The Performance of Rotating-Wing Aircraft Rotors By C. F. Toms, A.R.Ae.S.

I. Introduction

THIS paper presents a summary of the method and results of a general investiga-tion into the performance characteristics of 'single' autogyro and helicopter rotors, which was a preliminary to the establishment, by the firm the author serves, of a helicopter division.

Considerable data on the subject was available, but no generalized collection of working data appeared to be available; moreover, to one un-initiated, it was difficult 'to see the wood for the trees'. It was, therefore, decided to attempt a fresh analysis of the problems involved, based on simple 'strip' theory, disregarding the effects of periodic flow conditions on aerofoil section force coefficients, and using a system of force and torque coefficients and parameters similar to those employed in the case of propellers.

In its general form the analysis could be applied graphically to rotors having an arbitrary blade plan-form and an arbitrary blade twist. The analysis is applied analytically to two particu-lar blade plan-forms with zero twist, and to one of these plan-forms with a particular twist. The full development of the analysis (not given here) is complicated, but it results in charts which are quite simple to use.

It is assumed that rotor blades do not twist or bend under loadan assumption that can be justified in practice by a suitable choice of blade section and careful blade design.

Both the auto-rotational (autogyro) and pro-, pulsive (helicopter) regimes of operation are con-sidered ; the results for the auto-rotational regime agree tolerably well with such experimental data as are available and there is no reason to doubt that the results for the propulsive regime are equally reliable, although no suitable experi-mental data are available to check this.

The analysis could be adapted to cover the case where the blades are subjected by mechanical means to cyclic variations in setting angle.

II. General The lifting surface, or rotor, of a rotating-wing

aircraft of the type considered in this report, con-sists of a number of equally-spaced radial blades rotating about an axis which, under normal con-ditions of flight, makes a small or moderate angle with the perpendicular to the plane containing the direction of motion and the horizontal.

Rotating-wing aircraft supported by such a rotor fall into two groups: (a) The Aulogiro* or Rotaplane

In this type the whole of the power required to overcome the drag of the aircraft is supplied to a normal propeller, and, the rotor shaft being inclined slightly backwards from the perpendicu-lar to the direction of motion, the rotor is kept in motion around its shaft by the air forces acting on it. Since no power is supplied to the rotor by mechanical means, the condition of operation is known as autorotation. Its origin is, however, quite different from that causing the autorotation associated with spinning of a conventional aero-plane. (b) The Helicopter

In this type the motion of the rotor is main-tained by means of a torque applied to the rotor shaft by an engine. The forward motion of the aircraft may be maintained, as in the case of an autogyro, by a normal airscrew, but it is more economical to provide the propulsive thrust by

* This word is the registered trade mark o lthe Cierva Autogiro Co. Ltd.

putting all the power into the rotor and inclining the shaft forwards of the perpendicular to the direction of motion.

III. Notation D Diameter of rotor=2R NB Number of blades in rotor cR Blade chord at tip cr Blade chord at radius r S' Solidity ratio constant

wB Blade weight per square foot n Angular velocity of rotor in r.p.s. =

2TT v Forward speed of rotor shaft

aR Angle of attack of rotor=angle between rotor shaft and the plane normal to the direction of motion

a. Induced angle of attack = 0 63l(^f) (radians)

o. Blade setting angle=angle between rotor blade section no-lift-line and plane normal to rotor shaft. Assumed constant along blade in the analytical treatment; in the case of 'tapered' blades a particular twist is also considered, in which case a , is quoted at 07R. Arbitrary variations of a, along blade can be treated graphically

A2a, Change in blade section angle of attack due to inclination or rotor shaft

A3a, Change in blade section angle of attack due to flapping motion

8 Angular position of blade, measured from the plane normal to rotor shaft

>p Angular position of rotor blade measured from directly downwind

T Fraction of rotor tip radius TX Value of T below which the rotor forces are

negligibly small and are ignored T 2 Typical value of T for estimation of flapping

motion Tg Value of T for calculating solidity ratio kT Rotor lift c o e f f i c i e n t = ^ r < ^

kQ Rotor torque coefficient =gg9 o u e /6) . pn2Db

kx Rotor drag or thrust coefficient= Drag (>) or Thrust (T)

pn^D* a (^CL\ s l P e 0 I"D l a d e section lift curve for 0 \da I two-dimensional flow

IV. The General Nature of the Flow Pattern In order that it shall produce a lift force, any

lifting system, whether it be a fixed wing, a rotor of the type under consideration or cyclogyro ('paddle-wheel') rotor system, must impart a downward momentum to the air which comes under its influence. Interpreted physically this means that the air-stream affected is deflected through a small angle a, in the opposite direction to the lift force.

