationship. We have firstly to determine the diameter appropriate to the vurious degree8 of restriction.





THE SCREW I’KOPEI.I,ER. 303

The problem of determining the correct propeller diameter cannot in practice be separated entirely from the question of the number of !hdes. Thus, it is evident that if a certain diameter be oarreat for a four-bladed propeller, and only two blades be employed, it will be w c e w r y to make some increase in the diameter to compensate for the deficiency. The question involved is two-fold: firstly, provision must be made for the requisite mass of air per second to pass, that is to say, the disk area must be sufficient, calculated on the Newtonian basis; secondly, the propeller as an implement must be capable of dealing with the whole of the ma88 so provided; alternatively, the calculation must be rectified to take acbount of the deficiency of the propeller in this latter respect. The problem is complicated by the fact we are tied down to whole numbers so far as the actual quantity of blades is conoerned, whereas the conditions may call for odd fraotions; thua the designer is frequently under the necessity of ohoosing between a number in exams of the mquir8ment.a of theory, so that there is interference and so unnemssary loss of power, or a number which fa& short of theory, in which case he has to take a rather larger diameter to make up for the deficiency. Such a choice may be of value inasmuch t~ the restrictive conditions are not constant in every application, sometimes the diameter being a limiting factor in design, at other times the efficiency being the most important question; hence, the solution which we seek must be one which will indicate to the designer concisely where his choice lay, and it must be our final object to present the result of the investigation in the clearest form possible.

Reference has already been made to the fact that certain conventions must be adopted in order to allow of the results being put in concrete form, such, for example, aa the propeller diameter in relation to the blade radius of maximum effioiency, and the limits assumed as defining the aotive portion of the blade. No apologias are needed for the somewhat arbitrary proportions here adopted; the designer is at liberty to use any others he thillgs fit and reasonable-the method will apply equally whatever the particular conventions chosen may be. The author takes twiw the diameter as defined by the points of maximum efficiency aa the diameter of the propeller disk in every case; on t.liis basis, taking the inner blade limit aa being the point at which its effi- cienq is equal to its outer limit, we find, simply from trial OIL the o u m plotted in Fig. 20, that the blade length may be fairly



304 I H B INSTITUTlON OF AUT0MOHII.R ENGLNEERS.

taketi as p of the diek radius. The reason8 for taking the conju-

1 I

T t I

F I 1,

? LJ L,

P

u

gate blade limits aa approximate points of equal effioienoy will be found in the author’s “Aerodynamics,” 5 212.



THE SCREW P I O P E L L E H . 305

The whole matter, put in brief, is summarised by the questiou: How much of the efficiency curve is it policy to utilise, and how much of the central region of the propeller disk are we prepared to sacrifice? For the purpose of the present paper the convention above defined is the author's reply.

Q 23. The diagrammatic representation constituting Fig. 22 is an initial attempt made by the author to bring on to one sheet all the facts of importance relating to the design of a propeller; we may consider the examination of this diagram as a preliminary step before going on to a more elaborate effort in the same direction in which certain additional obstacles have been surmounted.

I n Fig. 22 abscissae represent propeller radius and ordinates effective pitch; the figure being made syrnnietrical about tlie origin, tho axis of y is in fact the propeller axis, and different propellers are shown as projections of cylinders whose diameters and heights (appearing in the figure as rectangles) give the data of the propellers they represent. The outermost rectangle represents the mse of maximum e5ciency, that is, the caae corresponding to K = 0'07 (compare Fig. 20), and shows the pitchldiu- m t e r relatiou to be slightly less than 1.5; this is, however, tlie effective pitch; the mean blade pitch will be in ex- of this by an ainount represented by the slip factor. In Fig. 22 anaddenduin is shown by a graph beneath the axis of x whiah is prssumed as added to the effective pitch; aasuming this as correctly plotted (which it is not) the blade pitch in terms of diameter bemmes morn nearly 1%. Now the propeller of maximum efficiency is the propeller which theory would indicate a3 right where no restrictions exist as to pitch or diameter, or as due to the weight of the propeller, or otherwise, and the diameter appropriate to any particular load and velocity for this particular case inay be calculated from the usual form of expression thrust = area x Cp Yr the value of the constant C being readily obtained, as will be exemplified later. The calculation of the diameter for the propeller of maacimum efficiency gives a scale to the diagram, a d for any given diaametral restriction the appropriate effective pitch, i.e., the best pitch under the restricted conditions, m y be read 08. On looking a t the figure, we sec at once that the pitchldia. ratio should be considerably less in the c a e of propeller diameters being adopted less than that described as the optirnuin; this is, the author believes, a fact well known to those experienced in

LANCHESTEK. U



306 THE INSTITUTION OF AUTOMOBILE ENGINEERS.

propeller design, but not one that has previously been made the subject of rigid or quantitative treatment.

An attempt was made in Fig. 22 to deal with the question of blade number. The basis adopted was geometrical rather than dynamic, although the underlying facts are, of cour50, dynamic. Thus, taking the cylinder of diameter equal to the blade length (2 the propeller radius) aa the equivalent of the periptery, the propeller race waa considered as a rope made up of a number of cylindrical strands, each of which represented the periptery of one blade; the number of strands waa calculated, firstly, on the assumption of no overlap or interference, and secondly on a certain degree of geometrical interference which, from other considerations, was deemed permissible. The result of this is given in the bands of heavy vertical lines; four lines indicate that four blades are permissible, three mean three, etc., the overlapping of two bands means that either number is appropriate. In Fig. 22 a curve of efficiency is given; this being a curve of

maxima it is higher than anything which can be hoped for in reality; the values given by the graph represent the extreme maximum transferred from Fig. 20.

Fig. 22 was never looked upon as more than an initial trial, and it is clearly deficient in the matter of information as to blade number-the method is too crude to be allowed to paas. Beyond this, the basis of portions of the work has been revised, and a lnore accurate expression has been adopted for the diametral relationship; also the blade number has been dealt with on a purely dynamic basis.

Q 24. As a mode of representing the smew propeller and the pitch-diameter relation in graphic form, Fig. 22 leaves little to be desired; it is true that the duplication of the curve on opposite Bides of the axis is unnewsary, but the diagram so drawn does more olosely represent the actual propeller, and is more easily read than a graph of less “pictorial” chareoter. In Fig. 23, p. 310, which repreaenta the result of the complete investigation, the graphs on opposite sides of the axis are made to give Merent data; that on the left hand is the curve based on Newtonian theory without reference to the number of blades, whereas that on the right hand represents the full solution. We shall now proceed to consider the plotting of Fig. 23 in

detail, and in the first place we will go through the actual calculations point by point.



THE SCREW PHOPEI,LER. 307

Pitchldiameter ratio. This is determined a~ a funotion of K aa due to the variation in the angle of maximum dXcienoy, illustrated by the various curves and the graph drawn through the maxima in Fig. 20. If for any given value of K we wish to adopt the most advantsgeous proportion of pitch to diameter, we must 80 design Ehat the blade includea the best portion of the efficiency ourve for that particular K value and that portions of the curve showing low efficiency are discarded; it has already been stated that in this we shall take it as a convention that the propeller disk diameter is twice that of maximum efficiency. On this basis and referring to the figure we have,

X

0.07 0.15 0.2 0.3 0-5 1.0

e max. efticiency)

43O 3 3 O 27'-30' 22O-20' 16'-10' 1 lo--20'

*410 9 9 0

tnn 0 'L -

*466 -325 .260 -205 -145 -099

P X ; r = - D

1.46 1-02

-65 -45 -31

The pitch/diameter ratio as above X 2 is, in Fig. 23, the tangent of the angle of lines passing from the origin through the oorners of the rectangle representing the "cylinderyy by which the propeller is defined (see scale to right and left of figure). We have next to determine the relative diameters as related to K in terms of the optimum propeller diameter =unity. This will be calculated on the Newtonian basis, the mass of fluid passing through the propeller disk being computed on the assumption that its velocity is the mean between the initial and final, i.e.,

0% u = v l + ; - 2 where u is the velocity in question. Now,*

15 Thrust = mtv8 = npv2 ( v I +:) . . . . . . . . , . (7)

but v-3=K v 1

* The factor +8 ie due to the deduction of the central " blind " area, 4 of the disk diameter.

T I 2



308 THE INPTlTUTlON OF AU'lOh10HlLE ENGINEERS.

0.07 0.15 0.2 0-3 0.5 1.0

It it3 oonvenient to express the thrust in the form usual for prwure on surfaces and other cases following the 1'-sq. law, thus,

Thrust = Capv, ' .................. (9)

*0725 -161 -220 -345 -6%

1.500

(10) c= 15 - (K+ -) K 2 ..................

16 2 For the m e of the propeller of optimum diameter, where

R = 0'07, we have, 15 .0049

c-,:(*O7 + )=*068 ............ (11)

this value of C may be taken aa the key value to Fig, 23, the disk area for any given thrust being calculated from equation (S), and the diameter so obtained gives the scale by which the diagram is to be read. Taking the above as unity the diameters appropriat,e to other values of K are obtained as follows:-

G.

4 -068 *151 -206 -323 -585

1.406

*068 G

-. I Rel. Diameter

1 1.00 -45 -33 -2 1 ~ 1 1 6 -048

1.00 -6 7 -575 -466 -34 -22

, _ _ _ _ - The values obtained in the last column of the above table, in

conjunation with the €'ID ratios for corresponding K values of the tab10 preceding, determine the plotting of the graph on th0 left hand of Fig. 23; the next problem is that of blade numbers. Q 25. We will first express the peripterd area in terms of the

disk area; on our assumption &S to blade length (= Q disk diameter), and assuming as usual the peripteral area aa that of a aircle diameter = span (blade length), we have at once, after allowance of 1/16th disk area a the central '' blind " region,

efective disk w e n - 15 peripterctl urea i 6

- - x ( ; ) ' = H a .

Now, if 42 be the mean angle of the peripteral axis, that is tu say



THE SCREW PROPEI,I.BR. 309

the angle described by the blade centre in its spiral path, the mass of fluid engaged by the disk area, in terms of that engaged by the peripteral area of a single blade will be 68 sine2. This expression gives A' the number of bladea required for the angle in question in order to fully saturate the propeller stream, in other words the expression in question defines the point at which the number of blades is in theory the least number sufficient to obtain from the disk axea the pressure reaction computed on the Newtonian basis; on tho other hand, it also dehea the point beyond which interference ,must take pleoe'in the event of the a c t d blade number being in excess. The angle OZ requires to be first calculated from the el of maximum efficiency, since the centre of the blade is 1'25 further from the axis than the centre of the radius, the latter defining (by our convention) the value of el. Thus,

hr.

c_

0.0i 0.15 0.2 0.3 0.5 1.0

c_

0 maximum efflciency.

43O 33O y'70--;30' 22°--t10'

110-10' 1 6 O - . 11'

out 8,

1 0 5 1.54 1 -92 2'434 3.45 5.1 ( I

1 34. I *OY 2.24 3.04 4 .3 6.3

36' 43' 27" 30' 2 2 O 37' 1 8 O 1.1' 13'

9 O

4.0 3-08 2.58 4.10 1-50 1.04

The X values of the final column are plotted in Fig. 23 close on the right hand of the axis line, and a curve is drawn which may be said to denote the capacity of the propeller expressed in liladea, the m o in this regard relating to the values of diameter and pitchldidmeter ratio, given by the single graph on the left hand of the figure. On the right hand, in place of the single graph representing the

disk capaoity, we have a number of graphs each representing the capacity of a definite number of blades; these graphs are plotted in accordance with the blade number curve and other dynamioal considerations.* These separate graphs are drawn not only over the range to which the particular number of blades is appropriate, but also considerably beyond in the direction in which choice of

* When there is admittedly interferaw, there i~ uome un-aiuty as to the -nut d i n m e neceesasy to cornpensake. This in Fig. 23 is represented by 8

thicknsing of the graph.



