Aerodynamics (Lanchester)/Chapter 4

by Frederick William Lanchester

Chapter IV.

 

Wing Form and Motion in the Periptery.^[1]

 

§ 107. Wing Form,—Arched Section.—The most salient characteristics of wing form are common to birds of widely different species and habit of life. In spite of variations in detail and in general proportions, there is a certain uniformity of design and construction that cannot fail to impress even the most superficial observer.

The features in common may be taken, on the doctrine of natural selection, as consequent on the form of the fluid motion essential to flight, although, physically speaking, it is the fluid disturbance that depends upon the form of the wing.

To be definite, we may say that the general nature of the fluid motion can be shown to depend upon the major function of the wing, i.e., the support of the weight; the wing form must then conform to the motion so derived, and the detail of the fluid motion in turn will depend upon the more minute character of the wing form. Such indefinite process of adaptation and readaptation as the above implies is one to which the methods of evolution appear to be eminently adapted, but to which the methods of calculation are ill suited; hence much of the difficulty of the subject.

One of the most remarkable, and it may almost be said unexpected, peculiarities of wing form is the dipping front edge or arched section. This is a characteristic in the wing form of all birds capable of sustained flight, but it is only within comparatively the last few years that this feature has been the subject of observation. It is scarcely credible that so marked a peculiarity should have escaped observation for centuries, but it would seem that such is the case.

The wing section given in Fig. 57 is that of a herring gull (Larus argentatus). The dotted line gives the form as plotted from templates made shortly after the bird had been killed; the full line gives the approximate form in flight when sustaining the weight of the bird. The direction of flight is supposed horizontal.

§ 108. Historical.—Historically, so far as the author has been able to ascertain, the credit of the discovery of the dipping edge is due to Horatio Frederick Phillips, whose publication is to be found in the specification of Patent 13,768 of 1884.
Fig. 57. The discovery appears to have been made as a matter of practical experience, and, as often takes place under these circumstances, the theory given by the inventor in his specification is erroneous. Just, however, as in patent law an inventor's theory, however unsound, is not held to invalidate an invention, so in the matter of discovery, the fact that a discoverer does not fully understand the fact that he has been the first to ascertain, does not in any way detract from the credit due. In a case such as the present the fact that the discovery is based on practical experience in the face of an imperfect and in reality hostile theory adds rather than otherwise to its value.

Fig. 58 is a reproduction of the forms of wing section given (as applied to artificial flight) in the specification cited. The motion is supposed to take place from left to right, as in Fig. 57. Fig. 59 illustrates a modified form given in a further specification by the same inventor, 13,311 of 1891. These two figures show clearly the nature of the feature under discussion. In both specifications the theory given is inadequate.^[2]

The advantages of the arched form of wing section were known to the late Herr Lilienthal at the time of his experiments in flight, 1890–94, and the discovery has been attributed to him by some writers.^[3]
Fig. 58. It is possible that Lilienthal was unaware of Phillips' previous work, and that discovery by him was made independently.
Fig. 59. There is no evidence to show that Lilienthal possessed more than a practical acquaintance with the arched form.

About the same time as Lilienthal was at work the author succeeded in evolving the arched form, or dipping front edge, purely from theoretical considerations, at that time having no knowledge of the previous work of Phillips or the experiments then being conducted by Lilienthal. The author first formulated his theory in 1892, the basis being the study of the special case of an aeroplane of infinite lateral breadth. Sections of the aerofoil employed in model experiments in 1894 are given in Fig. 60.

Fig. 60. The author gave a resume of his theory in a paper read at the annual meeting of the Birmingham Natural History and Philosophical Society on June 19th, 1894, a wall diagram of which Fig. 68 is a reproduction being exhibited. A more complete account of this work formed the subject-matter of a paper offered to the Physical Society of London, but rejected (September 3rd, 1897).

In the present chapter, on wing form and the motion of the fluid in its vicinity, the main argument and demonstration are taken without substantial alteration from the rejected paper, the subsequent work being a revision of the theory on more orthodox hydrodynamic lines.

§ 109. Dynamic Support.—Endeavours have been made in the past to apply the principles of the conservation of momentum—that is, the doctrine of the continuous communication of momentum (§ 3)—to estimate directly the efficiency of an aeroplane sustaining a load and the expenditure of power necessary. If the air were a fluid discontinuous after the manner of the Newtonian medium, then such methods would lead to immediate and reliable results, for we know that if $W$ be the weight supported, and $v$ the velocity of downward discharge, and $m$ the mass per second of the projected particles,

$W=mv\quad$ or $\quad v=W/m$

(1)

and if $E=$ energy expended per second,

$E={\frac {1}{2}}\ mv^{2}={\frac {W^{2}}{2m}}$

or for any given weight to be sustained ($W=$ constant) the energy is inversely as the mass of fluid dealt with per second.

Something by way of convention is necessary to connect the above quantities with the size and velocity of the wing member. Thus if we suppose the latter to be an elastic aeroplane of area $A$ and angle $\beta ,$ travelling at a velocity $V,$ we shall have:—

$v=V\times 2\sin \beta$ (where $v$ is the velocity imparted at right angles to the plane), and $m=VA\ \rho \ \sin \beta ,$ and (1) becomes

$W=\rho \ A\ V^{2}\ 2\ sin^{2}\beta .$

If we had taken for our convention that the surface of the aeroplane is inelastic,^[4] then, since the particles on impact would not bounce off, $v=V\ \sin \beta$ and

$W=\rho \ A\ V^{2}\ \sin ^{2}\beta .$

(2)

The above results are not altogether in harmony with experience. The weight sustained does vary approximately with the area of the plane and density of the fluid, and as the square of the velocity, but the relationship in respect of angle does not hold good.