In the case of an autogyro rotor it is necessary, in order that autorotation shall occur, to incline the rotor shaft backwards from the perpendicular to the direction of motion. The general path of the air affected by the rotor is as shown in the

upper part of FIG. 1; note that although the air stream is deflected downwards by the lifting rotor the air has a component of velocity up-wards through the rotor disk.

In the case of a helicopter rotor inclined to provide the thrust necessary to overcome the drag of the aircraft the rotor shaft is inclined in the opposite direction, so that the air stream is deflected downwards as before but has a down-ward component of velocity through the rotor disk, as shown in FIG. 2.

The other aspect of general flow pattern is the nature of the variation, in magnitude and direc-tion, of the resultant velocity with position along a blade and with the angular position of a blade. The resultant velocity (Kr) is the vector sum of v parallel to the direction of motion and rta normal to the blade axis, and it is clear from FIG. 3 that such variations do actually occur. F IG . 4 shows an alternative and more simple geometrical construction for finding the magni-tude and direction of the resultant velocity, and it has been used to construct FIG. 5, which shows the nature of the flow, in plan view, along a blade for the particular case of / = 1 *0.

Further investigation shows that the distribu-tion of velocity along a blade for a particular angular position (ifi) is the same as that which

" A ircraft Engineering

would exist if the blade were rotated with the same angular velocity, in still air, about a point on a circle whose radius, in terms of the blade radius R, is J/n. The angular position of the point with respect to the blade axis is variable, since it lies on the radius at ^=270 deg. FIG. 6 illustrates this point, and shows, for example, that when the blade is 30 deg. from the down-wind position in an anticlockwise direction when viewed from above this instantaneous centre of rotation is at P when J=l-Q. This fact is inter-esting in that it means that it is 'possible to simulate continuously on a ground test rig the flow conditions obtaining momentarily along a blade at any particular angular position >ji. The effects of a flapping motion could be incorporated by giving the test blade a suitable twist.

FIG. 7 shows the plan view of the paths of rotor blade elements at three values of //>

The effective component of the resultant velocity is taken throughout the analysis as the component normal to the blade axis, that is (tto+vsini/f). That this is the effective component is shown by the curves of FIG. 8 for which CJJ based on the forward speed of the aerofoil varies as cos2j3 where /$ is the angle of sideslip. The normal component of the forward speed (v) is v cos j3, and if the lift varies as the square of this component the CL based on the forward speed

should vary as fv-csff=cos2/?, a s it does.

V. Steadiness of Rotor Forces It is interesting to examine roughly at this stage

the probable nature of the total forces acting at the rotor hub, as far as any variation during one

revolution is concerned, as affected by the num-ber of blades in the rotor.

To simplify matters, consider the torque of a rigid rotor with its axis of rotation perpendicular to the direction of motion and with the blade

May 1947 151

setting angle zero, i.e. aB=as=0 deg. Under such conditions a rotor would, of course, produce no lift, would have to be driven round, and would give rise to a drag force.

Consider an element of one blade of area dS at radius r. Its effective velocity is

so that the drag force on the blade element is

and the torque

The only term in this expression which depends on the angular position of the blade element is the last one (1 +J/v sin i/i)2. If two blades are employed the total torque is then proportional to:

and the torque therefore varies as the blades rotate, since the angle ip appears in the expres-sion. If four blades are used the total torque is proportional to:

and the torque therefore is independent of the angular position of the rotor, as ip does not appear in the expression. This is also true in the case of three blades, five blades, etc.

Further consideration is given to this matter in more detail in Section XII.