3.1 0 THE 1NSTI'L'UTION OF AUTOMOH1J.B BNOLNEKHB.

number is permissible: evidently where diameter permits, it max be oonvenient to employ a two-bladed propeller instead of one of four blades, and not to utilise the whole of the propeller race. Q 26. A curve of efficiency is given in Fig. 23, so arranged as to

be read from the radial lines which correspond to different PID ratios; this curve is founded on an estimated average of the portion of the blade utilised, and is not the absolute maximum of the curve as given in Fig. 22, and thus it should more nearly represent the truth. It is found, generally qeaking, that these e5ciency ourves are higher than actual experimental figures, but the author iS by no means convinced that this is newsmily the case; there may be loeses not properly accounted in the theory as prasentad, but it may yet turn out that the real error does not amount to more than two or three per cenk.

By the aid of Fig. 23, the designer is able to select a propeller suited to any given conditions by inspection; he can, for example, when faced with a diameter limit, at once select the best possible PID ratio, he can ale0 see at once what falling off in e 5 c i e n g results from the limitation in question, and thus is in the immediate position to judge whether some modification is necessaq in a general design. He can see at a glance what additional diameter is necessitated by a reduction in the number of blades, and, in brief, knows beforehand the general possibilities of design whether the di5culty anticipated be diameter, revolution speed, efficiency, or number of blades. Q 27. There .me many questions both of a general and of a specific

character which remain, but which will form the subject of further discussion in Part I V . of the paper; the present section concludes the main investigation. Before leaving the subject, a comparison may be instituted between the results embodied in Fig. 23, and those of actual experience.

Firstly, comparison may be instituted with the ordinary practice of marine engineering. There is a rule which is found to apply with very fair exactitude in the case of the slow-going ocean steamer-the propeller disk area i s equal to one per cent of the area of wetted surface. The author has already given this rule in a report (unpublished) to the Advisory Committee for Aeronautics, and more recently in his James Forrest Lecture (Inst. C. E. 1914); it is not, so far 88 he is aware, a generally known rule, but it is nevertheless found to correspond with the average of current practice. It obviously ceases to apply, however, i f





THE SCREW PROIELLBR. 31 1

the speed be suflicient to bring in wave-making wistance aa a large pa& of the total, in such cases, in faat, it ie mtually found that the ratio is considerably higher-1’3 or 1.4 per cent being not uncommon.

Now the skin-friction coe5cient in the caae of a large vessel is somewhere in the region of 0*002 (single surf-), or C = 0.0012. On an axea 1 per cent of the total this becomes 0’12 as agahti C = 0‘068 ‘for our propeller of optimum diameter, Fig. 23. Thus, the propeller as ordinarily fitted is, according to our investigation,

of less than optimum diameter in the relation -0.75

approximately. This corresponds (Fig. 23) to a P/D ratio of 1.1 to 1’2 or therea,bouts; but this is eflective pitch, ant3 we must add, say, 10 per cent to this in order to bring our expression into line with the us& method of measurement for the blade pitch‘; we thus obtain a PID ratio of from about 1‘2 to 1’3, which agrees very well with the best practice. On reference to the chart also we find the indication as to blade number is quite in accordance with established practice. In brief, if we had no experience of m e w propulsion and had never seen a screw propeller, we should be able from theory done to arrive at something not materially different from that which has been achieved by the trial and error of some seventy or eighty years’ experience.

As a matter of fact, the aspect ratio on which Fig. 3 is founded is that appropriate to aeronautical design, n= 6; i f a lower value l i d been employed, say, n = 2’5 or 3, the constant C for the pro- pellor disk area of optimum efficiency would have been approxi- inately = 0.1, md the ordinary marine propeller would have proved to have been nearer the optimum condition than appears from the foregoing computation.

The next comparison will be with the author’s design of propeller given by way of an illustration in “Aerodynamics,” p. 326 (Fig. 139). This was laid out to represent the conditions of best efficiency, hence it is comparable with the optimum of the present investigaation. Velocitx 70 ft./sec. ; dia. 13 ft. = 130 sq., ft. wea, and employing the usual expression, and taking Q= 0’068 (w in Fig. 23)

PZ

-068 X -078 x 70 x 70 x 130 32-2 - 104 lb. Thrust (pounds) = .

against 100 lb. given in the original work. This example would only be of theoretical interest as showing



312 THE INST1T.UTLON OF AUTOMOBILE ENQINEERS.

that the results derived from the author’s older method and present method we in agreement, were it not for the fact that the older method has been for some years used by the staff of the Royal Aircraft Factory with conspicuous success. In brief, so f a r a4 the author has been a b b to test the

methods and results of the present investigation by com- pr ison with existing experience and data, it receives the most ample confirmation. There is, without doubt, a great deal of propeller inaterial published which the author has not yet had an opportunity of examining; also a still greater quantity in the hands of naval architects and constructors which has not been published; if the present paper has incidentally the effect of in- ducing propeller experts in shipbuilding circles to be a little more communioative than in the past it will have served a useful purpose additional to that for which it has been originally written.

PART Iv. CONCLUSIONS AND POINTS NOT OTIIERWISE DISCUSSED.

Q 28. I n the plottings given in Fig. 20 we have a number of curves of e5ciency, each belonging to a definite restriction defined by some particular value assigned to the constant K , these curvev constituting as it were branches from the curve of unrestricted maximum. The latter curve thus differs from the others inasmuch as its Il value is not constant, but is lower the higher the value of 8, that is to say the nearer the axis, and increases progressively as the distance from the axis is increased. It is for this reason that all the curves of K = constant tumble into the maximum curve sooner or later. This fact prevents us regarding the maximum curve aa definitely part of the series, but more than this, our description of this curve as the curve of Ii = 0’07 is only accurate at one particular point-the point of maximum, i.e., 8 = 43 degrees. Strictly epeaking, it would perhaps have been better to have termed the curve of maximum efficiency the curve of Ii = optimum. The difficulty which thus exists with regard to a variable K is due to the necessity for some real blade length; ,if the problem had been purely academic and the maximum of each curve alone had been under consideration the point would not arise. Thus it affects all curves of sufficiently low K value; in other words, such curves t ~ 8 fall into the maximum curve (the A’-optimum curve) before the ond of the blade is reached, and whose K value, therefore, is not actually constant over its whole length. Owing to the fact that




graphic treatment is adopted no real difficulty is encountered; it is, however, impossible to allow the point to pass without comment. In order to best appreciate the conditions, reference should be made to Fig. 16; here the vertical line r) = 0'07 (approximately) defines the curve K-optimum, and the iso-K lines running into this from the right hand of the figure are those proper to the other curves in Fig. 20. Apart from detail considerations it is quite in the usual order of things to find that between the unrestricted condition and the restricted condition the continuity is imperfect.

\ I FIG. 24.

When following through the series of diagram Figs. 16 to 19, i t ie helpful to think of Fig. 18 ria a projection of Fig. 16 wrapped on a curved surface of the form of the curve of Y values of Fig. 17; this is, in brief, the geometrioal meaning of the transformation employed.

29. The question of slip, or the difference between measured and effective pitch, is one on whit& it is important to be clear. On the one hand, efective pitch only has a definite value so long 88 the propeller is run at its appropriate thrust, that is to say at a thrust appropriate to the velocity of flight (the velocity of the



314 THE INBTlTUTlON OF AUIOMOHLLE ENOINEEHB.

vessel in the case of the marine propeller); on the other hand, the measured pitch of any given propeller is a purely arbitrary p a n - tity-it depends entirely upon the basis of meaeurement. Beyond this the measured pitch of a propeller may, and often does, vary from point to point along the length of the ‘blade; it is then impossible to speak of the (measured) pitch of the propeller at all, the term 088888 to have any meaning. When the propeller has no definite meamred pitch it becomes impossible to define the term slip as applied to the propeller as a whole, one can only speak of the &p of its different annular elements. Thus, if we must talk of slip, the first matter is to define the premises. The eflectise pitch is suffioiently defined by the maiq investigation-it ie the distance moved by the flying machine or vessel through the fluid when the propeller !is working under the conditions of its theoretid C value-the constant of the expression thrust = area C p Vi2. The measured pitch should in the opinion of the author be the pitch of the hypothetical chord. This, it is true, is not always a measurable quantity, but if the propeller has been designed by the author’s methods it is known to the designer. Investigation shows that on this basis the mean slip velocity v3 will be given by the expression,

n v 3 = v ,

sin e COB e + - ( 3 or a8 given graphically by the construction in Fig. 21.

Thus, the slip ratio will be equal to v&, and two typical examples of what may be termed a “slip curve ” are given in Fig. 24; the first of these is the w e of maximum e5ciency (K-optimum). It may be noted that the slip over the whole area is not hr from uniform; it is least at a point corresponding on the curve in Fig. 20 to that of highest e5ciency. When, however, we take the second example in which K =0*15, the slip curve is of a totally different type, the slip is far greater in and around the c e n t d part of the disk; so also in the other cases of K =constant. The only real interest this question of slip possesses from the author’s point of view is that when, as in a preceding section, comparison is instituted with established practice in marine propulsion, it is n e c m r y to be able to talk in the language of the naval architect. In the chapter on Propubion in the author’s “Aerodynamics ” the following opinion is expressed: “ The term slip, in its application to a screw propeller, is one that k d s to



THE SCREW PROPELLBH. 315.

confusion of thought; it is unscientific in ita present usage, and, would be better abolished.” This, as a concise statement of the author’s view, oannot be improved.

Q 30. Fig. M, of which conbiderable use has been made in the course of the investigation, perhaps requires a few words by way of explanation.

The velocity 2) is obtained by an ordinary parallelogram of forces, it being originally supposed that a stationary lamina, represented by the c w e of primary camber, caues a deflection of the stream at velocity V through an angle=w. It is next supposed that the whole system has superposed on it a velocity = V in the direction pf, and in the sen50 contrary to, that of the original stream; the resolution given then defines the motion of the fluid both as to direction and magnitude (u as given) on the basis that the lamina is in motion in a 0uid initially in repose.

8 31. Reference has been d e to a dynamical demonstration that the hypothetical chord is the correct measure of the blade pitch for any particular annular element. Referring to Fig. 21,

Let, as before, mt = mass per aecond acted upon. , I

Then,

and

but

or

9 , v = the velocity imparted to the fluid. and let w = total pressure reaction on blade element.

mp2 2

Rnergylt =

??3$ Resistance = __ 2V

w = mtv

Resistance = w sin 2 2

That is to say, the angle of the hypothetical chord is the measure of tho aerodynamic resistance coefficient.

This h~ some bearing on the slip ratio, Fig. 24. The solid lines represent fh0 slip ratio on the foregoing basis, w h e w the dotted lines repreaent the possible limit.

8 32. A question of some interest is that of the relationship which might be looked for between the screw propeller tind the “helicopter,’’ in short, is the helicopter to be considered a special case of the screw propeller, and can any continuity be traced between



316 THE IXSTITUTlON OF AUTOMOHILE ENOIN EBRS.

the generalisatioii expresscd by Fig. 23 and the extreme case where K=infinity! The author’s view is that there is no real continuity, at least, not without some revision of the line of treatment; the general theory is based deliiiitely on Hie question of efficiency,-in a sense that does not apply ai all to the case of the helicopter. Probably, also, the method of treatment breaks down before very high values of fZ are reachtul. On ilie other hand, the p h y s i d facts are such as would suggest continuity of some kind: the matter is left for future investigation.