Let us introduce an elementary notion of continuity into the fluid. It is evident that when the layers of air adjacent to the aeroplane are diverted these will react on the neighbouring layers of air, and so on, so that a stratum of some considerable thickness will be involved. Now the factor that must limit the thickness of this stratum is evidently the size and shape of the plane, for the more remote layers of the fluid only escape by the fact that a circulation takes place from the side of greatest to the side of least pressure, which circulation depends chiefly upon the size and shape, and but little upon the angle of the plane. The elasticity of the air might become sensible if the velocity were sufficient, but at ordinary velocities this factor is unimportant.

Let us then assume for our convention that the depth of the layer affected for a plane of given shape depends upon its linear dimension and is constant in respect of angle, the latter being supposed to be of small magnitude. Then, since under the pre- sent supposition the lines of flow will require to follow the surfaces of the plane (the fluid being unable to bounce off as in the previous case), we have

$v=V\ \sin \beta ,\quad m=\rho \ \kappa \ A\ V,$

where $\kappa$ is a constant, and by (1) we obtain:—

$W=\rho \ \kappa \ A\ V^{2}\ \sin \beta .$

(3)

This result for planes of certain general proportions, at small angles to the line of flight, agrees closely with experiment.

The quantity $\kappa \ A$ of equation (3) may be aptly termed the sweep of the aeroplane or wing. It is a measure of the effective cross-section of the horizontal column of air dealt with by the aeroplane or supporting member. It has been found, experimenting with superposed planes,^[5] that two planes fifteen inches by four inches in pterygoid aspect, and at angles less than ten degrees, do not suffer any sensible diminution of their individual sustaining power if they are separated by a vertical distance of four inches. It is therefore fair to assume that a plane of the dimensions stated is sustained by the inertia of a layer of air not more than four inches thick. That is to say, the sweep does not exceed the area of the plane itself, or we have $\kappa$ equal to or less than unity.

By employing this in conjunction with equation (3) an outside estimate may be made of the load supported by planes of the form stated. Such estimates generally fall short of the experimental value in the relation of about one to two. By substituting a fictitious value for the “sweep” of about twice that ascertained by experiment the results of the equation can be made to agree. It is evident, therefore, that all the conditions of the problem have not so far been included in the theory.^[6]

Fig. 61.§ 110. In the Region of a Falling Plane,—Up-current.—In the foregoing discussion the subject has been treated as if the air, coming into the immediate region of an advancing aeroplane, is in a state of rest, and as if the support is wholly derived from the downward velocity imparted to it. But it has been shown that if this were actually the case the weight supported could, as a maximum, be only about one-half of that found by experiment.

Let us take the simple case of a horizontal plane supporting a weight and allowed to fall vertically. There is at first a circulation of air round the edge of the plane from the under to the upper side, forming a kind of vortex fringe (Fig. 61), the air all round the edges of the plane being in a state of rapid upward motion. If now we impress upon the plane a simultaneous horizontal motion, it is evident that the air encountered by its leading edge will be in a state of upward motion, and it would appear probable that this up-current, in front of the advancing plane, would only cease to exist when the horizontal velocity of the plane becomes equal to the velocity of sound.

But if an up-current is encountered, impinging on the advancing edge of a loaded aeroplane, the downward momentum communicated to the air will be augmented, and may be regarded as consisting of two parts, to the sum of which the sustaining force is due, i.e., the part communicated in bringing the up-current to a state of rest and the part communicated to the air as velocity downwards.

It is evident that the problem as above presented is in effect identical with that of an inclined plane moving horizontally—that is to say, the relative direction of the horizon is not of importance. The force of gravity in the one case can be substituted by the resultant of the force of gravity and an applied force of propulsion in the other.

§ 111. Dynamic Support Reconsidered.—When we consider part of the support of a body as derived from an up-current, it is necessary to examine the origin of the up-current, for it is evident that the generation of such a current must give rise to a downward reaction, and everything depends upon whether such reaction is borne by the body itself or by the deeper layers of the air, and eventually by the earth's surface.

Reverting to the case of a body supported by the communication of momentum to a number of independent material particles, it is evident that the particles projected downwards eventually give up their momentum on striking the surface of the earth. We may follow the subsequent history of the particles in two extreme cases:—

Case 1.—If the particles or the earth's surface are supposed quite inelastic, the impact is accompanied by a continual loss of energy, which is given by the expression ${\frac {m\ v^{2}}{2}}$ foot poundals per second.

Case 2.—If, on the other hand, the particles and the surface of the earth be perfectly elastic, the former will rebound with a velocity equal to that with which they strike, and the system as a whole will not lose energy. If the body be arranged to deal continually with the same set of particles, none being allowed to escape, then it may be supported without any continued expenditure of energy—that is to say, without any work being done. Such a case is exemplified in the dynamical theory of heat when a loaded piston is supported by gaseous pressure in a closed cylinder. We could also suppose it to be effected by imbuing the supported body with sufficient intelligence and skill so to direct the particles that they would always rebound within its reach.

We have already seen (§ 4) that in Case 1 the weight supported is equal in absolute units to $m\ v.$ But in Case 2 the particles impinging on the body impart as much momentum as they do in leaving it; hence the supporting force $=2\ m\ v.$

In both cases it will be observed that the projected particles act as carriers of momentum between the earth's surface and the dynamically supported body, the weight of which is eventually carried down and distributed on the surface beneath; and, moreover, we are unable to conceive of any arrangement of material particles used for dynamic support, however complex, that will not eventually transmit the stress produced by the weight of the body down to the surface of the earth. (Compare § 6.)