VI. The Induced Angle of Attack Subsequent calculations show that the induced

angle of attack is calculated with sufficient accuracy for the present purpose by regarding the rotor as an aerofoil of circular plan form, of diameter equal to the rotor diameter, with an elliptic distribution of lift across the span.

For an elliptic aerofoil (a circle is the extreme form of an ellipse) the induced angle of attack is:

where A is the aspect ratio of the aerofoil. For a circular aerofoil the aspect ratio is:

and the lift coefficient is:

so that:

a{ may also be expressed in terms of kT and J as defined in Section III, and the expression is:

(It will be appreciated that the induced angle of attack is in fact not uniform over the whole rotor disk, but, as stated above, subsequent calcula-tions vindicate the assumption that it is so.)

The effective inclination of the rotor shaft is: a-RaRai

VII. The Particular Nature of the Flow Condi-tions over a Blade

The distribution of velocity along a rotor blade and the effect of the angular position (i/i) of the blade on this distribution have already been dealt with. It is now necessary to examine the effect, on the angle of attack distribution along a blade, of inclining the rotor axis to the perpen-dicular to the direction of motion. A rigid rotor is considered.

When a blade is rotating in a plane not parallel to the direction of motion the two velocity com-ponents (translational and rotational) are not in the same plane, and are at a periodically varying angle to each other, so that the angle between the resultant velocity vector and the plane of rotation also varies in a periodic manner.

FIG. 9 shows the geometry of the flow, and it is derived that:

where

and

The values of for various values of JL

have been evaluated taking a'fl=10 deg., and are plotted against tfs in FIG. 10. The effect on the results of using a'/j=20 deg. instead of 10 deg. is

also indicated and is greatest at ^=277 deg. and at high values of JL. This blade position and region are associated with small forces and any error involved in using the curves based on a'if= 10 deg. is negligible.

For the purpose of calculating the flapping motion it is found convenient to express the curves as a Fourier Series of the form:

and the coefficients a '0, a2, at, Z>n b3, are plotted on FIG. 11. FIG. 10 shows how closely the Fourier series represents the original curves up toJL=l 5. The agreement is fair at 7/y=2-0, but, with the number of terms used, hopeless at JL=3 -0.3 -0 is, however, an extremely high value and not likely to occur in practice, except at values*of T in the region of T1( taken as 0-2.

In the lift, torque and drag/thrust analysis it is

found more convenient to express at a

given value of I/I as an algebraic expression in /&. Only four values of

allowing the blades to 'flap' about axes parallel to the rotor axis of rotation and these 'drag flapping' axes must not intersect the latter, for if they did, the centrifugal force (C.F.) acting on a blade could have no moment about the drag flapping axis and hence there would be no position of equilibrium between the moments of the blade drag and the centrifugal force. This will be ap-preciated after reference to FIG. 13. Since there is very little aerodynamic damping associated with drag flapping, artificial damping is usually pro-vided.

FIG. 12 illustrates diagrammatically the arrange-ment of lift and drag flapping axes just described.

The problem of determining the nature of the lift flapping motion is approached by considering an isolated element of a rotor blade attached to the axis of rotation by a weightless, inextensible tie. Let the area of the element be ds, its weight dw and the radius r (other symbols are defined in Section III). The lift on this element is:

and Ci=ana=

A0 the following expressions, defining the necessary blade twist and weight distribution re> spectively, are easily derived:

The distributions of twist and weight given b> these expressions unfortunately change markedl) with the conditions of operation of a rotoi (/, a a^ j), and, for some conditions, the weighi distribution is quite impracticable. It is only evei possible, therefore, to cater for one particular sel of conditions, even if a built-in blade curvature is employed as an alternative to the weight dis-tribution prescribed by the above expression.

IX. Blade Plan-form Considerations, and Rotor Solidity

The simplest blade to produce, and one which has often been employed, is of constant chord, and this plan-form has been treated in the sub-sequent analysis, and is referred to as a 'parallel' blade.