Q 33. Certain other branches of the subject have been escluded from thc present paper. Thc limitation due to Cavitation, for example, as affecting design in the case of the marine propeller, hes not been dealt with; also the influence of the frictional wake straani as due to the body propelled, and as affecting the question of propulsion, ha9 also been excluded. I n respect of both these, the conditions are entirely ditferent in the two cases of the aeronautical and the inariiie propeller, and may be considered as legitimately separable from the theory of the screw propeller it9

such. Thus, in tlie flying iiiuchine the cavitation limitation does not exist; also contrary to the conditions of marine propulsion, the aeronautical propeller can rarely be so placed as to effectively capture the frictional wake. Both these matters have been dealt with to some extent in the author’s previous work.*

Q 34. One point which may be introduced as meriting discussion is the treatment of the central or “ blind ” portion of the propeller. The designer comnonly has the choice of supporting the blades either directly off a boss of large diameter, as in the well-known Gifford propeller, or of m r y i n g extensions of the blades inward to connect to i~ boss of ordinary size, the latter being the inore usual practice.

The relative merit of tlie one or tlie other plan, spurt from con- structional convenience, is solely a question of relative resistance. Thus for a well formed streamline boss the resistance in terms of normal plane (of the same projected area) is about 1/16th. For a well formed strut section, such as would bu suitable to form the arms of the propeller blade, the similar cocfficient is commonly about 0.15 or 0’2, never, i t would appear, less than 1,’8th; consed quently, given that arms are designed of sufficient strength, the

* “Aerial Flight” (Constable & Go., London), p f 12, S2 m d 216; also !$ 199 and 200.




matter reeolves itself into a question, roughly speaking, of whether the (spirally) projected area of the arms is greater or less than half of tho projected area of the (alternative) boss. Thus, in the case of a high speed-and, therefore, heavily loaded-marine propeller it may be distinctly advantageous to employ a large diameter boss, whereas in the aeronautical propeller any such feature may be deemed out of the question.

If , DS may sometimes occur, the boss forms naturally a part of the streamline section of the hull or fuselage, the circumstances are different, and the choice, in the sense discussed, is no longer in the balance.

8 35. In conclusion, a few words may be said on the subject of the foundation of the present investigation. When examining any piece of work critically, one of the first things which demanda scrutiny is the ground basis on which the main argument is founded, and it is well in every caae that this should be clearly stated. I n the present caae the said basis is singularly slight, a t least so far as the experimental side is concerned; the sum of this is contained in the assumed value of the coefficient of skin-friction; that is to say, the only experimental data utilised is that relating to the direct resistance of the blade as expressed by the augmented skin-frictional coefficient as it is known to experience. Along- side of this we have the assumption of a curve based on the theory of the compound nature of the total resistance as given in Fig. 17, and defining the relation of the pressure constant C and the resistance coefficient y ; as already stated, an experimentally aster+ tained curve may be substituted for this if preferred. In addition to the foregoing the author has drawn considerably on existing hydrodynamic theory, in the estimation of the type of the vortex. and its dynamic equivalent, aa embodied in the assumption by which the peripteral area is defined.

36. A few words may be said as to the deficiencies of present experimental data, to which reference has been made in the investigation constituting Part 111.

I t has not been recognised aq it should have been by those under- taking the work of experiment that no comparison is valid which is instituted between an aerofoil whose lines me conformable to its ficld of flow, and one whose attitude or camber does not conform, and which conseqiiontly is p o n e to give rise to flow of tlle type known as I t W L ~

shown by the author in 1907 in his Aerodynamics ” that there is a discontinuous ”; otherwise to set up eddies.



318 THE INSTITUTION OF AUTOMOBILE ENaINEERs.

definite relation between the aspect ratio n and the angle of attack! a and of trailB which define the camber, which corresponds to the condition of least resistance; these vdues were tabulated (Table V. p. 262) for different values of n with what turns out to have been considerable accuracy. The matter has been reinvestigated by the author in his recent paper with a similar result, and a simple approximate expression which in a measure defines this relation is deduced.* Now, in the experimental work which has hitherto been done and published, the aspect ratio is vsried without regard to the relation aforesaid, and consequently we have aerofoils which ‘ I fit ” their work and those which do not fit their work all tested, and the observation data tabulated and plotted, under the pretence that the resulta given properly represent the results of changing one condition at a time-namely, aspect ratio. Any set of observation data taken on the above basis is in itself useless and mis- leading, except it be considered as pure empiricism. If a number of seh of observations were recorded on the above lines and the resulk plotted a series of curves would result, and the envelope of these curves would be the curve we actually require.

Q 37. The quantitative theory of the periptery as laid down in the paper on “The Aerofoil,” enables a direct relation to be established between the span of any machine and the appropriate diameter to be assigned to its propeller. Thus the principles governing the expenditure of energy both as regards the propeller and the aerofoil being identical, it is at least plausibly obvious that the area of the propeller disk is related to the area of the periptery in the same relation as the thrust is to the load sustained. If , for example, the thrust required for my given machine is mlculataa as equal to 1/9th of the weight, the area of the propeller disk for the optimum condition will be lj9th of the oircle whose diameter is the span; in other words, the optimum propeller diameter wil l be one-third the span of the aarofoil. This relation may be demonstrated more rigidly as follows:-

Let W = weight of the machine, t’ = its velocity, the flight velocity proper to least resistance. 8 = aerofoa span, “monoplane” basis. u = velocity (downward) impressed on periptery, = nc , .

* The Aerofoil in the Light of Theory and Experiment ; Appendix XV.




mt = mass of content of periptery per second. P = density of air.

w = mtv

or

or if weight be inpounds, and assuming r ) = 0.07,

but from expression given with regard to Fig. 23, if T = thrust optimum propeller diameter

= P - 3 2 * 2 T - zr X 0.068~~~

or

The difference between the constants 0’07 and 0’068 is in part due to the blind central region of the propeller and in part due to other c&use~, as may be gauged from 24. In any case it mill be seen that the law holds good with su5cient exactitude.

It may be pointed out that where the aspect ratio of the main foil differs seriously from that of the propeller blade, the value of r) for tho propeller blade and foil proper to least resistance will not be identical, and there will be a corresponding difference in the propeller pressure constant, so that under thew conditions it may be desirable to apply a correction if the present simple rule is adopted; in any case the departure, will not be great. In praatiae the propeller is invariably less in aiameter than

the foregoing rule would indicate; the rewon for this is two-fold. Firstly, as already noted, the optimum value is never employed; the diameter is commonly from about two-thirds down to half that of the optimum, with a pitchldiameter ratio (corresponding to this degree of restriction) from about unity to 0.7 (compare Fig. 23). Secondly, the flight speed for least resistance is usuully taken somewhat below the customary or ordinary maximum flight speed, as this tends to greater stability and also provides for landing at low speed, an admitted necessity in the aajority of



cases; the design of the propeller, on the other hand, is usually based on the ordinary lnaximum speed, and so the propeller even for optimum condition will be of less diameter than if the two -amofoil and propeller-were designed for the same velocity. Appropriate allowance may be made for both these causes of departure.

Thus, in Fig. 25, (a) represenk the optimum diameter of propeller in relation to the span (monoplane), (b) a propeller having a diametral restriction two-thirds of optimum, the corresponding pitch/diarneter ratio being unity (pitch=dia.). In Fig. 25 (c), it is assumed that the aerofoil is designed for least resistance at a velocity 10 per cent less than that assumed in the design of the propeller.



I H E SCREW PRC)PEI.I,EK. 32 1

In Fig. 26,* a comparison is given of propeller diameter as between monoplane and biplane. In Fig. 26 (a), the case is given as in Fig. 25 (b). Fig. 26 (b) gives the saiiie propeller employed on a biplane of equal resistance; in Fig. 26 (c) the

E

FIG. 16.

biplane of equal lift is shown. to the original are given by the expression:-

The relation of these as to span

' 1 81 = ~

I r/a

"3 = q z B

In Figs. 25 and 26 the gliding angle is taken at 0'13; that ia to my Zifl/dhft ratio of 7-7. The monoplane ia taken as basis. In Fip. 26, in the case of the biplane, thie oorreaponds to about 0.16.

LANCEESTEB. s



322 THE INBTITUTLOK OF AUTOMOBILE ENGINEERS.

where q is the lifting efficiency of the two members of the biplane aerofoil in terms of their value as two separate foils.*

In Fig. 27, the case is taken of the well-known type B.E.2C. Here, taking the actual propeller diameter and comparing with the theoretical optimum, the pitchldiameter ratio is found to correspond closely with that actually adopted, and the propeller efficiency given by theory, 80 per cent, is not materially in excess of that reported as actually reached. The gliding angle for this machine has been msumed for the purpose of discussion as 0‘13, i.e., a 13 per cent gradient.

I n Fig. 28, a study has been made of a sp&al type of rapid climbing machine for airship attack as illustrating the application

I

. - ,I /- - I

!+ - 3 7 f d

\+’

FIQ. 27.

of the present method. Here, the normal resistance coe5cient has been taken =0*25, it being assumed that the propulsion is to be studied on the baais of continuous climbing at maximum rate.

The machine is taken as a ‘’ one-man ” type of 1,600 lb. gross weight, designed for climbing at flight velocity 60 m.p.h. = 88 ft.,’sec.; 400 lb. thrust.? I n Fig. 28 (a) it is shown as monoplane, and in 28 (b) as biplane, in each case with the appropriate propeller diameter on a basis of diametral restriction corresponding

* Compare “The Aerofoil,” Appendix VI., arttc. A h “ Jamea Forest hture,” Proc. Inst. Civil Engineers, 1914. Also Rep. Adv. Committee for Aeronantica, 1911.-12, p. 73. t With a gliding angle, say 0.011, the angle of ascent wil l be 0.14, repre-

aenting about 750 feet per minute at an expenditure of SO b.h.p.




to pitoh/diaameter ratio = 0’75 (as actually in use in the B.E.2C.). It wi l l seen that as a single propeller job $he diameter is prohibi-. tive; it beoomes quite possible, however, if twin propellers are

The method of the present section, i.e., treating the propulsion problem by a direct comparison of propeller diameter and span, is d y in no way different from that of the sepsrmte and independent calculation of the aerofoil and the propeller, It is, how-

employed.

ever, of use aa showing that there is a de6nite and well-founded relation, which must in some degree determine and dominate &e design of any and every machine, and which cannot be neglected or evaded. It is, perhaps, the simplest basis on which to specify or prescribe the general I ‘ lay-out” of any new type, and as such has been deemed worthy of detailed consideration.

# 38. I n spite of certain pretensions to completeness in the conclusions of the present work, as exemplified, for example, in the solution given in respect of the “helicopter,” and more fully in

S’L



324 THE INSTITUTION OQ AU'I'OMOHILR RNBINEERS.

the results embodied in diagrammatic form in Fig. 23, there are many points of detail which may turn out one way or another to be of material importance, and which have not even received men- tion. In applying, aa has been done, the theory of the aerofoil to the propeller blade, one must not lose sight of certain differ- en- in the conditions which need to be fully investigated either by theory or experiment before we are justified in considering ourselves sure of our ground. Thus, for example, there is the phenomenon of the centrifugal shedding of the '' dead-water I' which must take place in the case of the screw blade, and for which there is no parallel in the ordinary theory of sustentation or in the case of the aerofoil under normal conditions. Again, the symmetrical pressure variation along the two wings of an aerofoil, which gives rise to the two lateral surfaces of gyration or vortex sheets in ordinary rectilinear flight, can have no exact equivalent in the case of the propeller blade, where the pressure intensity, in order to accord with the regime, must on the whole incrrnsc from the axis outward.