§ 112. Aerodynamic Support.—We may now examine and discuss the behaviour of an incompressible and frictionless (inviscid) atmosphere with respect to an aerofoil^[7] traversing it.

When a loaded aerofoil is dynamically supported by a fluid, we know that its weight is eventually sustained by the surface of the earth, and that the transmission of the stress is effected by the communication of momentum from part to part, and is thereby distributed over a considerable area as a region of increased pressure. But, as is usual in fluid dynamics, there is a certain ambiguity in the application of the principle of the continuous communication of momentum,
Fig. 62. and we as yet lack some definite statement as to the application of the principle to the case in point.

In Fig. 62 $A\ B$ represents an aerofoil, supporting weight, $W$, dynamically, under the conditions of the hypothesis. Consider a fluid prismatic column formed by imaginary vertical surfaces touching the edges of the aerofoil and continued downwards to the earth's surface and upwards indefinitely. Adopting the hypothesis that the fluid is inviscid, all forces acting on the column from the surrounding fluid must be normal to its surface, and have no vertical component. The only vertical forces acting on the column are therefore the weight of the loaded aerofoil, $W,$ acting downwards, and the pressure on the base of the column $C\ D,$ due to the distribution of the weight W on the earth's surface. Let this latter equal $w;$ there is then a downward resultant $W\ {\boldsymbol {-}}\ w$ acting on the column. (The weight of the column itself and the pressure produced thereby on $C\ D$ are obviously in equilibrium, and require no consideration.)

When the aerofoil has a horizontal motion through the fluid the conception of the prismatic column will not thereby be altered. Although its contents are constantly passing out on one side and being renewed on the other, the instantaneous condition of the forces acting is not in any way affected; the downward static resultant $W\ {\boldsymbol {-}}\ w$ remains. Consequently the downward momentum imparted per second to the fluid leaving the prism plus the upward momentum received per second from that entering must be equal to $W\ {\boldsymbol {-}}\ w.$

When the height at which the aerofoil is sustained is great in comparison with its own dimensions, the area over which the weight is distributed on the earth's surface is obviously also great, and the quantity $w$ becomes negligible. Under ordinary conditions this would usually be the case, so that the weight may be regarded as in no part statically supported. In special cases, however, $w$ may become of sensible magnitude, and it is probable that results obtained with a very large aeroplane near the surface of the earth would be found not to hold good for the same aeroplane at any considerable altitude.

§ 113. Aerodynamic Support,—Field of Force.—We have already (§ 60) learnt to regard the lines of flow of hydrodynamic theory in the light of “lines of force” and the region occupied by such lines as a “field of force.” The definition may be given as follows:—

A line of force in a fluid is defined as a line lying everywhere in the direction in which the particles of the fluid are undergoing acceleration, and in the case of a fluid initially at rest at the instant of its being set in motion the lines of force are identical with the lines of flow of mathematical theory. The whole region occupied by the lines of force is termed a field of force whose intensity is everywhere proportional to the acceleration of the particles.

In the case of the field proper to a force of stated direction applied to a given body in a quiescent fluid, it follows from considerations belonging to hydrodynamic theory that the form of the field is unique, that is to say, its geometry is absolutely defined by the conditions.

In the case of a fluid in an arbitrary state of disturbance, the field of force will not generally be of the same form as for the quiescent state. Where there is pre-existing motion in the fluid we may speak of the field as a distorted field.

The form of the field in the case of a fluid initially at rest for such forms as a sphere, an ellipsoid, or a circular or elliptical cylinder, is perfectly well known (§ 79), and in an infinite expanse of fluid extends indefinitely in every direction. If, however, the region is bounded as in the case of the atmosphere, limited by a rigid boundary constituted by the surface of the earth, the field will be modified as represented diagrammatically in Fig. 63, in which the continuous lines are the lines of force, and the dotted lines, normal to the former, are lines or surfaces of equal pressure.^[8]

It is a necessary consequence of the definition of lines of force that all lines in the immediate vicinity of a stationary boundary surface must be parallel to it, and therefore that surfaces of equal pressure, if they meet the ground, must do so normally, as indicated in Fig. 63. This figure will consequently represent diagrammatically the spreading out of the pressure area and its ultimate distribution as a region of increased pressure on the surface of the earth.

§ 114. Flight with an Evanescent Load.—We will now suppose that the aerofoil that gives rise to the field of force is in flight, that is to say, it possesses a horizontal velocity. Now we know at present very little of the nature of the disturbance created. We cannot even assert that the form of the resulting flow is geometrically similar for different load values; in fact, it will be hereafter shown that it is not.

We will in the first instance direct our attention to the case where the load is supposed to be very small indeed, so small in fact as not to be measurable in finite units, a small quantity of the second order.
Fig. 63.

In order that there shall be no ambiguity in respect of the proposed conditions, let us imagine a number of aerofoils of equal area carrying different loads that vary from some finite value down to zero, and suppose that each aerofoil is of the best form possible for deriving the support necessary from the atmosphere. Then the form of the aerial disturbance may vary in the different cases, but as the load approaches zero the aerofoil approximates more and more closely to an aeroplane, and the disturbance approximates to its evanescent form, that which we now propose to investigate. We therefore base the initial argument upon the case of an aeroplane gliding horizontally and edgewise, supporting a load smaller than can be specified infinite units.