The other plan-form considered is referred to as 'tapered' andhas a chord varying inversely as the radius, so that if CR is the chord at the tip and cr is the chord at radius r:

The plan-form is selected on a basis of rotor efficiency when hovering" or in vertical flight (which states of operation, incidentially, appear to have been regarded as the alpha and omega of the helicopter by early workers). In order that the kinetic energy- loss in a rotor 'slipstream' shall be a minimum, it is necessary that the lift de-veloped shall be uniformly distributed over the rotor disk. * Consider an annular ring of the rotor disk, of radius r and width dr. The lift developed by each of the A^ blades falling inside this an-nulus is:

The area of each blade element in the annulus is cr dr and its speed is rco. Hence the lift coefficient at which the blade elements work is:

* This argument was first propounded by Schrenk and is published in NA.C.A. T.M. 733.

t The constant 0-96=0 n ' ) = 0 -0-2') . It is assumed that no lift is developed inside r^O-Z

Now there is an optimum value of CL for any given aerofoil section, and it is clearly desirable that this value shall obtain all along a blade. It can be seen from the above expression that this requires that rcr be constant, i.e. that:

or cr=ci( where cR= chord at tip (r=R). r Although based on conditions during hover-

ing etc. this conclusion is taken as applicable to forward flight at low values of/.

In view of the fact that such a relation between chord and radius results in a very large chord' at small radii, and the fact that in forward flight the flow conditions at some values of blade position (tfi) are poor, it is not advisable to adhere to the relation below 0-2R, i.e. below ^ = 0 - 2 .

Leaving the value of TX as a variable, the effec-tive area of the Ny tapered blades in a rotor is:

The corresponding effective disk area is:

The rotor solidity is defined as:

a may also be expressed as the ratio: S chord perimeter

at some value of T(TS), and this value is easily shown to be:

In the case of parallel blades:

and

so that

The values of the constant S' and T, are sum-marized in the following table, for various values ofrj .

* 1

0 1

2 3

Tapered

S'

1^ 07 0-84

Blades

*

545 615

Parall<

S'

0-64 0-58 0'53 0-49

1 Blades

T j

705 740 775 805

NOTE: TO obtain a uniform distribution of thrust over the rotor disk it is necessary to employ a tapered blade, as demonstrated, twisted to give an increasing angle towards the root. For further information see Section X.

X. Rotor Lift Analysis The lift on a blade element (i.e. the forces at

right angles to the local direction of the airflow relative to the element) is given by

Considering tapered blades for which this becomes

and the instantaneous value of the total lift on one blade is:

This can be put in coefficient form, introducing the solidity (a) at the same time, thus:

(see Section III for definition of KT and Section IX for definitions of 5" and a). This expression dis-regards the fact that the lift vectors at different points along a blade act in different directions, and that the mean direction is inclined to the normal to the direction of motion. A subsidiary investigation has shown that a sufficient cor-rection for this is to introduce a factor cosa^ in the final result.

The lift coefficient of each blade element is given by:

the values of f(JL) being those given in section VII. This is substituted in the preceding ex-pression, giving:

which is the general expression for calculating the instantaneous value of the 'lift' coefficient of one blade. It will be shown subsequently that the total lift of a rotor consisting of three or more blades varies but little as it rotates. For the purpose of an analytical evaluation of the total lift a four-bladed rotor is considered 'with the blades in the positions i/r=0, 90, 180 and 270.* The results are applicable to rotors having three blades, five blades, etc. Zero blade twist is as-sumed. Integrating the general expression above

* The lower limit of integration (n) has been taken as 02 for ^=0, 90 and 180, but as Jin for v = 270, since whilst the forces inside (Jin) K at 270 are small, the analysis would interpret the reverse flow conditions obtaining as very large angles of attack, corresponding to larger forces.

154 A ircraft Engineering

for the four values of ip, adding, eliminating a( etc. gives a final result in the form:

whereC'=Cforafl=10deg.andJ=-,andi>'=i) T 2

for JL and JsJu are explicit functions of J only, and their values are plotted in FIG. 17. The analysis for parallel blades is very similar to that for tapered blades, and the final expression is of the same form, but the constants /8/12 have different values which are plotted in no. 18.