The author believes that the present paper will be found to be of considerable service to those engaged on the design of propellers, especially where the conditions are foreign to the ordinary run of experience; in particular, the case of the aeronautical designer should be made more easy, where (as is usually the case) them is a conflict of interests to be served; in brief, where engine, propeller and ground clearance are at loggerheads. He hopes that tho promising character of the results obtained will in some modest degree act tw a spur to those engaged on aeronautical research, and give to the theoretical aspect of the subject a greater interest and weight m affecting the conduct of experimental work and the framing and control of the experimental programme.



T U R SCKEW PKOPELLElt.

THE DISCUSYlON.

Mr. L. BAIRSTOW, in opening the discussiou, said: As far as I know, aeroplane propeller design is at the present time wholly based on such experimental data as have been obtained, and up to now the theory of aerofoils as put forward by the author has not been used. I think that a good deal of the complication of the present paper is due to the fact that it is based rather on the theoretical treatment of the aerofoil than on the more practical experimental information. The blade element theory, which is

the essential part of the existing method of design, probably dabs back to the time of W. Froude as stated in the paper, bat as the construction of the fiying machine improves, this theory as it has been adopted in the drawing office has not been found to be quite sufficient for practical purposw. The greatest difficulty is not so much to get efficienoy, as to get a propeller which will give the necessary thrust for a given engine. One of the assumptions underlying the ordinary use of the iheory is that the aerofoil element moves in still air. I n his paper, the author has removed that restriction. I think it is quite easy t o see that it is an important restriction which he thus removes by giving different values to K as defined in equation (3), p. 300. The importance of axial velocity, which is Ule physical idea underlying the use of X, can most easily be seen in connection with



3-26 (Mr. L. Bairstow.) the “helicopter” (Fig. 29). Suppose the blade is moving in the direction BE so that the relative motion of air in its path is in V direction and makes a known angle of, say, 2 degrees with the aerofoil AB. If the air has not got any axial velocity, then the angle at which the air strikes the blade is this known angle, and from experimental results the lift and resistance of the aerofoil are known. Usually an aerofoil gives its best ratio of lift to drift, which may be as high as 20, when the angle ABE is between 2 degreas and 4 degrees. The corresponding inclination of the resultant force OR to the vertical is about three degrees, so that the farce which is giving torque is only Ij2Oth of that conhi- buting to the thrust. Imagine that, due to previous movement of the propeller through the air, axial velocity has been produced, and that the axial velocity u is only 1/20th of the circumferential velocity v ; the motion of air is appreciably altered, so that the relative wind instead of lying along BE lies along RD. To get the best performance out of the aerofoil, i t must now have its angle of inclination increased as indicated in BC, and when that has been done, the additional inclination of the resultant force bringing OR to 0’3‘ is very nearly three degrees. I n other words, the inclination of the resultant force to the vertical is doubled for an axial velocity of only l/2Oth of the rotational velocity, and corresponding with this change of inclination the resistance to motion is doubled. The author’s K is doubled, and that is the equivalent of doubling the angle of inclination of the resultant. I t will be seen that the existence of an axial velocity accounts to a large extent for the large powers n e c e s w in the “helicopter.” If propeller theory be taken directly froan experimental work, i t is quite easy to modify the ordinary methods of design so as to introduce an axial velocity into the calculations, and the trouble is that the amount of such axial velocity is not known. I t is not determined by the authorin his papepr, but the choice of K is left to the designer. A t the National, Physical Laboratory we are now arranging experiments for the purpose of determining experimentally the value of the axial velocity. A certain final velocity is given to the wake of the propeller stream, and the author states that the velocity through the propeller disk itself will lie between half that final velocity and full velocity. I am not quite sure that I can accept half as being the leaat value, but for the moment it may be accepted &B

probable, and I do not see any special means of determining it

THE INSTITLTlON OF AUTOMOBILE ENQINEEKP.



THE SCREW YKOPELLBR. 327

with reasonable muracy except by experiment. There are many other problem relating to propeller design which are under consideration experimentally at the National Physical Laboratory. The theory of aspect ratio, for instance, is not satisfactory. We do not know yet how, in detail, neighbouring elements affect the efficiency and performance of the propeller. In addition, there are probably some small effects due to centrifugal action on the air, and difEculties due to the position in which the propeller is placed on the aeroplane. It is scarcely ever free, and it is usually in front of an elongated body. Certain experiments at the Royal Aircraft Factory have suggested that the interference between the propeller and the body may be very considerable, and experiments to determine in greater detail any such effects axe also under consideration. In conclusion, I shoild like to express my regret that the author has made his theory so complicated that it is very difficult to follow, a complication which appears to be entirely due to his neglect of experimental data in preference to his theory of the aerofoil.

Mr. A. E. BERRIMAN: I am not in any way qualified to follow Mr. Bairstow in discussing, far lew in criticising, the paper that I think all will agree-whether the theories put forward are actccepted or not-certainly has been a masterly exposition. Nevertheless, I have some sympathy with Mr. Bairstow’s remarks on the subject of aerodynamic theory. The author’s standard work mas one of the first that I ever read seriously, but inasmuch as it is only now that he has himself discovered a vital connecting link in his chain of argument, which hitherto was miwing, the mere student may be forgiven if some chapters of the book were not so clearly impressed upon the mind as in future they will be.

Those of us who, in an elementary way, tried to define the cross section of the mam of air deflected by a wing in flight, in tenms of the span and the chord, r e d i d soon enough that there waa a misaing link somewhere, and were proportionately grateful to the National Physical Laboratory for providing the experimental data that havc alone been available to the aeroplane designer throughout an extremely critical and very historic period in the development of the science and art of aeroplane construdim.

If the author has now given us, in the two papers that he htw read before the Institution, the keystone of the arch in his reasoning, he unquestionably throws a new and very interesting light on the theoretical side of the subject, but he does not on that



328 THE INSIITUTION o y A U T ~ M O H I I . F : ENGISEEKS.

(Mr. A. E. Berriman.) account detract from the importance of experimental research, tLnd I would even venture to suggest that there are still some esplana- tory passages to be added before the theoretical and experimental sides of the subject are thoroughly welded together for the use of the practical man.

In particular, I refer to the virtual and real geometry of the aerofoil section. I n the records of experimental research, as published in the Government Blue Books, all we can find about the aerofoil section is the real geometry of its profile and the r e d angle of its chord to the datum line representing the relative wind.

In the author’s theory, i f I understand i t properly, the virtual geometry of the aerofoil section, both as regards profile and angle, differs appreciably from the real geometry as illustrated by the actual construction of the wing itself. He has explained, for example, how, starting from a certain theoretical camber, it may become necessary to curtail the entry edge and extend the trailing edge, two operations that may cause a very great change in the apparent angle of incidence. I can follow the general line of his synthesis, but I am not correspondingly clear how to set about analysing an existing wing section, and that, after all, is a very important point to-day, when the whole of the existing data consists of characteristio graphs for wing sections, about which nothing is known save their actual form. I t would be desirable, in short, to be able to take any of the results published in the Blue Books and work them out on a theoretical basis according to the author’s method.

I have ventured to criticise the absence of what I regard m a n important connecting link between existing experimental data and this new theory, but I would follow this by. congratulating the author on the dominant practical note of his paper in itself. Starting from a theoretical bmis of a highly complex character, he evolves a series of diagrams of an oinineiitly practical nature. Fig. 23, for example, is a chart that sonieone usually prepares about five years after the publication of theoretical work such a8 constitutes the major portion of the present text.

Mr. G. 8. BAKER: I t has been my business for a good many years to settle propeller dimensions for ships, and in more recent years to settle the form of the ship as well. I n dealing with the propeller for lifting weights, the author givee ccluations for the different soiirces of loss of energy-some in the direct resistance



‘L‘HB SCREW PROPELLER. 329

of the blades, and some in the loss due to the rotation of the wake, and in the latter he includes no loss as due to “‘ shook.” When rotation is imparted at the propeller, there is always liability to this loss, which may amount to as much as that due to the rotation of the water. By loss due to shock I mean that the energy required to set up motion in a particle is greater if the motion is imparted suddenly than if it were set up gradually. The question is treated very completely in one of Rankine’s early papers, where he deals with the particle set iu motion by shock as distinct from that set in motion under the supposition of R. E. Froude’s theory, where shock is entirely eliminated. But neglecting this loss in shock, the equation given for energy loss is, I think, only true for an annulus. The equation is derived from the energy required t o set the particles in motion in a circular orbit, sad' it o q y applies to those particles traversing that particular orbit; it does not apply to any particle all over the race oolutm unless the equation is integrated. He says this equation applies to four cores or ropes twisted so that together they form a cylindrical race, and in that I think he is entirely wrong, and the equations require modifying in ,order that they shall comply with a complete and solid coliunn. The actual loss due to rotation of a solid column of water can be obtained f r m Mr. R. E. Froude’s paper read in 1892, where he gives equations for loss of thrust. This loss is not very important for low angles of rotation, but when h 9 h pitches and high slips are employed tho loss is very considerable, much more than the loss shown by the difference between the two diagrams Figs. 3 and 6.

The author does not do that.

Mr. LANCHESTER: It is a stationary propeller. Mr. BAKER: I t applies to any propeller in which rotation is set

up in the race itself. Turning now to the question of the number of blades which

axe necessary to make a complete column in the race of B propeller, the author assumes what I should call an “effective area of influence ” for each blade of his propeller, and working on that basis and the quantity of air dealt &tli by the screw, he obtains the number of propeller blades required. I think, however, that the method given by Prof. Cotterell in his paper in 1879 is much better. He works from experimental results, and his method is free from some of the assumptions which the author has to make, when he adopts this theory of a circular sphere of influence around



330 (Nr. G . S. Baker.) the propeller blades. Prof. Cotterell takes the equation for normal pressure, and by equating that pressure to the momentum put into the race per second he obtains the thickness of a layer of uniform disturbance, which gives the normal distance apart of the blades of his propeller. He then works out the widths a t each radius of a propeller of a given pitch, and shows how the method may be adapted to any other pitch. I t is a practical way of doing it, and not one based upon supposition such as that m d e in tllis paper. Passing on to page 289, where the author first deals with the theory of the maximum efficiency of a propeller, special attention should be drawn to the fact that there is no attempt to work out a rigid theory for a solid screw race, but I gather that he takes the simple equations which can be derived from Mr. Froude’s theory, and by the use of certain ‘‘ conventions ” which may be practical or otherwise, he obtains certain results, and amongst these is Fig. 17, which gives the relation of the pressure coefficient C and y. This, together with Fig. 16, decides the assumed relation between slip, pitch and thrust, but these figures either represent mere assumptions, or are based upon experimental results.

Air. LANCHESTER: Fig. 17 can be arrived at experimentally or theoretically, but both curves are wanted. Fig. 18 is derived from Figs. 16 and 17.

hfr. BAKER: These figures and Froude’s ordinary theory for an annulus form the basis of the practical part of the paper. There are two conventions which the author makes in applying the theory to the screw race. With regard to the first, I would like to ask him what effect he has found in any marine propeller by taking a diameter twice that which gives the greatest efficiency-his first convention. So far as my experience goes, there is a very wide range of diameter which can be adopted in the ships of the class which the author has mentioned i n his paper, without any material influenoe on the efficiency of the propeller, provided that each diameter is associated with the proper pitch and blade area. With regard to the point raised in the last pages of the paper aa to the relation of t h e aspect ratio of th6 propeller to the front and rear angles of the blades, I made 80me experiments some years ago with a single screw ship i n which the four blades of the propeller were so made that they could be shifted round and turned into two blades, i.e., changed from an

THE INSTITUTION OF AUTOMOBILE ENCtINEEHS.




aspect ratio of about 6 to 1 with four blades to an aspect ratia of 3 to 1 with two blades.