§ 115, Aeroplane of Infinite Lateral Extent.—We have in the previous chapter become familiar with the simplification that results from the consideration of cases in which the motion takes place in two dimensions only,
Fig. 64. and with the conception of bodies of infinite lateral extent as a special case involving such a condition.

In Fig. 64, let $A$ represent the forward and $B$ the after-edge of an aeroplane extending to infinity in the direction at right angles to the plane of the paper; or, if preferred, we may consider the plane to be of finite extent, but bounded laterally by two continuous parallel walls rising vertically from the surface of the earth.

Let us examine a portion of the field $e\ e\ e$ enclosed between two adjacent lines of force, $J_{1},J_{2}.$ Then the intensity of the field in the region $e\ e\ e$ is inversely proportional to the distance between the hounding lines of force. For let $q_{1},q_{2},$ be the normal distances at any two points $G$ and $H,$ and let us suppose a small displacement to take place in the direction of the lines of force. Let this displacement at $G$ and $H$ be equal to $s_{1}$ and $s_{2}$ respectively; then, since the flux is everywhere equal, ${\frac {s_{1}}{s_{2}}}={\frac {q_{2}}{q_{1}}}.$

But the acceleration of the particles is proportional to the rate of displacement, and therefore to the displacement itself.

Hence ${\frac {{\mathit {Acceleration\ at}}\ G}{{\mathit {Acceleration\ at}}\ H}}={\frac {s_{1}}{s_{2}}}={\frac {q_{2}}{q_{1}}},$ that is, the intensity of the field is inversely proportional to the distance between the boundary lines of force.
Fig. 65.

Taking the velocity of the fluid through the field as $V,$ let $k$ be the intensity of the field (Figs. 64 and 65), where $q$ is the normal distance between two adjacent lines of force, $J_{1},J_{2}$ (so that $kq$ is constant), and let $p$ be the distance in the line of relative motion, and $\theta$ be the angle at which the path of the particle cuts the lines of force. Then the time taken by the particle to traverse the “tube of force” $J_{1},J_{2}={\frac {p}{V}},$ the momentum imparted in the direction of the lines of force $={\frac {k\ p}{V}},$ of which the vertical component $l$ is:—

${\frac {k\ p\ \sin \theta }{V}}$

But	$p=q\ {\mbox{cosec}}\ \theta$
∴	$l={\frac {k\ q\ {\mbox{cosec}}\ \theta \ \sin \theta }{V}}={\frac {k\ q}{V}},$

which is constant. Therefore, if the particle, after cutting the tube at $G$ (Fig. 64) and continuing its course, recut the same tube at $H,$ the upward momentum communicated at $G$ will be equal to the downward momentum communicated at $H.$

But a particle of fluid traversing the field of force of the aeroplane may be regarded as passing through a series of regions bounded by adjacent lines of force, to each of which the foregoing result may be applied.
Fig. 66. Consequently the upward velocity acquired in traversing the ascending field to $C_{1}$ will be given up in traversing the descending field to the medial line $P\ Q$ (the line separating the front and rear portions of the field), and the downward velocity imparted to the particle in cutting the descending field to $C_{2}$ will be given up in traversing the corresponding ascending field, so that, in respect of the vertical component of motion, the final state of the fluid will be the same as its initial state.

Again, since the conditions determining the form of the field are symmetrical, the field itself must also be symmetrical about the plane of which the medial line $P\ Q$ (Figs. 64 and 66) is the trace.

Let $\ h_{1},h_{2},h_{3},h_{4},h_{5},$ etc. (Fig. 66), be points on the path of a particle of fluid cutting $P\ Q$ at $h_{4},$ corresponding to equal intervals of time. In the elements $h_{3}\ h_{4}$ and $h_{4}\ h_{5}$ the horizontal components of the forces acting on the particle are equal and opposite; therefore the loss of horizontal velocity along $h_{3}\ h_{4}$ is equal to the gain along $h_{4}\ h_{5},$ and the horizontal velocity at $h_{3}$ is equal to that at $h_{5}$. Similarly the horizontal velocities at $h_{2}$ and $h_{6}$are equal, etc., and in general the horizontal velocity component of any particle on one side of $P\ Q$ is equal to that of the similarly situated particle on the other side. But the original state of the fluid is one of no horizontal motion.^[9] This, therefore, is also the final state.

We have consequently shown, in a system such as we have established by the present hypothesis, that the motion imparted to the fluid is eventually given up by the fluid both in respect of its vertical and horizontal components, and consequently there is no continual transmission of energy to the fluid, and no work requires to be done to maintain the motion or to support the plane. The fluid in the vicinity of the aeroplane is in a state of motion, and consequently possesses energy, but under the conditions of hypothesis the quantity is less than any assignable finite magnitude, that is to say, infinitesimal, but the motion remaining in the fluid and the continued energy expenditure are of zero value considered as infinitesimals of the same order. Therefore, adopting a method of expression common in mathematical work (but not so frequently employed in direct physical demonstration), we may say that if we take as hypothesis a small finite load, so that the actual motions of the fluid be small finite quantities, the expenditure of energy in sustaining the load will be zero, neglecting small quantities of the second order.

§ 116. Interpretation of Theory of Aeroplane of Infinite Lateral Extent.—The system of flow deduced in the foregoing article in the case of an aeroplane of infinite lateral extent in an inviscid and incompressible fluid is one that may be classified as a conservative system, the energy of the fluid motion being carried along and conserved just as is the case in wave motion. The motion round about the plane may thus be considered as a supporting wave. When the amplitude of motion becomes sensible, there is no doubt that the streamlines react on one another in a manner not accounted for by the form of field contemplated, i.e., that of the quiescent state. Under these conditions the method of investigation is not strictly applicable, but there would appear no reason to doubt the validity of the main inference. This conclusion is confirmed by a subsequent investigation conducted on different lines.