The charts given cannot be used for the hover-ing condition, which corresponds to J=0, be-cause Jt and J12 are then both infinite and C" is zero. (There is, of course, no question of a flapping motion when / = 0 , but the blades do 'cone.)' When / = 0 the velocity of flow () through the rotor disk is given by the ordinary airscrew momentum theory as:

which causes a reduction in effective blade angle of amount :

and a simple analysis shows that, u being as-sumed constant over the whole rotor disk, the hovering lift coefficients for untwisted blades are given by the expressions: Tapered blades

Parallel blades

FIG. 19 is constructed according to these ex-pressions and should be used in conjunction with FIGS.17 and 18.

A more exact analysis of the hovering con-

dition can be made if the blades are twisted in the manner which, combined with the tapered blade planform, actually gives the uniform dis-tribution of lift over the rotor disk implied by the expression for u and Aa above. From Section IX:

(

and also

from which it is readily deduced' that:

and

The difference between the values of kTo given by this expression and the one above for un-twisted tapered blades is indicated in FIG. 19 for a,=10 deg. at JR, and the associated twist is shown in no. 20.

In the above expressions for kT and kTo, a0 is 'per degree'.

XI. Rotor Torque Analysis The torque is due partly to the profile drag of

the blades and partly to the inclination of the lift vectors relative to the axis of rotation. The torque due to a blade element is given by:

Considering tapered blades for which this becomes, for the instantaneous torque of a whole blade:

In coefficient form this becomes:

which is the general expression for calculating the instantaneous value of the torque of one blade; the expressions for CL, A2a, and A3a, given in Section X are also applicable here.

After integration for the same four values of tp as used in the case of lift, and adding etc. the final expression for the total torque of a rotor having more than two blades is found to be:

wherea'B=aBa

question. For preliminary use a value of 0-015 is suggested. These remarks also apply to the thrust/drag calculations, and a low value of CD0, i.e. a very clean blade, is of paramount im-portance in an autogyro.

In the above expressions for kQ and k^, a0 is 'per degree'. XII. Further Discussion of Steadiness of Rotor

Forces It is now possible to examine more closely the

variation of the lift and torque experienced by the individual blades of a rotor, and by a rotor as a whole, as it rotates, and the effect of the number of blades on such variation.

A typical set of operating conditions, viz:

*J=06 ait=-10deg. a,=8 deg. a=-05. C D 0 = - 0 1 2 has been taken for examination, assuming

(a) a rigid rotor {b) a rotor with blades free to flap about a

lift axis only. The curves of KT and KQ^^ appropriate to

the above conditions determined by graphical integrations according to the general lift and torque expressions, given in Sections X and XI, are shown in FIG. 24 for (a) and in FIG. 25 for (/>). FIG. 24 confirms the approximate argument put

* The thrust and torque variations at a higher value of /would be greater than those shown.

forward in Section V (which relates to a rigid rotor) by showing that the thrust and torque vary with tfi for a two-bladed rotor, but are constant for one having four blades. However, as is pointed out in Section VIII, a rigid rotor is sub-ject to an (undesirable) rolling moment, and the individual blades to fluctuating bending moments. FIG. 25 shows, firstly, how the, lift flapping motion greatly reduces the variation of lift on each blade as it rotates, and how at the same time it increases the amplitude of the torque variation. FIG. 25 also shows that for a two-bladed rotor, in spite of the much smaller variation of lift with tft for each blade, the variation of total lift with i/r is of the same order as that for a rigid rotor (FIG. 24), though of different form. The lift and torque for

156 Aircraft Engineering

three- and four-bladed rotors, however, whilst not exactly constant, are sensibly so. As stated in Section VIII it is, in practice, desirable to allow the blades of a rotor freedom to flap about a second (drag-flapping) axis. Such a flapping motion will not reduce the amplitude of the torque variations to anything like the same extent that the lift-flapping motion does in the case of lift, and some form or mechanical damping must be provided, due to the virtual absence of aero-dynamic damping.

XIII. Hovering Efficiency The term 'hovering efficiency' is used to mean

the ratio of lift developed to power absorbed when hovering. The following relation is easily formulated:

Lift _ 275 kro Horse-power vnD' ICQQ ,

and this quantity is plotted in FIG. 26 (for CDo= 0-01) using the data in FIGS. 19 and 23.