Mr. LANCHESTER: Is that a propeller in which the blades are on one boss or are there two propellers?

Mr. BAKER: The blades are on the same boss, so arranged that the two at the rear could be taken off and turned round so that the leading edge of the after blades just fitted agaimt the trailiug edge of the forward blades, so 819 to make a continuous driving face for eaoh pair of blades. Thus, the diameter was not altered, but a different aspect ratio was given although the leading and the trailing angle of the blades were identical in the two propellers. No perceptible difference in eiciency was found between the two, but there was a difference of 2 per cent in the revolutions for the same thrust. Another experiment was made to test the effect of throwing the blade area out towards the tip of the blade instead of having a blade of ordinary shape.. The first propeller tried had blades which were pure ellipses in plan, the minor axis being at one-half the radius of the propeller tip. The second set tested had the larger portion of the area thnolwn out towards the tip. Both sets of propellers had the same blade thickness at any radius, irrespective of length of blade section at that radius. The angles which the author calls a and fi were therefore altered in tho inverse ratio of the alteration of blade width. The man aspect ratio of the whole blade was the same as before, although it wm virtually decreased at and near the blade tips, but this change only produced a 2 per cent reduction of efficiency. These experiments do not support S 36 of the paper.

Mr. LANCHESTER: The suggestion appears to be that any twisted piece of hoop iron is good enough.

Mr. H. GRINSTED: The paper, I am afraid, will not give a very great amount of information to the ordinary designer. For instance, Fig. 23 appears to be a most useful diagram, but the ordinary man will be sure to ask what is the basis of it, and I think there are several things that the author himself would hardly like to accept without further question. If, however, rJ1 the preliminary hypotheses could be properly established and experimental data embodied in it, that figure would be mod useful.

At present, when a designer has to produce a propeller he usually finds that the velocity of advance is given by the chan-



332 THE IBSTI1'FTlON OF AUTOMOHILLE ENGISEEHS. (Xr. H. Grinstad.) acteristios of the aeroplane, i.e., the resistance of the amoplane and the power available, and that the engine speed is fixed; 110 generally has to use a particular engine, and the diameter of the propeller is practically fired by the clearance, and although I think that the Figure would be of great interest in showing him what could be got, it is not of much use looking to a diagram to find the effective pitch to be used when everything is already dekr- mined. Then for the detail design the author's old theory must be used.

Mr. LANCIIESTPR: You mean to say that if the diameter is fired by the clearance and the pitch by the engine speed it is no 11x1

making any calculations at all. Mr. GRINSTEIJ: Then me have to return to the old method of

design, which must be used in any case. At the Royal Aircraft Factory propellers have been designed on the theory which ~ v a a elaborated by the author in his book, and the results have been most remarkable, although everyone admits that the theory, as a real basis of design, is very loose. There are all sorts of points to be considered, such tw the scale effect, the aspect ratio, the effect of which we know nothing about for the conditions unrlur which a propeller blade is working, and a certain amount of difference might be expected due to variation of plan form. By this method the width of the blade and the most suitable angles of attack and so on are determined for maximum efficiency by trial and error. By working out propellers on these lines, using tho National Physical Laboratory results for pressures and resistance of aerofoils corrected for scale effect as far as is possible at present, the results obtained have been remwkable. Very rarely have propellers had to be altered to accommodate them to the power available; they usually come out fairly right for tho power and also for e5ciency when the slips are not large, but when the slips are large there is a discrepancy in the efficiency. That may not be due to the propeller itaelf or to a fault in the theoqy, but possibly a3 Mr. Bairstow suggested, and as the author has also stated, it is due to mutual interference between the body and the propeller.

I t is possible to design a stationary propeller to absorb a given power by treating its blade as a series of aerofoil elementa working at angles of equal attack to the blade angles, that is, treating it as though the air were not accelerated at all before reaching the




propeller. The author waa the first to put into a very useful forni the method of treating the blade as a series of elemental &em- foils, and since then propeller design has been remarkably easy.

Mr. H. BARLING: There is one point I do not follow in the paper. The author said there were three losses: a loes due to the backward going wake, a loss due to the rotationd wake, and a loss which we might aall the drift loss.

Mr. LANCHESTER: Direct rasistance, not drift, because that includes aerodynamic resistance.

Mr. BARLING: My point is that the drift did include gliding, and that the whole loss is included in the drag loss. Again, Fig. 10 would suggest the use of a very emall diameter, because by wing a tiny diameber large average blade anglw csn be ub- tained, and on the other hand, upan the momentum theory, w0 ought to use a very big diameter.

Mr. LANCHESTER: The diagram only gives the dimneterlpitch ratio.

Mr. BARLING: But the angle of the helix in an actual propeller is usually found to be less than 45 degrees. It is usually abouk 12 to 20 degmes at the most important part of the blade, but the best angle according to this kind of theory is 45 degrees, minus y/2. That would lead, in an actual design based on these assumptions, to the adoption of a very small diameter, to avoid using the tip where the helix &gle is very mall.

Mr. LANCHESTEB: No. Mr. BARLING: Thus a better blade angle and better efficiency

would be obtained, whereas from the point of view of momentum a very big diameter should be used, and the great mass of air be allowed to leave it at a slow speed. That is a point that has often arisen. Calculated on the strip theory of the propeller, it will be found that by decreasing the diameter from 10 ft. to 9 ft. or 8 ft. the calculated eEcienay is increased, but looked at from the general momentum theory, a large maas of air must be dealt with and allowed to give the thrust by sending the air b d at slow speed, which means a large diameter. I think that it is a pity that there is not a more praot id connection between the primary camber diagram, Fig. 7, and our ordinary aerofoils. I should be grateful if the author would give us this relation to definite aepofoils, for RAF 6, for example.

31r. F. w. L m c i r m T E R . in replying on thc diucussion, said: 1



334 T H E lKSl'1TU'llON OP AUTOMOHlLE ENGINEERS. (Nr. I?. W. Lanche~ter.) propose in the first instance to deal with one of the points raised by Mr. Grinstd, because it relates to the question of the bnsC of the paper, and if I clear up any doubt on that point at the outset, it may assist my replies on other points, and amid any, possible misunderstanding.

Mr. Grinstad has asked me to specify more fully tlie data forming the basis of Fig. 23, and also whether I should be prepared myself to accept all the assumptions made in this paper. I thought that in the paper I had made the basis of the Fig. 23 q u i b clear, but it is a long paper, and perhaps a brief rericm of the basis may not be out of place.

The foundation assumptions are as follows :--FirstlF, the peripteral area of the blade is definitely that deduced and adopted in my paper on "The Aerofoil,"* i.e., the peripteral area is representecl by a circle whose diameter is equal to the eff'mtive length of blade, this corresponding to the span in tlie casc of the aerofoil. It is not pretended that the area so defined applies with precision to every design of aerofoil, but i t does apply very closely to all ordinary and useful designs of foil or blade.

Secondly, the effective blade length is taken as three-quarters of the radius of the propeller. This agrees or accords with ordinary practice, but the method is applicable i f any other ratio be selected or adopted. I have taken three-quartcrs, another designer may select any other ratio: there is no compulsion. For the purpose of illustration an absurd proportion may be chosen, say, a propeller with arms 9 feet long, or a boss 18 feet diameter, carrying a one foot blade; it is true that such a blade may be designed to give practically the maximum possible e5ciency over its whole length (compare Figs. 8 to 13), but the economy or saving thereby effected will be lost many times over owing to the excessive boss skin-friction or arm resistance; also the propellel: will need to be of unduly large diameter owing to the small proportion of area usefully employed. A n assumption of some kind has usually to be m d e in giving the solution of any practical engineering problem: so long as the method is flexible, it is, of course, absurd to take exception to the particular quantitative msumption made.

Thirdly, the assumption of an aspect raLio about 6/1 and tlie curve given in Fig. 17 forming the basis of the graphic trans-

* Seep. 171, nwte.



formation. Here the designer may assume any data he thinks fit; he mill select an aspect ratio appropriate to the conditions of the case-for aeronautical work a much higher ratio may be employed than for marine work. Having chosen his aspect ratio, he may either calculate his curve as I have done in the example given (and as at present I prefer to do), or he may take the result of laboratory experiment conducted with an aerofoil of the particular aspect ratio and in other respecta equivalent to the blade form which he intends to adopt. The propriety of the present assumption is on the same footing as that preceding; for the purpose of illustration something had to be assumed, and I took fi, wres corresponding to ordinary practice in aeronautical design for the aspect ratio, and a curve which is sufficiently typical for the purpose of quantitative illustration; I in no way tie other de- signers to my own particular selection ; as already stated, the method is quite flexible.

Fourthly, the assumption of R. E. Froude’s law. This, perhaps, is questionable. It is necessary to assume some law as representing the degree of contraction of the slip-stream at the point where it passes the propeller ‘‘ disk,” otherwise we have no means of computing the mass of fluid dealt with; my view here is that Froude’s law will do as a start, later on it may become possible to extend this to include the coefficient Q, which I have introduced to take account of losses not contemplated in the regime studied by Mr. R. E. Froude. On this point, reference should be made to the Transactions of the Institution of Naval Architects, Mr. Froude’s papers of 1889 and 1911, and my own paper read at the spring meeting in 1915.

Having made the position as to the basis of the method as dear a5 seem possible, I will proceed to deal with the discussion in the order of the speakers.

In reply to Mr. Bairstow. I am not in a position to say definitely whether Mr. Bairstow is correct when he states that as far as he knows aeroplane propeller design is entirely based on experimental data, but I suspect that he is appreciably at fault.

The difficulty is that, m demonstrated by my James Forrest lecture of last year, my own theoretical results given in “Aero- dynamics ” in 1907 have been so accurately confirmed by modern experiments, and in particular by the work done at the Nationd PhyRirsl Laboratory, that without entering into the designer’s



336 ’L’H E INSTITUTION OF AUTt~IIWBLLE ESUINEEHS. (-Ur. F. CV. Lanchester.) confidence it is impossible to decide on what basis he has worked or whose b t a he has employed. In defence of theory I can only remark that I was there first.*

Owing to the variation in the velocity through the air of various “elements” of the propeller blade at different radii, it is only possible to apply the results of aerofoil experiment to blade design by the aid of a liberal amount of interpretation; in other words, whatever the basis data taken, theory must play the predominant part in the actual process of design. As demonstrated in “ Aemf dynamics,” the condition of least resistance defined by P / W = constant must apply approximately to each annular element.

Now this asymetrical distribution of the pressure system cannot fail to have some modifying influence on the behaviour and properties of the blade considered as an aarofoil, and thus whatever use be made of experimental determinations made with aerofoils in rectilinear flight, the application is by no means as simple as might be imagined; however, this point can be left for discussion in conuection with some of Mr. Bairstow’s later remarks.

Mr. Bairstow says that, “The greatest difticulty is not so much to get efficiency as to get a propeller which will give the neceasary thrust for a given engine.” Frankly, I find it difEcult to understand what Mr. Bairstow’s point hem is; = a phrase the sentence quoted looks aJl right, but when it is examined with a view to reply the simplicity vanishes. I t may be presumed, firstly, that the given engine is a given engine in the sense that it is intended to run at m e given speed arid develop some given h.p.-it is not merely a given assemblage of components. Secondly, I suppose it is contemplated that there is a definite propulsion problem, i.e., the normal velocity of flight is also part of the initial data. Now Mr. Bairstow says the diffioulty is to obtain the necessary thrust rather than to get the best efficiency. But a given engine speed and flight velocity means an equally definite effective pitch, since the effective pitch is the velocity divided by the number of revolutions per second. Hence the effecfive pitch is part of the initial data; there seems no escape from this. Now, if we tag, a number of propellers of the same effective pitch but of different efficiency, it is clear that the propeller with the highest efficiency will give

Whea all ha8 been said, in the original exposition of the aerofoil theory, a certain amount of experimental knowledge sud mnterial w w employed. aerodynamic^, Ch. IX )




the greatest thrust: thus, if the thrust of a propeller of the pit& in question as calculated from the brake h.p. with no allowance for propeller loss be 100 lb., that of a propeller of efficiency = 0% will bc 60 lb., and one of efficiency = 0’7 will be 70 lb., and one of efficiency = 0.76 will be 75 lb., and so on. Thus, the problem of obtaining the necessary thrust with a given engine is tanta- mount to that of obtaining the maximum e5ciency. Perhaps I have mistaken Mr. Bairstow’s intention; possibly he has some special problem in his mind in which the revolution speed and h.p. of the engine are taken as variable, or the velocity, perhaps, is a matter in which a certain amount of selection is permissible. If this is so, there is nothing in the statement as put to indicate the intention.