Considered in the light of wave motion, the peripteroid system must be regarded as a forced wave, the aerofoil supplying a force acting from without.

 

§ 117. Departure from Hypothesis.—Before proceeding to the further investigation, it is of interest to note briefly the consequences of a departure from the initial hypothesis.

If we suppose the aerofoil to be of finite lateral extent, it is immediately obvious that neither the lines of force nor the lines of flow can be represented by a single section through the field. The former, being no longer constrained to lie in parallel planes, diverge laterally, some portion of them escaping, as it were, and passing round the ends of the aerofoil through the regions marked o, o, o, (Fig. 67), in which R and L are the right and left-hand extremities of an aerofoil whose direction of motion is perpendicular to the paper. The fluid traversing the regions o, o, o, will have upward momentum communicated to it during the whole time that it is in those regions, and will be finally left in a state of upward motion.

Now, owing to this lateral spread of the ascending field forward of the aerofoil, the upward velocity imparted to particles in that region is less than the downward velocity imparted in the corresponding portion of the descending field, and the fluid crossing the medial line P Q (Fig. 64) will have, on the whole, a downward velocity. Similarly the downward momentum imparted by the descending portion of the field aft of P Q will be greater than the upward momentum imparted by the corresponding ascending field aft of the aerofoil. Consequently the portion of the fluid traversing the regions f, f, f, f (Fig. 67) will be ultimately left with some residual downward momentum, which must be equal to the total upward momentum received by the fluid traversing the regions o, o, o, for otherwise there would be a

Fig. 67.

continual accumulation, or else attenuation, of the fluid in the lower strata of the atmosphere, which is impossible. (This otherwise constitutes an application of the principle of no momentum of § 5.) Thus in the case of a loaded aerofoil of finite lateral extent, there is a continual loss of energy occurring, and a source of power is consequently necessary to maintain the aerofoil in horizontal flight.

In addition to the residual vertical motions of the fluid, of which the causes have just been discussed, there must also be horizontal counter-currents formed simultaneously with those in a vertical direction, the horizontal and vertical motions being the horizontal and vertical components of the actual resultant motion of the fluid. We may regard the latter as in the main consisting of two parallel cylindrical vortices, having right and left-handed rotation respectively, which are being continually formed at the flank extremities (as in Fig. 61, reading this figure as an end-on presentation), whose energy is being continually dissipated in the wake of the advancing aerofoil.

From another point of view, this loss of energy may be looked upon as a gradual spreading out and dissipation of the wave (§ 116) on the crest of which the aerofoil rides, and it becomes necessary that the aerofoil should constantly renew the diminished wave energy in order to maintain sufficient amplitude and support the given load.

The first of these conceptions, i.e., that of the vortex cylinders, is not, for a perfect fluid, compatible with hydrodynamic theory, for such vortex motion would involve rotation, and could not be generated in a perfect fluid without involving a violation of Lagrange's theorem (§ 71). In an actual fluid this objection has but little weight, owing to the influence of viscosity, and it is worthy of note that the somewhat inexact method of reasoning adopted in the foregoing demonstration seems to be peculiarly adapted, qualitatively speaking, for exploring the behaviour of real fluids, though rarely capable of giving quantitative results. The problem in three dimensions will be again examined after reviewing the subject on more rigid lines.

 

§ 118. On the Sectional Form of the Aerofoil.—We are at the present juncture in a position to draw certain elementary inferences as to the form of aerofoil appropriate to the motion of the air in its vicinity. The two aspects of form which are of most interest are, firstly, cross-section by a vertical plane in the direction of motion; and secondly, plan-form or projection on a horizontal plane.

The immediate function performed by the sectional form of the aerofoil is to receive a current of air in upward motion and impart to it a downward velocity, the whole air being dealt with possessing relatively to the aerofoil a superposed motion of translation. It would appear that any appropriate smoothly curved form, whose leading and trailing angles (Fig. 68) are conformable to the lines of flow, might be regarded as fulfilling the necessary conditions, the essential feature evidently being that neither edge shall give rise to a surface of discontinuity.

Fig. 68.

Since the amplitude of the motion may be regarded, for a fluid of given density as a function of the load on the aerofoil and its velocity of travel, the steepness of the lines of flow must also be a function of these variables, and for a given sectional form of aerofoil there is some critical velocity at which the advancing edge may be taken as conformable. When the aerofoil is supposed of infinite lateral extent, then if the sectional form be made symmetrical, at the velocity at which the leading edge becomes conformable, the trailing edge will also be conformable. If, however, the aerofoil be of finite lateral extent, we do not know what the relation ought to be between the angles $\alpha$ and $\beta$, and we have as yet no means of ascertaining same, either for any particular point in the length of the aerofoil or generally for all points. The partial solution of this problem is reserved for a later chapter. It is probable that in nature the conformability of the trailing edge is substantially ensured by the extreme flexibility of feathered construction, an incidental advantage of this method being undoubtedly an automatic adaptability to variation of velocity and load.