A low tip speed, a low solidity and a large blade angle (with the proviso that Cy0 remains con-stant) and are seen to be desirable in this con-nexion.

XIV. Rotor Thrust Analysis The magnitude of the thrust (positive or

negative) produced by a rotor is most easily de-termined from the overall energy equation in con-nexion with the lift and torque determined from the kT and kQ charts.

To a close approximation the energy equation is: (Work done by engine per sec.)=(Work done per sec. by lift in deflecting air)+(Work done per sec. "by thrust)+(Work done per sec. against profile drag of blades), i.e.

Dividing through by pn3D5:

or in general terms: kx (positive forward) =/0/:(jJ^kr2 J^OCDQ where J0, Jx and J2 are explicit functions of / only, and their values are plotted in FIG. 27. (The double integral has been evaluated graphic-ally.)

XV. Autorotation and the Determination of Auto-giro Operating Conditions

The circumstances under which a rotor system will rotate freely under the action of the air forces on the blades can be understood by con-sidering an element of a rotor blade in the position t/i=90 deg., i.e. when at right angles to the direction of motion and travelling up-wind.

FIG. 28 shows a blade element in such cir-cumstances, with a positive value of aR (which is essential to autorotation) and with velocity vectors v (due to forward speed) and ro> (due to rotation) of such relative magnitudes and in such relative directions that the resultant velocity vector is in such a direction relative to the rotor shaft that the resultant aerodynamic force (R) on the element is inclined forwards of the shaft, and therefore tends to increase the rate of rota-tion. At some (greater) rate of rotation the re-sultant force vector will become parallel to the rotor shaft, and a steady condition of autorota-tion will then exist. For a complete blade this argument is true of the mean resultant force vector and the inner 2/3 of the blade acts as a windmill driving the outer 1/3 round, while the whole blade contributes to the lift developed.

The importance of having a low drag is now obvious since a reduction in D means an increased forward inclination of R at a given rotational speed, which in turn means a higher autorota-tional speed. A corollary to this is that it is necessary to estimate (guess), the mean drag coefficient of a rotor blade quite closely if the performance of the rotor is to be estimated to a satisfactory degree of accuracy.

On account of the increase in drag coefficient with angle of attack, autorotation is only possible over a very limited range of blade-setting angles; on the other hand autorotation can occur over a large range of positive angles of attack, between 5 deg. (say) and 90 deg.

Using FIGS. 17/21 and 18/22 charts can be pre-pared giving directly the conditions of operation

of an autogyro rotor as a function of aR and a if the appropriate drag coefficient is known. The process is simply to discover the value of / for which a rotor of given solidity and blade plan-form, at given values of aR and a, has zero torque, and then to calculate the lift coefficient A'r. The result of such a series of calculations is presented in FIG. 29 and a number of such FIGS. can be prepared for different solidities and as-sumed blade drag coefficients. The value of the drag coefficient to be used for any particular standard of blade finish, etc., must be based on experience.

XVI. Autogyro Vertical Descent The analysis for an autogyro in vertical descent

(i.e. for aB=90 deg.) can only be related by theory to the velocity of flow through the rotor disc. The relation between the velocity of descent and the velocity of flow through the disc is not correctly predicted by a simple momentum theory, because the flow conditions above the rotor disc are similar to those in the windmill brake state of operation of an airscrew, with some of the air temporarily moving in closed circuits, after the manner shown in the upper part of FIG. 30. Recourse is had to the results of model airscrew experiments to bridge this gap where theory fails.

