# Aerodynamics (Lanchester)/Chapter 6

Chapter VI.
The Inclined Aeroplane.
§ 144. Introductory. Present State of Knowledge.—The problem presented by the inclined aeroplane is of very great complexity, and no general solution has at present been found. Our knowledge of the behaviour of the plane inclined to its direction of motion is in the main confined to the immediate results of experiment, extended it may be by the drawing of smooth curves through the observed points plotted on a co-ordinate chart. In certain extreme cases theoretical solutions have been found, and in other instances empirical formulae have been proposed, in fairly close agreement with the results on which they are based.
In addition to the considerations that weigh in the case of the normal plane, we have now not only to deal with some unknown law correlating pressure and angle, but we have also to take account of the remarkable effects due to the influence of aspect.
The early writers on fluid dynamics did not draw a proper distinction between an aeroplane and the surface of a solid of similar form, such for example as the wall or roof of a building; this has resulted from a too literal application of the impact theory of Newton. The pressure on a circumscribed area of the surface of a solid cannot be given by any formula, or, in fact, at all, unless the form of the remainder of the solid be known; any equation or theory that attempts to give a solution for the individual elements of the surface of a body independently of its ​whole form, is of necessity unreliable and in general entirely misleading.
An aeroplane may be regarded as the special case of a body whose whole form is defined by the shape of its face in presentation, and consequently in stated aspect its pressure reaction can be expressed as a function of its angle and velocity.
§ 145. The Sine$^2$ Law of Newton.—The difference between the behaviour of a real fluid and the Newtonian medium, sufficiently evident in the case of the normal plane, is further accentuated when the effect of inclining the plane is taken into account. Fig. 92.
According to the hypothesis of the Newtonian medium the pressure is due to the impact of the particles of which the medium is composed. In the present case it is simplest to presume, in the first instance, that the plane and particles are perfectly elastic. Let Fig. 92 represent a plane the pressure on which is due to the momentum communicated by a Newtonian medium, whose relative path is that indicated by the arrows making an angle $\beta$ with the plane itself. Then if $A$ be the area of the plane, and $V$ the velocity of the “medium” whose density is $\rho ,$ the total momentum of the stream per second $=\rho \ A\ V^{2}\sin \beta ,$ and component normal to plane, $=\rho \ A\ V^{2}\sin ^{2}\beta ,$ and on the assumption of perfect elasticity the total momentum communicated per second is:—$2\ \rho \ A\ V^{2}\sin ^{2}\beta ,$ or, if $P_{\beta }=$ pressure per unit area on the plane we have—
$P_{\beta }=2\ \rho \ V^{2}\sin ^{2}\beta .$
But for the normal plane, denoting the pressure by the ​symbol $P_{90},$ we know that for the conditions of the present hypothesis—
$P_{90}=2\ \rho \ V^{2}$ (§ 136),
or we have—
| $P_{\beta }=P_{90}\times \sin ^{2}\beta .$ | (2) |
If we modify the hypothesis to the extent of supposing the plane inelastic both $P_{\beta }$ and $P_{90}$ are diminished in the ratio 2:1, :
| or, | $P_{\beta }=\rho \ V^{2}\sin ^{2}\beta ,$ | |
| and | $P_{90}=\rho \ V^{2},$ | |
| ∴ the relation | $P_{\beta }=P_{90}\times \sin ^{2}\beta$ | still holds good. |
We may view this problem in another light with the same result. If we regard the motion of the plane as compounded of its edgewise and normal components, then the former can be neglected since it does not involve any reaction on the plane. Now if $V_{2}$ be the value of the normal component, the mass dealt with per second is $\rho \ A\ V_{1}$ and the momentum per second is $\rho \ A\ V_{1}^{2},$ or (on the elastic hypothesis),
| $P_{\beta }=2\ \rho \ V_{1}^{2}$ | (3) |
which is the same as (1), for $V_{1}^{2}=V^{2}\ \sin ^{2}\beta .$
So that the pressure in the Newtonian medium is independent of the edgewise component of motion, and is the same as for a normal plane of velocity equal to the normal component of the actual motion.
An important consequence of this is that if we had to do with a Newtonian medium, or if a real fluid behaved as such, then the time of falling of a horizontal plane would be independent of any horizontal motion impressed upon it. The “falling plane,” therefore, becomes the experimentum crucis in respect of the “sine square” law.