 

§ 119. On the Plan-form of the Aerofoil: Aspect Ratio.—In the experiments of Professor Langley and others, planes of long, narrow plan-form, in pterygoid aspect, and at moderate angles, have always been found to give a greater lifting effort, ceteris paribus, than other forms, or than the same form moving end on. The reason of this is at once evident when it is considered that the amount of the fluid traversing the regions o o o o, Fig. 67, or "stray field," is relatively much less when planes of great lateral extent are employed, and every increase in the lateral extension of the plane makes the relative loss of field still smaller, the behaviour of the plane approaching more and more nearly to the ideal case in which the conservation is complete, and the plane reaps the benefit of the whole up-current generated.

Wherever flight has been successfully achieved, advantage has been taken of the influence of aspect; the aspect ratio varies amongst birds from about 4:1 (as in the lark, also scops owl) to about 14 or 15:1 (in the albatros). The wing spread with which Lilienthal successfully experimented had an aspect ratio of about 8:1, similar proportions being adopted in gliding machines subsequently by Pilcher, Chanute, and others. The author, experimenting in 1894, successfully employed a ratio of 13:1, and Phillips in his captive flying machine, about 1893, succeeded, by his "Venetian blind" method of construction, in employing a ratio of more extreme proportion still.

 

§ 120. On Plan-form (continued): Form of Extremities.—The form of the extremities of an aerofoil exerts a considerable influence upon the dissipation of energy, irrespectively of the aspect ratio. It is evident that if, as a provisional assumption, we suppose the pressure distribution to be uniform over the whole area, the circulation will be much more rapid in the immediate vicinity of the edge than at some distance away; and since the fluid in circulation in the stray field represents energy lost, we can minimise this to a considerable extent by adopting a pointed, or acutely rounded, extremity, as in Fig. 69; so that the stray field is not contiguous to the edge of the aerofoil except in one spot at each extremity. If we neglect other factors that have weight in practice in determining wing form, and endeavour to rationalise on purely an aerodynamic basis, we can lay it down that for uniform load distribution, if we take the

Fig. 69.

extreme wing tip as origin, the form of the wing extremity will be a surface that can be generated by a straight line passing through the origin. This law may be taken as holding good for such a length of the aerofoil at each end as may be regarded as inconsiderable in comparison to the total length, and follows from the absence of any scale factor in the problem; a surface as above defined may be regarded as a segment of the surface of an irregular cone.^[10] It is possible that in a viscous fluid some departure from the form above prescribed may be anticipated from the fact demonstrated in the previous chapter, that the existence of viscosity is sufficient to give a scale to a fluid.^[11]

In practice, it will be shown later in the work, the question of wing-form, especially with regard to the extremities, is not decided by aerodynamic considerations alone, and that the question of equilibrium is involved.

It is evident that we are not bound to our assumption of uniform load distribution, and that if we suppose the pressure difference (between the under and upper surfaces) to be less towards the extremities, the latter may be made proportionately fuller without seriously disturbing the relative distribution of the stray field; we might thus take an elliptical form as a standard, with a pressure distribution appropriately proportioned. In general, the wing-plan of a bird has ordinates that approximate more or less closely to those of an ellipse. The discussion of the practical aspect of this question will be resumed in a subsequent chapter.

§ 121. Hydrodynamic Interpretation and Development.—We may recognise in the foregoing investigation (§§ 115 and 116) an elaboration of the theory initially put forward in § 90 (Chap. III), where the forces acting on the fluid were dealt with in bulk, instead of as in the present instance being studied in detail.

In § 90 it was shown that the disturbance peculiar to the neighbourhood of the aerofoil possesses angular momentum, and it was inferred that this being the case, the disturbance comprises a cyclic motion, for otherwise it must involve rotation, which is excluded by the nature of the hypothesis. We are consequently confined, in an inviscid atmosphere, strictly to the case where the aerofoil is of infinite lateral extent, for a cyclic motion is only possible in a multiply connected region.

The problem, then, from the hydrodynamic standpoint, resolves itself into the study of cyclic motion superposed on a translation. We have already devoted some attention to such a combination, and we have traced the field in a simple case for values of the functions $\psi$ and $\phi ,$ Fig. 48. In Fig. 70 we have the stream lines for this particular case plotted over a considerably greater area, the internal system of flow being replaced by a solid of substitution. We may look upon this figure as representing in section a theoretical wing-form, or aerofoil, appropriate to an inviscid fluid with its accompanying lines of flow; as such it is merely one of an infinite number of possible forms, its only virtue being that of representing the simplest possible case of peripteroid motion.

 

§ 122. Peripteroid Motion.—An infinite cylinder, of any sectional form whatever, divides infinite space into a doubly connected region, and in such a region cyclic motion becomes possible. From the hydrodynamic standpoint irregularity of contour is no detriment, as obstructing neither the cyclic motion nor that of translation. The consequence is that peripteroid motion is theoretically possible in the case of a cylinder of infinite extent, no matter what its cross-section. This conclusion applies naturally only in the case of the inviscid fluid; in a real fluid we are threatened with discontinuity. The position is analogous in every way to that of simple translation. In the inviscid fluid all bodies are of stream-line form, in real fluids only those that in their motion do not set up a discontinuity. Again, just as in the simple translation only certain simple cases are capable of solution by known analytical methods, so in peripteroid motion the cases capable of solution are very limited in number.

In order that a case of peripteroid motion should be solvable, the boundary conditions (both internal and external) must, generally speaking,^[12] be such that their lines of flow for both translation and cyclic motion are separately known. The author has succeeded in plotting the stream lines in the following cases:—

Fig. 70, a filament of infinite lateral extent in an infinite expanse of fluid.

Fig. 70.

Fig. 71.

Fig. 71, a plane of infinite lateral extent moving edgewise, may be taken as an aeroplane at evanescent angle. Fluid infinite.