Consider tapered blades with constant a,. Let u be the velocity of flow through the rotor disc (assumed uniform). The torque due to a blade element (see FIG. 31) is given by:

May 1947 157

where

Now J'v is a small quantity and only its first and second powers need be retained; also, Cm, is small compared with a0. Hence, after integrating, we get:

Putting a0=5-73 and CJo=0-015 gives: 7'2 )=-0-944a,+V0- 892a,2 +0-0107

The lift due to the blade element is given by:

which, after integration and manipulation, neg-lecting the higher powers of JD, gives:

kT=2-3lcr0(

Book Reviews by Prof. W. J. Duncan. D.Sc, F.R.S. Travaux du Laboratoire Aerodynamique. Vol. 1,

1938. By E. Carafoli. (Imprimeria Nationala, Bucarest. No price given.) Although this volume has only just come to

hand it bears the date 1938 and it is, therefore, not surprising that most of the contents are con-siderably out of date. The book, which is written in French, begins by giving a general description of the wind tunnel at the Ecole Polytechnique at Bucarest. The general layout of the wind tunnel is very similar to that of the compressed air tunnel at the N.P.L. but the tunnel works with aif at normal atmospheric pressure and the main structure is built of wood. The exterior casing of the tunnel is in effect an independent building exposed to the weather and takes the form of a horizontal octagonal prism with truncated pyra-midal ends. The working section is l-5 metres in diameter and a driving motor of 40 kw. gives a top speed of 46 metres per second or very roughly 150 ft. per second. Results of tests on a consider-able variety of Joukowski aerofoils carried out at a Reynolds number of a little below half-a-million are given in the report. On account of this low

Reynolds number and of the fact, that the sections are not of types now used in practice, the data will be of little interest to the aircraft industry. One minor point of terminology which appears to be worthy of comment is that profit a diedre means a profile having a finite trailing edge angle, the idea being that there is a dihedral angle between the tangent planes at the trailing edge.

Theorie des Ailes Monoplanes D'Envergure Finie. By E. Carafoli. (Imprimeria Nationala, Bu-carest. No price given.) The author of this monograph, which is written

in the French language, is a Professor at the Ecole Polytechnique at Bucarest. He is well known as an aerodynamicist on the Continent, particularly in France, and has collaborated with Professor Toussaint of the Sorbonne in aero-nautical work. The monograph under review is entirely concerned with the 'Prandtl theory', or what is now usually called the 'lifting line theory', of monoplane wings. The author avowedly bases his work on-Glauert's treatment of 'lifting line

theory' but he introduces a new method in the detailed solution of the problem of the load distri-bution. The method depends on expanding an expression, which in the usual British notation

would be written - sin 6, in a finite Fourier c series in the cosines of 20 and its multiples; this device is due to Irmgard Lotz, as the author acknowledges. A number of detailed examples show that wings of all common plan forms can be adequately represented by a few terms of this series. The novelty consists in obtaining the coef-ficients in the expression for the lift distribution by solving finite difference equations by the Laplace integral. Some other topics discussed are the lift distributions across the span in horizontal circling flight, continuous rotation about an axis parallel to the span, and the effects of deflected ailerons. This publication will be of some interest to those who are concerned with the finer points of aerofoil theory, but 'lifting line theory' of the monoplane is now somewhat vieuxjcu. It is to be hoped that the author will devote his undoubted talents to 'lifting plane theory' and to problems of unsteady motion, where there are great opportunities for research.

ROTOR PERFORMANCE (Concluded from page 158) measured and estimated characteristics compare. The value of CD used (found by trial and error to give best agreement) was 0-012. The agreement is good except for k* at'Values of aR less than 8 deg. The 'hump' in the measured kT curve does not occur in the case of the N.A.C.A. tests, and since the R. and M. tests showed appreciable scale effect it is thought that the experimental curve shown may be abnormal at low values of OR, where the mutual interference between the blades is likely to be greatest.

There is no reason to doubt that the theory is equally reliable when applied to helicopter con-ditions of operation.

XIX. General Conclusions No extensive application of the charts pre-

sented has been made, but a few general con-clusions can be drawn: 1. It is very desirable to allow the blades of a

rotor to flap about two axes lying in planes at right angles, to eliminate rolling moment on the rotor as a whole and to eliminate fluctuat-ing bending moments in the'individual blades. The latter requires in addition a specific blade twist and curvature or weight distribution, and it is only possible to completely eliminate fluctuating bending moments for one parti-cular condition of flight.

A possible alternative to the lift-flapping motion would be a cyclic variation of pitch im-posed mechanically on the blades.