§ 146. The Sine$^2$ Law not in Harmony with Experience.—It has long been known that in actual fluids the sine square law does not hold good. Probably the first experimenter to ascertain this fact was Vince in the year 1797 (Phil. Trans., 1798); later we find an explicit statement by Robinson (System of Mechanical ​Philosophy, 1822), that: “The resistances do by no means vary in the ratio of the squares of the sines of the angle of incidence; and for small angles the resistances are more nearly proportional to the sines than to their squares.” This is a very important statement, from which a vast amount of inference may be drawn; it has been fully justified by the subsequent work of Wenham, Dines, Langley, and others.
The most direct disproof of the Newtonian law is to be found in experiments with the falling plane. It is found that if a horizontal plane, suitably mounted in vertical guides, be allowed to fall freely, the time of fall may be increased almost indefinitely by imparting to it a simultaneous horizontal motion. This was pointed out by Wenham in the year 1866, and has more recently been brought into prominence by the experiments of the late Professor Langley. Langley employed an appliance which he termed a “plane dropper,” mounted upon the arm of his “whirling table” (§ 233), for making his determinations. It is of interest to note that although Langley took occasion more than once to comment upon the defects of the Newtonian law, as a deduction from his other experiments, he did not apparently appreciate that the falling plane really constitutes a direct disproof.
§ 147. The Square Plane.—The nature of the Newtonian discrepancy and the extent of agreement between the work of different investigators may be exemplified in the case of the plane of square form.
The square plane may be taken as the type of greatest simplicity which includes generally planes of square proportion; such planes are not affected seriously by considerations of “aspect,” although doubtless a square plane will not give exactly the same results in diagonal as in square presentation. The latter is always assumed in the absence of an explicit statement to the contrary.
In Fig. 93, in which ordinates represent the relative pressure ​on the plane for all angles from degree 0 to 90 degrees, we have curves plotted as follows:—
| (A) | The Newtonian or sine$^2$ law— |
| $P_{\beta }=P_{90}\ \sin ^{2}\beta ;$ ||
| (B) | According to an empirical formula proposed by Duchemin |
| $P_{\beta }=P_{90}\ {\frac {2\ \sin \beta }{1+\sin ^{2}\beta }};$ ||
| (C) | From the experiments of Langley; |
| (D) | From the experiments of Dines. |
Fig. 93. It will be noticed that the Newtonian curve does not accord with any of the experimentally ascertained curves, which latter do not even agree very closely amongst themselves. Perhaps the most salient facts connected with these curves are the close agreement between Langley and Duchemin (an agreement pointed out by Langley in his Memoir), and the remarkable disagreement in the curve of Dines, characterised by an erratic “kick up,” a maximum being recorded at or about 55 degrees angle, at which point the pressure is actually greater than when the plane ​is normal. This is one of the many experimental disagreements that at present are far too common in aerodynamics.
Fig. 94. § 148. The Square Plane.—Centre of Pressure.—According to the Newtonian hypothesis, the centre of pressure on an inclined plane should be coincident with the geometric centre. In real fluids this is found not to be the case.
This point has been fully investigated, so far as the square plane is concerned, by Joessel, Kummer, and Langley. It is found that for the normal plane the geometric and pressure centres are coincident, but that as the plane is inclined the latter is displaced towards the leading edge, the displacement of the centre of pressure increasing as the angle made by the plane to its line of flight becomes less and less. This is shown in the form of a diagram in Fig. 94, in which it is supposed that the plane is swung through a quadrant, from zero to 90 degrees, the locus of its centre of pressure, as determined by the different observers, being indicated in the figure, in which also are given the position of the plane at every 10 degrees angle, and one-tenth divisions from which the position of the centre of pressure may be read in terms of the width of the plane.
The general character of the curves of the square plane, both as to magnitude and location of pressure, are shared to a greater or less extent by planes of other proportions.
§ 149. Plausibility of the Sine$^2$ Law.—The general acceptance of the experimental fact that the sine$^2$ law is in error, has without doubt been delayed by the very plausibility of the law itself.
If we suppose (as is quite customary in dealing with physical problems) that the diagonal motion of the plane is compounded of its edgewise and normal components, then, as in the previous discussion (§ 145), we may, neglecting skin friction, regard the former as of no influence and the pressure as due entirely to the normal component. In greater detail, if we suppose the motion of the plane to take place in steps, i.e., alternate edgewise and normal movements, and if we assume the former to take place with infinite rapidity, and the steps to become infinitely numerous, then it would appear that the pressure due to the inclined motion has been demonstrated to be, in effect, exactly that due to the normal component of the whole motion.