I am glad to find Mr. Bairstow so appreciative of the feature introduced into the aerofoil theory of the screw propeller in the present paper, namely, the taking into account of the general motion of the fluid as apart from that more immediately due to the action of the blade. Mr. Bairstow is quite right in pointing out that it is in the case of the stationary screw or “ helioopter ” that the di5culty of the old line of treatment is most acute; I think that his diagram and clear exposition of the importance of the point will be found of assistance to those who wish to study this question closely. 1 am not quite sure,‘however, that Mr, Bairstow is not attributing to my factor K virtues which really belong to my me5cient Q (comp. p. 272). It may be pointad‘ out that the factor or constant K represents the slip stream velocity in terms of the flight velocity (p. 297), and thus only defines on? of the factors by which the velocity through the propeller “disk” is determined, the other factor is the coefficient Q. If, as in the latter portion of the paper, we aaaume R. E. Froude’s law, we have &=unity; in this case we know that u, the veb- oity at the propeller, is equal to the velocity of flight plus half the slip stream velocity, and more generally

t’ t4=.u1+ q

At some future time, when our general knowledge has been extended, we shall doubtless be in a position to sssign a value to Q with some assuranca of being near the truth; in the meaatime I have thought it best to assume the result of Mr. Froude’s analysis without qualification, that is to say, except in t h ~ w e of the helicopter (where both maximum and minimum valuw of Q we

LAXCHRSTEH. Y



338 THE INSTITUTION OF AUTOMORILE ENOINEERS. (Mr. F. W. Lsnchester.) employed), Q has been taken as unihy. I note that Mr. Bair- stow expresses doubt as to the Froude condition representing the least possible value of u; in other words, he apprehends a possibility of Q being less than unity. If this is his intention, I feel sure that it would interest a great many who have followed Mr. R. E. Froude’s work (as expounded in his 1889 and 1911 papers), to hear more from Mr. Bairstow on this subject. In the discussion on the later of the two papers cited, the Hm. C . A. Parsons expresses the diametrically opposite opinion. *

Mr. Bairstow mentions the possibility that the neighbouring elements may not be without interaction, as is tacitly assumed in the treatment ; he also mentions the possible influence of centrifugal force. I am fully alive to both pomibilities; as to the first, i t is admitted that the method of treatment of the whole blade as the sum of its annular elements is somewhat in the nature of a convention. If I remember aright, some of the earlier ,efforts in this direction did not include any allowance for the fact that the annular elements are part of the blade as a whole; in other words, the original treatment w&s based on the blade being .considered as a number of separate (square) planes. I think there were actually experiments conducted with individual small planes at different radii and angles and the whole blade assumed as the sum. In my own treatment, as given in “Aerodynamics,” although the annular elements are separately dealt with from the point of view of their relative efficiency, the gliding angle on which this efficiency must actually depend is taken as that of the blade considered as a whole; in other words, in dividing the blade up into (imaginary) annular elements these elements are supposed to retain unchanged the properties belonging to the aspect ratio of the blade of which they form a part. I do not think that sufficient stress has generally been laid on the importance of this feature in the method of treatment. It is still possible that there are other and more subtle effects of the interaction of the different parts of the blade than the broad effect already taken into account aa above. There is room for considerable error to creep in between the determination of data experimentally with a model aerofoil and the application of the said data in the construction of a pro-

* Compare “A Contribution to the Theory of propulsion and the Screw Propeller,” F. W. Lanoheater, Proc. Inst. Naval Architech ; Spring Meetings, 1915.




peller. Thus, in the aerofoil the pressure difference is greatest in the central region, and the regular and symmetrical generation of vortices is closely connected with this fa&. In the propeller blade, on the other hand, the pressure difference (under the conditions satisfying the equation of least resistance) increases progressively along the blade from the root outward nearly to the tip, and so we cannot feel by any means assured that the conditions are so nearly parallel m it is now customary to assume. The quantitative resulk of aerofoil experiment are not with any certainty applicable without considerable “ interpretation ” to the propeller blade. On the question of centrifugal force, I believe the main effect is related to the fate of that which is by custom termed the dead-water. It is evident that when, as in the screw propeller, the aerofoil is represented by a blade in rotary motion no fluid which is carried with it can remain for long, owing b the action of centrifugal force; thus, anything in the nature of dead-water will be shed almost as fast as it is formed. This fact is mentioned in my “Aerodynamics” (5 215), where it is suggested that since this means the continual regeneration of dead-water, it will probably result in a greatly increased resistance in cases where the type of blade is one definitely involving motion of the discontinuous type. Since writing the passage in question, I have slightly modified my view; it seems probable that in cases where the blade is any approach to a streamline section, the rapid shedding of the dead-water will tend to continuously restore the Eubrian form of flow, and so it is to be anticipated that a form of section which may just fall short of being satisfactory as an aerofoil or strut section may be perfectly satisfactory as a propeller blade. Further, it is conceivable that the difference between what will work as an aerofoil section and what mill work as a propeller blade section may be far greater than eng- gested, in fact under the conditions of the propeller blade it is possible that motion of the f r d l y discontinuous type may never in practice be fully developed.

I note that in conclusion Mr. Bairstow e x p r e m regret that my theory is so complicated and difficult to follow. I would like to point out that the result as represented and summarised in Fig. 23 is one whioh has never been previously aohieved. That is to say, the theory of the screw propeller under restridad acm- ditions has never before been generalised in a form available to the engineer or designer; I believe, in fact, it has never been

Y 2



340 (Mr. F. W. Lanoheater.) generalis& before in any shape or form. I do not pretend that the present work is complete, inasmuch as the restricted condition is only one of several possible kinds of restriction. I n t h e future it may be found possible to simplify the method of procedure, but at the same time I anticipate that, with the addition of methoda of solution for other forms of restriction, the treatment will on the whole tend to become more complex before many years have elapsed. 1 think that when Mr. Bairstow attributes any complcaxitp of the treatinelit to the employment of theoretical instead of experimental data, he is entirely wrong. I cannot conceive that the subbtitution of experimental data for the theoretical data, employed could possibly result in the slightest simplification. If, for example, an experimental curve be substituted for the theoretical curve in Fig. 17, no simplification whatever results, the procedure is exactly as before. If, more broadly, any attempt be made to assess the peripteral area experimentally, I fear that the complication that must inevitably accrue would render the investigation indefinitely complex or even impossible, at least so far %s a summarised conclusion is concerned. Tersely put, the subject itself is one of great complexity, and it is not possible to expound a complicated subject by a simple treatment except by a sacrifice, in some measure, of the truth.

I fully agree with Mr. Berriman that there are still some ex- planatory passages to be added "before the theoretical and experimental sides of the subject are thoroughly welded together for the use of the practical man."

I can quite sympathise with Mr. Berriman's difficulty as to the virtual and real geometry of the aerofoil section. The basis of the diffirulty is, to put it in a nutshell, that although in designing an aerofoil i t is quite easy to start with the virtual geometry, and assign in place of this an actual form, it is by no means easy to take an existing form; that is to say, it is difficult to start with the r e d geometry and work backwards. The inference to be drawn from this is that if theory and experiment are in the future to be welded together into one homogeneous whole, the aerofoil models upon which experiments are conducted should be designed on the basis of theory. If this is done, and I hope it will be done, i t will very soon become apparent to what extent the theory is to be relied upon.

The dificulty of working backwards, that is to say, from the real mrofoil to its theoretical basis, is in a sense comparable to the

THE INS'TITIJTION OF AUTOMOHILE ENQINEEKS.



T H E SCREW PROPELLER. 34L

difficulty of an integration of a inathematical expression; it is, generally speaking, only possible as the recognised inverse of an established differentiation. A better illustration, perhaps, is that of the ichthyoid or balloon form; it is not difficult, as shown by Prandtl, to build up a form from a given “source and sink” distribution so that the theoretical stream lines are calculable ; it is, however, impossible to reverse this process and deduce the correct “source and sink” system from a given form.

I t is to be recognised that any final application of soientifio knowledge to the practical problems of engineering is largely in the realm of art, in which personal temperament and experience will always play a considerable r81e; in all probability the experienced or gifted designer will learn with a considerable degree of assurance to correctly give in his designs the material equivalent of what Mr. Berriman has terined the virtual geometry of the aerofoil, that is to say, he will learn to produce an actual aerofoil which will correctly reproduce the calculated performance as based on the virtual geometry.

I n reply to Mr. Baker, I will say at once that I am very pleased to find myself taken to task for not including losses due to so-called “shock”; I have sometimes heard this strange and imaginary form of loss spoken of, but I have never previously been a b b to ascertain the origin of the expression; I have hitherto regarded the term as a thinly veiled disguise for an admitted ignorance of the true nature of an otherwise unaccounted loss. Mr. Baker states that the origin of the idea of a loss due to I ‘ shock ” i s to be sought in the work of no less an authority than Rankine himself.

I assume that the paper referred to by Mr. Baker is that numbered XXXIII. in the published volume of the “ Miscellaneous Scientific Papers.” Now in this paper, which is really very elementary, the word “shock” is not employed, but, assuming Ran- kine’s statement of the case as accurate, it may be admitted a legitimate to paraphrase his description of a loss due to “morlr wasted through suddenness of action of propeller on the water,” and further, “work wasted if transverse motion is produced suddenly” as included by the one word “ shock.” I would like to suggest, however, that in this particular matter the work of the late Prof. Rankine is a trifle out of date, and his views on the point at issue would be expressed in an entirely different manner in the l ight oi modern knowledge.



342 THE INSTlTUTlON OF AUTOMOBILE ENGINEERS. (Mr. F. W. Lanchester.) In the first place, when any blade or aerofoil acts on the fluid

there is (unless cavitation is taking place) nothing either in the nature of “shock” (in the words of Mr. Baker) or sudden action as supposed or suggested by Rankine, so that if Rankine’s suggestion is to be taken seriously, we must find some construction in his writing other than that implied literally by the words he1 employs above cited. To begin with, it is to be remarked that although water is always mentioned in the paper in question, the wholo basis of the Rankine method depends in its essence upon the fluid being something of the character of the medium of Newton, the property of continuity of any actual fluid is completely

---.- FIQ. 30.

ignored; the whole argument and mathematical context is rigidly accurate i f applied to a discontinuous medium consisting of an initially uniform distribution of particles, it is inexact and of the nature of an approximation only when any real fluid is oon- cerned; it must be regarded likewise as inexact if applied to any theoretical fluid in which the property of continuity is part of the hypothesis.