2. It is necessary to provide mechanical damping of the drag-flapping motion since there is very little aerodynamic damping in this plane.

3. A rotor with three or more blades is smoother in operation than one with two blades.

4. Rotor blades should have a very 'clean' finish to keep down the power required for sustenta-tion and forward flight, and to make sure that autorotation is easily maintained after engine failure.

5. To avoid poor flow conditions over the blades in the retiring positions and to keep the rotor torque low, the highest rotational tip speed, consistent with the avoidance of undesirable compressibility effects, should be used. But from the point of view of power economy a low rotational tip speed is necessary, so that a compromise must be struck on this matter.

Acknowledgment The author is indebted to the Bristol Aeroplane

Co. Ltd. for permission to publish this paper, and wishes to acknowledge the assistance given by Mr. J. N. B. Percy during the detailed develop-ment of the theory.

ADIABATIC FLOW IN PIPES To the Editor, DEAR SIR,

I should like to draw attention to the following misprints in my article on 'Adiabatic Flow in Pipes', which appeared in the February and March issues: Page 86, equation (11) should read:

Page 86, 3rd column, 22nd line: delete (12) Page 89: In order to correspond to the description

given, the four photographs of Fig. 1 la should be numbered, from top: (iii), (ii), (i) and (iv). The equation in the second line of the title of Fig. 12 should read:

1/V7= -0-8+2 log10(/teV/) Page 92, equation (A39.2) should read:

p{\ +yMz)=const. Yours faithfully,

21 Woodville Gardens, J. LUKASIEWICZ London, W.5

STRESSES I N STREAMLINE SHELLS DUE T O UNSYMMETRICAL LOADING

In the article under the above title by A. M. Binnie on pp. 125-6 of our last issue the blocks for Figs. 2 and 3 were made without notation and, unfortunately, the error was discovered too late to be corrected. The correct drawings are given below.

162 Aircraft Engineering

PROFESSIONAL PUBLICATIONS Under this heading are given each month the principal articles of aeronautical interest appearing in the current issues of the journals of the leading Professional Societies and Institutions.

The Royal Aeronautical Society JOURNAL {Monthly)

Vol. 51, No. 436, April 1947 The Development of the Spitfire and Scafire. J. Smith Development of Air Transport During the War. Air Marshal

Sir Ralph Cochrane Mechanical Vibration and Acroclasticity. P. B. Walker

The American Society of Mechanical Engineers (U.S.A.) MECHANICAL ENGINEERING {Monthly)

Vol. 69, No. 4, April 1947 Recent Developments in Gas Turbines. A. Meyer

Institute of the Aeronautical'Sciences (U.S.A.) JOURNAL OF THE AERONA UTICAL SCIENCES (Monthly)

Vol 14, No. 3, March 1947 Investigations of 24 S-T Riveted Tension Joints. R. L. Feffcr-

man and H. L. Langhaar Room Temperature Tensile Properties of Aluminium-Alloy

Sheet following Brief Elevated Temperature Exposure. J. T. Lapsley, A. E. Flanigan, W. F. Harper and J. E. Dorn

Some Ballistic Contributions to Aerodynamics. A. C. Charters Beam-columns. W. R. Osgood An Application of I.B.M. Machines to the Solution of the

Flutter Determinant. E. L. Leppert, etc.

Research Studies Directed. Toward the Development of Rational Vertical-Tail-Ioad Criteria. L. A. dousing

Aerodynamic Centre and Centre of Pressure ot an airfoil at Supersonic Speeds. H. W. Sibert

Simple Analytical Equations for the Velocity of an Airplane in Unacceleratcd Level, Climbing and Diving Flight. H. B. Freeman

Wind-Tunnel Turbulence Effects. D. P. Riabouchinsky

AERONAUTICAL ENGINEERING REVIEW (Monthly)

Vol. 6, No. 3, M.uch 1947 Civil AviationA New Economic Frontier. The Hon W. A.

Harriman The Army Air Forces Development Programme. Major-Gen.

L. C. Craigie The Engineers' Part in the Civil Aviation Expansion Pro

gramme. The Hon W. A. M, Burden