The above reasoning is manifestly in error, since the result does not accord with experience. The fallacy has been pointed ​out by Lord Rayleigh, whose explanation is substantially as follows:—
When the plane undergoes the edgewise component of its motion, it abandons air which has been set in motion (normally) at its trailing edge, and embraces air that has not been set in motion at its leading edge. This exchange obviously results in an augmentation in its resistance. This reasoning applied to the “step by step” motion evidently continues to apply when the steps become infinitely small and the motion continuous, consequently the pressure will be greater than that due to the normal component alone, as is found experimentally to be the case.
It appears to the author that the augmentation of pressure will be greater than might be supposed from the foregoing reasoning, for the abandoned air, having motion in the same direction as the plane, will impede the flow of air round the following edge and so maintain a greater pressure difference between its two faces; likewise the new air seized by the advancing edge being already in circulation round that edge has a higher velocity relatively to the plane than the normal component of motion, so that the pressure it will develop will be greater than if it had been merely new air coming into the grasp of the plane. We are now evidently touching on the subject of Chap. IV., and dealing with the pressure due to the cyclic disturbance; this aspect of the subject will be resumed later.
§ 150. The Sine-Squared Law Applicable in a Particular Case.—It is evident from the foregoing reasoning that planes of different aspect ratio will have their normal pressure components augmented to different degrees, inasmuch as the relative extent of their leading and trailing edges differ.
If we consider the case of a plane of extreme proportion it is ​obvious that in apteroid aspect the augmentation will be very small indeed, and if we go so far as to suppose the plane of infinite length, then the augmentation vanishes.
Thus the infinite parallel lamina, in apteroid aspect, affords the case of a plane that will conform to the sine$^2$ law, and the pressure on its faces will be given by the expression: $P_{\beta }=P_{90}\times \sin ^{2}\beta ,$ or in full: $P_{\beta }=C\ \rho \ V^{2}\ \sin ^{2}\beta$ where $C$ is the constant of the normal plane, or in absolute units for a plane of the form under discussion in air, $C=.76\ {\mbox{or}}\ .78$ (about).
The above result becomes self-evident from the point of view of relative motion. The conditions of the problem will be fully represented if we suppose an infinite parallel lamina in normal presentation to slide along in the direction of its own length. It is evident that such sliding motion, presuming no skin-friction, can have no effect whatever upon the pressure reaction, and therefore by § 145, the sine^ law holds good.
We might go so far as to suppose the above experiment to be tried on a “whirling table” (Chap. X.), the plane being extended to form a complete ring bounded by two concentric circles. Assuming the method to be that of the falling plane, it is evident that the time of fall of such a ring will be substantially independent of its velocity of rotation.
§ 151. Planes in Apteroid Aspect (Experimental).—In Fig. 95 we have plotted to a common maximum value: (A) the curve of sine$^2$ as deduced in the preceding article for the special case where the plane extends to infinity; (B) the Duchemin curve for the square plane, the Dines curve also being shown dotted; (E) curve as plotted by Langley for plane, 6 inches by 24 inches; (F) curve as plotted by Dines for plane, 3 inches by 48 inches. If, as there is every reason to suppose, the normal pressure is a continuous function of the aspect ratio of the plane, then as we suppose the latter to undergo variation from the square to the infinite ​lamina the curve will pass gradually from the form given by (B) to that given by (A), (E) and (F) being intermediate stages, and we may expect that the whole intermediate series will be in most part included within the area between the curves A and B, and in their character the intermediate curves will form a homogeneous series; thus a few accurate plottings from planes of known proportions would enable the curve to be drawn for any intermediate plane with a reasonable degree of certainty. Fig. 95.
The curves as plotted in Fig. 95 are to some extent misleading, each curve being plotted in terms of the common maximum ordinate. In Fig. 96 the necessary correction is made to reduce the curves to a common scale, the maximum values being assigned proper to each particular proportion of plane in accordance with Fig. 89 (Chap. V.).
In Fig. 96 abscissae represent angles of inclination as before, and ordinates give the values of the constant $C$ generalised so ​that (in absolute units) $P=C\ \rho \ V^{2}$ where $P$ is the normal pressure on the plane for any angle.
There is some doubt as to the correct plotting of the Langley curves owing to the fact that this observer was unaware of the variation to which the $C$ of the normal plane equation is subject, as dependent upon the shape of the plane. Fig. 96.