Fig. 72, the same as Fig. 71, but with more powerful cyclic component, showing form of motion in greater detail.

Fig. 73, combined system due to two superposed planes, separated 1⅛ times their width. Planes and fluid infinite.

Fig. 74, elliptical cylinder of infinite lateral extent, in infinite expanse of fluid.

Fig. 75, an aeroplane of evanescent angle in vicinity of boundary surface.

 

§ 123. Energy in the Periptery.—A body in motion in a fluid is known to carry with it kinetic energy due to the fluid disturbance in addition to that due to its proper mass (§§ 81, 84). A superposed cyclic motion adds to the energy so carried. A cyclic motion around a cylinder or cylindrical filament, or round about a plane, in an infinite expanse of fluid contains an infinite quantity of energy (§ 85), and the resulting peripteroid motion for these cases will consequently require an infinite quantity of energy for its production. We must consequently regard Figs. 70, 71, 72, 73, 74 in the light of types of motion, rather than an actual form of motion that we could produce if the circumstances of hypothesis were materialised. If, however, we limit the expanse of fluid by a boundary, such as in Fig. 75, the energy of the cyclic motion immediately becomes finite, for the number of squares is limited (§ 86), so that the flow as here depicted is not open to the same objection.

The quantity of energy in the particular case given in Fig. 75 is equal to that of a body of fluid moving with the aeroplane, whose area is approximately one-seventh of that of the square on the aeroplane section.

The quantity of energy contained in peripteroid motion, and its relation to the load supported, is a matter that awaits more complete investigation.

Fig. 72.

Fig. 73.

Fig. 74.

Fig. 75.

Fig. 76.

Fig. 77.

§ 124. Modified Systems.—In the examples given in Figs. 70, 71, 72, 73, 74, there are in all cases abrupt motions of the fluid at certain points, such as could not occur in practice where a real viscous fluid such as air is concerned; the stream lines that most nearly fulfil the necessary conditions are those belonging to the elliptical cylinder (Fig. 74).

If we select from Fig. 74 a pair of stream lines possessing the requisite smoothness of curvature as the boundary of a supposed aerofoil, and, having truncated the fore and aft extremities,

Fig. 78.

proceed to whittle away the abruptness of the ends so formed (Fig. 76), we obtain a possible wing section whose form, derived entirely from theoretical considerations, bears an unmistakable resemblance to an actual section taken through the thick of the wing of one of the larger soaring birds. The whittling process is supposed carried out just as would be done in the case of a plank, originally sawn with square edges, to which it is desired to give a stream line form (Fig. 77).

A different and perhaps not quite so legitimate subterfuge is employed in Fig. 78, in which the space enclosed within the dotted line is supposed to contain uniform rotation. This requires that the load should be distributed throughout the region in question (compare § 92), a condition that could be only approximated in practice by the employment of a number of surfaces, such as indicated diagrammatically by the stouter lines shown in the figure. The form of these surfaces for the conditions stated is that of a series of concentric cylindrical sections.

Fig. 79.

 

§ 125. Peripteroid Motion in a Simply Connected Region.—The problem presented in the case of an aerofoil of finite lateral extent is, from the present standpoint, one of some difficulty, inasmuch as the region under these circumstances becomes simply connected, so that cyclic motion can no longer exist, and rotation in some form constitutes the only solution. It is, of course, conceivable that flight in an inviscid fluid is theoretically impossible.

Let us first study the case of a viscous fluid, and then, by supposing the viscosity to become less and less, endeavour to approach the conditions of the inviscid.

We have seen in § 117 that the lateral terminations of the aerofoil give rise to vortex cylinders, which trailing behind gradually dissipate their energy in the wake. Such a supposition presents no difficulty in viscous fluid, for the core of the vortex cylinders can then be formed of a mass of fluid in rotation.

Now we know that two parallel vortices, such as we have here, possessed of opposite rotation, in the first instance attract one another, and by their mutual interaction move through the fluid parallel to one another in the direction of motion of the fluid that lies between them (§ 93). Consequently in the present instance they will precess downwards as fast as they are formed, so that the aerofoil and its accompanying vortex train will appear somewhat as shown diagrammatically in elevation and plan in Fig. 79.

But if the dissipation of the vortex motion takes place sufficiently slowly, as when the viscosity of the fluid is not great, the vortices may persist until they reach the level of the ground. Under these circumstances one of two things will happen: either the vortices will spread apart as they approach the ground surface, each acting under the influence of its own "reflection" in the well known manner, or the ends of the vortices will attach themselves to the surface in the manner suggested by § 93.

If it be supposed that the aerofoil and its load were created in some upper region, and set in motion away from the earth's surface, the former assumption would be perhaps the most academically correct: if, however, we suppose the loaded aerofoil to be launched from the earth beneath, the vortices would naturally grow out from the surface, and would remain attached to the surface as they travel with the aerofoil to which they belong.

In the case of real fluids, the existence of these vortices can be traced experimentally by the employment of an aerofoil under water and inverted (Fig. 80), the pressure region being on its upper surface, and the vortices being evidenced by the dimples in the surface of the water. This experiment shows that in practice the vortices are continually breaking up and being left behind as fragmentary eddies. If the experiment is tried in a comparatively narrow vessel the eddies are actually found to have retrograde motion, owing to the influence of their own "reflexion" in the sides of the vessel. If the experiment were tried in an open expanse of water on a large scale it would probably give more perfect results.