When, as in the case of the Newtonian medium, there is na continuity, wo can easily form a regime in which there is loss of energy aa due to shock. Thus, if the particles are acted upon by a.n instrument of some kind which involves impact, such BS

an inclined plane, Fig. 30a, then if the instrument and the par--




tides are absolutely inelastic there must be a loss of energy equal to the kinetic energy imparted to the medium; this is just as stated by Rankine where he says: “ . . . so that the lost work, instead of being simply equal to the actual energy of the water discharged per second, is increased to double that quantity OP energy; and thus, etc., etc. . . .,’ Unfortunately Rankine everywhere uses the word water; Newton was more careful, he h t defined his medium and everywhere is careful to qualify his re- m k s as referring thereto. If, still discussing the Newtonian medium, we imagine the implement not to act by impact, as when we substitute a foil of curved section for the inclined plane, Fig. 30b, or alternatively, if we suppose the plane and particle@

FIa. 31.

perfectly elastic, Fig. 300, then there is no loss or expenditure of energy other than that represented by the kinetic energy of the particles set in motion: in the latter example, the “angle of attack ” of the plane of course only requires to be half that necessary where by hypothesis elasticity is excluded for a given sus-, taining reaction, or, in the case of the propeller, thrust reaction. In the case of a real fluid, or in the perfect fluid of mathematical

theory, the conditions are totally different; to start with, there is definitely no such thing as impact, If we are considering the case of a plane in its action on a fluid, it is true that if we ignoxe skin-friction or assume it not to exist,* we must consider the re-

* Without otherwipe modifying the syetem of flow aa proper to a real fluid.



344 THE INSTITU1 I O N OF AUTOMOHILE ENGINEERS. (15. F. W. Lanohestar.) sistance as acting normally to its s u r f m and therefore to ba related to the thrust or supporting reaction by a simple resolution of forces, and as such it will be just double as great aa would be the case for the same angle of trail if we assume a regime OP the character depicted in Fig. 31s. However, this latter regime is purely imaginary; the actual form of flow, either as deduced by theory or as known to experiment is more of the character given diagrammatically in Fig. 31b, and the appropriate section of foil is arched with dipping front edge as indicated; i f this be the regime taken for comparison, then the plane may be considered as at a still greater disadvantage; in other words, we might be led to infer a greater relative loss than the maximum given by Rankine as proper to the Newtonian medium.

As a matter of fact, all this kind of argument is a delusion,; we have no more business to assume a plane as an organ of sustentation, or a true helis as a propeller blade, than we have to take any other equally unsuitable form, or to a u m e a coffin aa an example of a form on which to found a theory of ship’s re- sistanw. Actually, if a plane or true helix be taken, there is lost energy aa due to the discontinuous system of flow set up, there is a dead-water region on the back of the plane or blade, which is the seat of eddy motion, but even here there is nothing which can be described as “shock.” When we deal with more rational forms of aerofoil or blade, the word “shock” becomes if possible still more unsuitable, for, m I have many times demonstrated, the resistance may be correctly computed on the basis .of the actual work done on the fluid plus a skin-frictional allowance ; the latter is based on the ordinary coefficient of skin2 friction plus an addendum comparable to that found to be required in the case of ichthyoid forms and spar sections. This addendum xesistance is mainly due to the failure of the pressure region in the rear of the section or body owing to a degeneration of the streamline system; this degeneration it is generally agreed is ansequent on skin-friction aa an indirect effect. It is never suggested that in them other cams of fluid resistance there is any portion of the total to be accounted for by “shock,” and considering the proved nature of the phenomenon the word seems impossible of application in the sense employed by Mr. Baker.

It is evident moreover, in view of the fact that these resistances are fully taken into account in my work, and are represented in their totality in the resistance coefficient r , that Mr. Baker, in




quoting the rather inexact work of Rankine of half a century ago, cherishes a belief that there is some special kind of resistance in the case of a propeller blade not associated with other of the well known cases of fluid resistance.

I think that it is clear that I am not misinterpreting Rankine in any way, since he says himself in the paper in question that ho regards Woodcroft’s “ gaining pitch ” screw and Mangin’s screw as having for their object to diminish waste of the kind under discussion. Evidently, these are efforts intended to provide a blade more conformable to tlie character of the streamlines than the simple helix.

In any actual fluid, there are many additional complications which invalidate the accuracy of any estimates of loss based upon Rankine’s suggestion. In the casc of an inclined plane, for example, the skin-friction in the dead-water region may be wholly or partly relieved owing to the return current of the eddy system; this was first pointed out in niy “Aerodynamics ” in 1907. From more recent experiments it would appear that the same may be true in the case of aerofoils whose attitude is not that of (3011-

formability to the system of flow. Thus, some of the ensvgy wasted according to Rankine may be in effect restored, or at least may reappear as a credit against an 0therwi.w necessary loss of a totally different kind.

A point of some interest in connection with Rankine’s investigation cited is that it in a ineasure anticipates Mr. 1%. E. Froude’s more recent result as given in his paper of 1889. It is virtually the same analysis by which Rankine has shown that the sudden acceleration of the particle (it must be supposed by an inelastic agent) results in tho losr of energy equal to that spent dynamically, which has becn since utilisecl Ly Yroude to Show that if no energy be lost the force inust act on fluid having the mean velocity of the slip stream; resulb which may be interpreted as being identical.

Proceeding to Mr. Baker’s further remarks, 1 cannot quite understand hi? illcaning when hc says that the results as given by m y equation only apply to an annulus, and that I have neglected to integrate thc equation. I think Mr. Baker’s difficulty must be that he does not appreciate that it is the saiiic thing to plot a n equation as a graph and to integrate the graph m to integrate the equation itself. As I apprehend tho position, it is scarccly pos-



346 THE INSTITUTION OF AUTOMOBlLE ENGINEERS. (Nr. F. W. Lanchester.) sible to integrate the equation as an equation with any useful result; take the case, for example, of the condition of maximum efficiency, the equation applies for each value of e to some specific annulus, and when plotted for a number of values of e we have o graph representing the efficiency at different distances from the axis, that is to say, for an indefinite number of annular increments: such are the graphs given in Figs. 8 to 13 inclusive a d also in F i g . 20. Now if the work done on each equal radial increment were the same, it would be merely necessary to integrate the curve in question in the ordinary way, either mathematically or graphically, but this is by no means the case.

The work done by the different annular increments is defined by the “ thrust grading curve,” which is a matter to a great extent under the control of the dwigner; it is a curve which should in some degree approximate to a triangle, the ordinate being everywhere proportional to the area and therefore the circumference of the annular increment; but a further condition is that the curve tails off to zero at the inner and outer limits of the effective portion of the blade. This is all as laid down in my “Aerodynamics,” Ch. IX.,* the method being actually employed at the Royal Air- craft Factory. If the thruet grading curve were known, we could then calculate the theoretical efficiency by integrating the product of the ordinates of the thrust grading and efficiency curves, and any designer when he has provisionally settled the proportions and thrust grading curve of his design may easily do this; the procedure, of course, will be to plot the product of the two sets of ordinates as a graph and then use the planimeter. To have introduced this and other equally important kindred matters into the paper, however, would have been to have obscured the main issue in a maze of detail. For the immediate purpose of the paper, the maximum value was taken in the preparation of Fig. 22, it being plausibly assumed that the integration would, generally speaking, bear some fairly constant relation to the maximum, sufficiently so a t least for the purpose of illustration. In Fig. 23 the efficiency given is in fact a rough intqration of the curves of Fig. 20: there is no pretension to accuracy inasmuch as no particular thrust-grading has been specified or deduced as appropriate; in the words of the paper:-“ . . . this curve is founded on an estimated average of the portion of the blade

* Compare “Aerial Flight,” Vol. I., $ 4 206 and 213, Fig. 129.



THE SCHEW 1’HOPEI.LER. 347

utilised, and is not the absolute maximum of the curve as given in Fig. 22, and thus it should more nearly represent the truth.”

I now pass to Mr. Baker’s ramarks as to the number of blades. He says that he would like to draw my attention to a much better method of arriving at tha number of blades than that given by me; he then proceeds to disinter for my benefit .the niethod given by Prof. Cotterall in 1879. The issue appears quite simple; the question is that of estimating the effective thickness and content of the layer of fluid on which each individual blade operates; I have in the present paper appealed dimct to the vortex theory of sustentation. Prof. Cotterell’s method consists in a dynamical calculation founded on pressure plane experiment. Now any- dynamical calculation of the kind in question depends upon an. assumption that we know the velocity imparted to the fluid mted upon; in brief, we know the momentum per second aa being equal to the weight sustained, and presuming to know the velocity imprewd on the fluid, we h v e at once the means of calculating the mass per second, and thence the effective area of the column of fluid handled. Now I do not happen to be in possession of the paper by Prof. Cotterell to which reference is made, neither have I the necessary reference to ldcate it, so that I have no definite information as to the basis on which the velocity imparted to the fluid was assessed. If, as is probably the case, the velocity was taken as that due to the angle of trail of an experimental plane or foil, then the result would be so greatly in error as to rend= the calculated mass totally unreliable; the extent of the probable error depends mainly upon the aspect ratio; if this be very low (say unity or less), the result may be reasonably near the truth, but if the aspect ratio be high, the estimate may be hopelessly at fault. Perhaps Prof. Cotterell may have had some other method of ascertaining the velocity, but it is improbable that it can have been at all reliable, since at the date in question, 1879, the type of streamline system associated with an inclined plane or aerofoil in motion had not been worked out; this is, in fact;, the precise piece of work which formed the main subject of my “Aerodynamics ” and of the preceding paper.

I scarcely think, therefore, that Mr. Baker will be in a position. to substantiate his assertion that the procedure of Prof. Cotterell is a m w h better method than that disclosed in the present paper. I may, perhaps, put my reply in another form; any method of investigating the number of blades must depend upon assessing



3'48 THE INSl'lTUTIOIU OF AU'lO,\lOtlIl.E EKGINEERS. (Mr. F. W. Lanchester.) in some manmr the mass per second dealt with by one blade, and in calculating the total nl~lss passing through the propeller disk, the appropriate number of blades being given by the latter in krms of the former. Cotterell does this by a lriethod which lias been applied to the problem of sustontatioii in flight and invariably failed to give results which accord with espcricnce. I hare given a method which, likewise applied to the problem of snstenta- tion, has invariably given results which do accord with experience; which in 1894 enabled me to design and construct an aerofoil which, tested recently, has beaten all records for economy, and which, further, enabled me to tabulatc pressure values proper to least resistance, best values of Ziftldrift, and angles defining the aerofoil camber as far back as 1907, all of which have since been closely verified. In spite of this Mr. Baker refers to Prof. Cot- terell's method as niuch better; I gravely question whether he has uscd the one, or attempted even to use tho other.

Mr. Baker proceeds to criticise my results on the ground that in his experience there is a wide range of propeller diameter which can be adopted without inaterial influence on cfficiency. This, however, if indeed not otherwire obvious, is clearly brought out in my diagram Fig. 23. On reference to this it will be seen that in tho region of the optimunz the efficiency curve reads nearly constant. It is, of course, quite self-evident that where efficiency is maximum, the rate of change of efficiency in terms of the rate of change of diameter is zero, this is in fact the usual condition of all maxima or minima as used in the mathematical solution,

Finally, Mr. Baker quotes an experiment directed to determine the influence of aspect, ratio, which seems to prove that which I have more than once stated, and which I firmly believe to be true, that one can prove anythivhg (right or wrong) by experiment. Aiiyuiie thoroughly acquainted with the intricacies of aerodynamic es- periment or with the screw propeller would hesitatc before drawing any general conclusions from an experiment conducted on the lines indicated. For the present purpose it is sufficient to point out a few of the possible sourcea of error; I do not suppose thab even Mr. Raker, however much he may believe in the accuracy of the experiment he describes, redly wishes to suggest that we are to conclude that the influence of aspect ratio is of little or no account where the blade of a screw propeller is concerned. Firstly, unless the number of blades were clearly not resulting in intcr-

d y l d z = 0 .