Fig. 80. It would appear probable that in a fluid of very small viscosity vortices springing from the extremities of the aerofoil and terminating on the boundary surface may be permanent; in fact, we might regard the whole system as a single-vortex filament, with both its extremities situated on the boundary, and enclosing the aerofoil as an incident. Following out this idea, we should obtain, for an inviscid atmosphere, a system consisting primarily of a vortex hoop or halfring, loaded in the centre by the aerofoil (Fig. 81), and whose energy will be perfectly conserved, the aerofoil and its supporting vortex lying in a plane at right angles to the direction of flight. Such a system in a fluid that is truly inviscid would be uncreatable and indestructible, just as in such a fluid a vortex ring is uncreatable and indestructible. The system of static forces called into play is represented diagrammatically in Fig. 82, in which the tension due to the vortex motion is represented by an irregular polygon following the vortex core, the forces at right angles being those due, on the one hand to the load on the aerofoil, and on the other to the cyclic motion round the vortex core in translation, that is, the force that prevents an ordinary vortex ring from collapsing on to itself.

Pending the complete hydrodynamic investigation of such a system as above sketched out, it must be regarded somewhat in the light of a speculation in which there is nothing actually

Fig. 81.

improbable. The conception suggests that if we had been called into existence surrounded by an atmosphere destitute of viscosity our natural method of locomotion would have been to glide horizontally sustained on the crest of a vortex hoop, a structure}}

Fig. 82.

which from its immutability would require to be specially created at birth, and would after death continue to pervade the world for all time like a disembodied spirit.

 

§ 126. Peripteral Motion in a Real Fluid.—In dealing with a real fluid the problem becomes modified; we are no longer under the same rigid conditions as to the connectivity of the region.

The whole subject of cyclic motion in the case of a viscid fluid has not been thoroughly investigated. It is evident that to a certain extent the restrictions proper to the inviscid fluid must apply, but since we can generate rotation we are able to induce vortices with a freedom not possible when viscosity is absent.

Basing our argument on the facts as already ascertained, it is evident that if we continuously generate vortices at the right and left hand extremities of the aerofoil, as in Fig. 79, we can regard these vortices as forming in effect, taken in conjunction with the

Fig. 83.

aerofoil itself, an obstacle to connectivity, so that, although the vortex dies away after a while, it persists as long as is necessary to permit of a cyclic system being established and maintained.

It is probable that these terminal vortices do not each actually consist of a single vortex but rather of a multiple system of smaller vortices; especially should this be the case with the larger birds, and similarly for mechanical models of any size.

We can conceive that these vortices are formed after the manner indicated in Fig. 83, in which an aerofoil is represented in end elevation with the flow indicated diagrammatically. We may suppose that the air skirting the upper surface of the aerofoil has a component motion imparted towards the axis of flight,
Fig. 84. and that skirting the under surface in the opposite direction, so that when the aerofoil has passed there exists a Helmholtz surface of gyration. This surface of gyration will, owing to viscosity, break up into a number of vortex filaments or vortices after the manner shown.

§ 127. Peripteral Motion in a Real Fluid (continued).—The cyclic flow of the vortices to the right and left hand of the aerofoil finds itself superposed on the main cyclic system of the aerofoil,
Fig. 85. so that the axes of these vortices will not be parallel to the axis of flight as might be supposed, but will take up a resultant direction and may be conceived to spread out as shown in Fig. 85. The compounding of two cyclic systems into a resultant system is illustrated diagrammatically in Fig. 84, in which the circle $a\ a\ a$ represents the main cyclic system, that whose supporting reaction is concerned in sustaining the load; $b\ b$ represents the cyclic system of one of the vortex filaments, and $c\ c$ the resultant.

Representing diagrammatically the relative strengths of the cyclic systems as the sides of a parallelogram (Fig. 85), we arrive at an indication of the manner in which the vortices will spread as they are left behind by an aerodrome in flight.

Following the matter further we may represent the interaction of the vortices on each other in the manner shown in plan in Fig. 86. This figure is merely a diagram, the motion indicated being based on the known properties of vortices (§ 93).
Fig. 86. The filaments will evidently wind round one another like the strands of a rope, being involved in common in the resultant cyclic disturbance. The two vortex trunks springing respectively from the right and left hand wings, owing to their rotation being opposite, do not wind round each other but precess downwards as in Fig. 79. The motion is represented as becoming incoherent in Fig. 86, as undoubtedly must sooner or later be the case.

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1. Gr. περι and πτερόν (see footnote. Preface).
2. In his 1884 specification Mr. Phillips says:—
   “... so arranged that a current of air striking the forward edge of the blade at an acute angle is deflected upwards by the forward part of the surface, and a vacuum (or partial vacuum) is formed on the after-surface, substantially as described.”
   Further, in the patent of 1891 he writes:—
   “The particles of air struck by the convex upper surface A at the point E” (compare Fig. 59) “are deflected upwards, as indicated by the dotted lines, thereby causing a partial vacuum over the greater portion of the upper surface.”
3. See article “Aeronautics,” “Encycl. Brit.,” O. Chanute.
4. Compare “Principia,” prop, xxx., Book II.
5. Langley.
6. Compare §§ 160-1.
7. From Greek ἀέρος and φυλλον (lit. an air leaf). Compare § 128.
8. Compare § 60. Lines of equal pressure only for initial motion otherwise correspond to $\phi =$ const. of mathematical theory.
9. Relatively to the earth.
10. Compare §§ 190-192.
11. Compare §§ 36 and 56.
12. A case, such as Fig. 70, is an exception. Here neither system is known separately for a cylinder the form of the shaded section. In a case of this description, where a body is substituted for a self-contained system of flow, we have an exception to the fact stated.