TI3g SCREW PROPEI.I.ER. 349

ference, in every case the comparison between two blades and four blades may entirely discount any advantage in the latter case due to high aspect ratio. Secondly, since the peripteral area of the b W is almost entirely dependent upon its length and almost independent of its breadth, assuming no interference, the four bladed propeller would be adapted to give best efficiency at something near twice the thrust of the two bladed propeller. Thirdly, the sectional form cannot have been in every case con- forniable to the system of flow; on the face of things it cannot 1iit1-e been conformable in either case; the consequences of this deficiency can only be vaguely conjectured. Fourthly, the experiments may not have been conducted in either case a t or m a r the condition of highest efficiency, in which case no possible inference can be legitimately drawn. If we for an instant imagine any one of our aerodynamic laboratories making an analogous test to determine the effect of aspect ratio in the case of an aerofoil, i.e., testing two foils separately or in tandem, we can imagine the shouts of derision which would arise on publication ; the experiment cited by Mr. Baker is no better directed to its object.

I am pleasad to note that Mr. Grinsted is able to speak per- sonally as to the utility of the theoretical work presented in my “Aerodynamics” as a basis of practical design, though I think no one is more conscious than I am myself that i t has its short- comings; i t has bean part of the object of the prcsent paper to so. extend and amplify the theory as to diminish the said short- comings and render it of greater immediate value to the designer.

I t is, however, to some extent news to me to be told that ‘‘ everyone admits that th0 theory as a real basis of design is very loose.” I think that most of the thmries that are of the greatest value to the engineer and designer would be described by the student of rigid dynamics as “very loose,” but to have my theory thus criticised by everyone is more than I am prepared to accept.

In taking a birdseye view of the history of scientific development, it appears to me that we have waves which include periods in which the prevailing tendency is to pin our faith to theory and other per ids in which the tendency is to accept results as established by experiment. I sometimes feel that the swing of the pendulum has to-day gone too far in the direction of discrediting everything which is not demonstrated by direct experiment; I have, in fa&, of late years time after time seen conclusions given as the result of experiment, which have no more foundation in



350 T H E INSTlTUTION O F AUTOMOBILE ENGINEEHP. (Mr. F. W. Lanchester.) reality than the loosest and mast ramshackle theory ever put on paper. Like the man who once stated that he could prove anything by statistics, I am equally prepared to assert that I could prove anything by experiment.

The best proof of any theory is its utility when applied, thus the theory of hhe unchangeability of the atom and the immutability of the chemical element is just as useful to-day, when we know that it is not strictly speaking trm, as it was before the days of the discovery of radio-active matter and the disintegration of the atom.

I hardly take i t that Mr. Grinsted is serious when he suggesh that since the velocity of advance is given and the engine speed is fixed by extraneous considerations, and the propeller diameter is practically fixed by considerations of clearance, the results of the present investigation cannot be of value. It is perfectly evident that if this statement were true, the designer of the propeller, having his pitch and diameter determined in advance, cannot possibly be expected to interest himself in the theoretical question conosrned in propulsion; in brief, his job is virtually gone. I infer that hlr. Grinsted does not intend his remarks here to be taken too seriously; I rather take it that he is calling attention in a picturesque manner to the fact that the scope of the designer, in the matter of popeller design, is unfortunately very much set about by restrictions.

Unquestionably, the design of a pl-opeller should be taken into consideration in the general lay out of a machine, and, since the addition of six inches to the height of the alighting chassis will give an additional foot to the diameter of the propeller, it is clear that the conditions am not so closely circumscribed as suggested. Again, in the case of the revolution s p e d there is considerable latitude, since the propeller may be driven off the crankshaft direct, or, as in the case of the Renault and R.A.F. engine, it may be driven off the two-to-one gearing; there is, in fact, nothing to prevant any gear ratio desired being employed. On this point, it must not be forgotten that the pmaent wopk

is intenkd to be applicable to marine propulsion, and one of the important points at present to the fom in this conneation is that of direct z;%rm8 geared turbines.

Mr. Barling at the outset somewhat misrepresents the case presented in the paper. He refers to three kinds of losses; the third of those which he refers to as “ a loss which we might cal l



THE SCREW PROPELLER. 35 1

the drift loss.” In the terminology of the p p e r , it certainly is not the drift lorn, sinoe this includes losses of all kinds. Mr. Barling, I think, has entinely misread Fig. 10 to which he refers; this is similar to the remainder of the series of Figs. 8 to 13. He ‘is entirely under a misapprehension aa to the interpretation of theire diagrams; the abscissae represent radius in terms of pitch, not radius aa an absolute quantity. Thus, he must not for an instant suppose that because tha radius in terms of pitch is in one case smaller than another that any inferenoe is to be drawn aa to the actual radii or diameters of the two propellem. Gene- rally speaking, as shown by the diagram Fig. 23, the bigger the diameter of the propeller chosen the less the diameter/pitch ratio proper to highest efficiency.

Mr. Barling is entinely at fault also in his further remarks as to dkagreament between general practice and the propeller as deduced from the present theory. 27 at the conclusion of Part In. of the paper, he will see that when the results of the invmtigation a m properly compared with those of p w n t day experiences the agreement is almost closer than would be expeoted.

Mr. aarling is, I think, again under a misapprehension where he refers to a momentum theory and what he calls the “strip” theory of t b propeller being at variance. The momentum theory alone is incompbte, inasmuch as the bigger the propeller the greater the efficiency to an unlimited extent. The complete theory indicates that if the propeller diameter exceeds a certain optimum value, the gain in efficiency as due to the larger mass of fluid handled, is more than outweighed by losses due to the skin- frictional and other direct blade resistance; the deterniination of the best diameter is in fact the basis of the theory of the real screw propeller, whether rn represented by the present investigation or by the previous work of Froude, Rankine and others.

If he will refer to



352 1 H E INSTITUTION OF AUTOMOHII~E ENGINEERS.

CObfiMUNICATIOK. Prof. J . U. HENDIXSON wrote: 1 ndmiie the author’s courage

in boldly treating the problem as if the whole phenomena were confined within the circumscribing cylinder. I t is an idea I have had for many years, and it naturally occurs to anyone like him 11 ho is familiar with the few problems in the mathematical science of hydrodynamics which can be solved rigorously and which have M bearing on the subject, such as the rotation of an elliptic cyliuder in n liicli thc angular momentum within the circumscribed cylinder is equal to the angular momentum in the whole fluid. I would like to believe that it is true in every case, but I could not see my way to draw such a general conclusion from such scanty premises. If I understand the author rightly, he boldly states on the evidence available that it is true in general, and learns it to the mathematician to disprove it. By this one step he opens up a great field of work, because by means of it Rankine’s graphical methods can be applied to many practical problems which are beyond mathematical treatment at present.

I agree with the author as to the distribution of vortices in the propeller race, but I cannot agree with him in attributing the acceleration ahead of the screw to the thrust of the scmw at the moment. This is, however, a controversial subject which he has done his best to avoid in the present paper by means od his flexible coefficient Q which can be made to fit any particular theory.

I should like to ask one question. All bodias moving on straight courses through fluids are accompanied by energy systems the effect of which virtually increases the mass of the body. The blades of a screw propeller follow spiral paths, and no energy system will follow a spiral path unconstrained. The screw propeller as a whole follows a straight path, and the sum of the effects of the individual blades on the fluid as a’ whole is determined by the screw disk, just as the magnetic field a t some distance from a magnet depends only on the magnetic moment of the magnet and not on the shape of the pole facoas. I n view of these facts, does the author deny the possibility of the screw propeller having a virtual m s as a whole apart from any it may have due to the individual blades ?




Every method of treating the problem, if correct, should lead to the same results. In the circumscribed cylinder method the datum of reference is not necessarily the still watar through which the screw is moving, but may be an advancing coluinn of water started during the initial acceleration of the screw. This point should eaaily be decided experimentally.

In reply to the written communidation Mr. LANCIIESTER wrote: Firstly, as to the specific interrogatory of Prof. Henderson on the general fact of the virtual w s of a body in motion in a fluid, as stated by him the energy system of a body in motion in a finid is virtually equivalent to an appropriate increase in the mas6 of the body. I see no difficulty in the case of the blades of a propeller on the ground mentioned, i.e., that “ no energy system will follow a s p b d path unconstrained.” I believe that a sphere, for example, if moved in a circle (or spiral) in the perfect fluid of mathematical theory, would carry its fluid energy system with it in its curvilinear path-the boundary surface of a sphere con- strained to move in any form of path will in turn exercise the neemsary constraint on the fluid. The meaning of this in its application is that each propeller blade may possess a certain virtoal mam which will follow it just as if it were a spar of “faired” section in rectilinear motion; I do not imagine for an inettsat, however, that this is the energy system that Professor Hendersor. has in mind when he puts his question.

Assuming it to be true (in accordance with my theory and treatment) that there is cyclic motion (i.e., a vortex system) associated with the propeller blade, as in the cas0 of an aemfoil sustaining a load, there will be a corresponding energy system (as also in the case of the aerofoil) which will give to each blade of the propeller, and in effect to the propeller as a whole, a virtual added mam, as in fact already stated in my “Aerial Flight.” * Thus, categorically, my answer to Professor Henderson i s that I do not deny the possibility of a screw propeller having a virtual i m s as a whole. Whether this energy is viewed as “apart from arag it m y have due to i t s individual blades” or otherwise, is a matter of definition.

My view is that the energy system in question is something quite apart from and additional to that which is fundamental to the Rankin Froude theory of propulsion, and this difference is

* Vol. I., 4 123. LANCHESTEH. z



354 THE INSTIT.L!TION OF A IITOMOHILE ENGINBEKS. (Mr. F. W. Lanchester.) clearly brought into relief in this and the preceding paper. The Nuid motion as satisfying the momentum equation (thrust= momentum/second) is that defined by the priwiry camber acting on fluid having only two degrees of freedom (radial and axial i n the case of the propeller). The cyclic component of motion (by Continental writers sometimes called thc “ circulation ”) is that which in effect constitutes a conservative system and is that for which the secondary camber is required. Whereas the former motion involves a continuous expenditure of work, and represents the momentum communicated to the fluid and thus is w e n - tial, the latter motion neither contributes to the momentum nor represents an expenditure of energy-it is merely incidental to the particular nature of the machinery or instrument of propulsion.

I am acquainted with other writings of Professor Henderson in which the question of the virtud mass of the propeller race is insisted upon. The truth is, I think, that there are nearly always two views to any problem in hydrodynamics. Wherever any constant energy distribution or configwation is found to remain unchanged, whilst the actual fluid undergoes continual reneval, it is from one point of view possible to regard the energy as c0n- served-to think of it, in fact, as “ear-marked.” So long as in any such system there is definitely no continuous expenditure of work, no other view is possible; when, however, work is being done and fluid is being permanently accelerated, this view may result in error. Whereas in an efflux jet and i n the system set up (in theory) by the Froude actuator, the energy of the contracting jet is only conserved in the sense that the water in a reach oif a river is conserved, there is as much being fed in at one end as k lost at the other. We cannot argue of the water of a river that because its quantity and distribution is constant it therefore undergoes no change. Likewise, we cannot argue of the energy in an efflux nozzle, or in the head ram of the Froude actuator that because its quantity and distribution remain constant i t must therefore constitute a conserved energy system.. This would appear to me to be an error and one to which i t seems Froeesw>r Henderson invites me to subscribe.

I feel a little uncertain as to whether Professor Henderson rightly apprehends the basis of my method when he refers to the rotation of an elliptical cylinder. The basis of my quantitative treatment is the elliptic cylinder in motion of translation as =ore definitely laid down in my paper on “The Aerofoil.”



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