§ 152. The Infinite Lamina in Pterygoid Aspect.—The case of the inclined infinite lamina in pterygoid aspect has been examined by Kirchhoff and Rayleigh on the Helmholtz hypothesis. According to this investigation the pressure is given by the equation: $P={\frac {\pi \ \sin \beta }{4+\pi \ \sin \beta }}\ \rho \ V^{2}$ from which the curve (Fig. 97) ​is calculated and plotted. The ordinate scale is given in terms of maximum value = 1, and in terms of $C,$ value in which case becomes .440.
According to the theory advanced by the author (Chap. IV.), the case now under discussion is indeterminate; the reaction on the plane is a function of the strength of the cyclic motion and its velocity of translation, and is not dependent upon the angle in the particular case of the plane of infinite lateral extent. Fig. 97. If the plane, although of great aspect ratio, be of finite length, then a dispersal of the energy of the cyclic motion will take place, and in order that a steady state should exist this energy must be continuously renewed by the work done in proportion, which requires some specific angle in order that a stated load shall be sustained.
In a real fluid it is evident that the type of motion depicted in Figs. 71 and 75 could not exist in toto; the abruptness of the motion round the sharp edges of the plane would give rise to discontinuity. From our knowledge of problems of this kind we ​may predict the general character of the resulting flow (Fig. 98) (a), which in all probability would be accounted for by the Kirchhoff-Rayleigh analysis. There is, however, some probability that when the inclination of the plane is small the viscous drag ejects the dead-water from the region above the plane, as in the case of the stream-line body, so that the motion will be approximately as represented in Fig. 98 (b), in which a small remnant of the dead-water alone remains immediately above the front edge of the plane. The resulting type of motion from a hydrodynamic standpoint is somewhat obscure; Fig. 98. that a cyclic component exists there can be no doubt, but it is difficult to frame a régime which is in strict accord with hydrodynamic principles. It is possible that the surface of junction of the two streams, when they meet at the after edge of the plane, contains rotation, there being a finite difference between the velocities, and that this region of rotation modifies the lines of flow of the cyclic system in a manner that remains for future investigation.
§ 153. Planes in Pterygoid Aspect (Experimental).—The experimental information at present available relating to planes in pterygoid aspect is very unsatisfactory and conflicting.
In Fig. 99 we have a plotting given by Langley in the case of a plane 30 inches by 4.8 inches, with the curve for a square plane given for comparison. It was pointed out by Langley that the pressure on a plane in pterygoid aspect is greater for small angles the more extreme the proportion, Fig. 99. but that this rule does not hold good when in the comparison of any two planes the angle exceeds a certain critical value. Thus in the figure the 30 inch by 4.8 inch curve crosses the 12 inch by 12 inch curve at an angle of about 23 degrees.
Curves as plotted by Langley are not fully comparable, in view of the fact, discovered by Dines, that the shape of the normal plane affects its pressure.
In his investigations on the influence of aspect. Dines has failed to show any trace of the “reversal,” or crossing, of the ​curves, so clearly brought out by the experiments of Langley. It is possible that the form of the so-called “planes” employed by Dines is responsible for much of the disagreement. Dines employed slabs of triangular section (Fig. 100) (a), whereas Langley adopted a flat section (b), his “planes” having square edges, and being of about one-eighth inch thickness.
In Fig. 101 the curves are plotted B, B for a “plane” of square form, and (F) for one measuring 48 inches by 3 inches in pterygoid aspect, from Dines' paper (Proc. Royal Soc., Vol. 48). These curves incidentally cross one another, but there is nothing resembling Langley's reversal. Fig. 100.
In experimental aerodynamics we are used to encountering discrepancies of various kinds, but a disagreement of the present extent is most unsatisfactory. On the whole, for planes at small and moderate angles, the author is disposed to accept Langley's data as the more reliable.
§ 154. Superposed Planes.—The effect of the proximity of one aeroplane to another has been investigated experimentally by Langley. In a series of experiments carried out by the aid of his whirling table and “Plane-Dropper,” Langley showed that two parallel planes, one above the other, will, at a given angle, support as great a load as if they were entirely independent, so long as they are separated by a certain minimum distance. In the actual experiments two pairs of planes, each 15 inches by 4 inches, were employed in pterygoid aspect (Fig. 102); it was found that so long as the angle did not exceed a certain maximum value a separation of four inches (i.e. equal to the fore and aft dimension ​of the planes) was sufficient to prevent any sensible interference between the two pairs, so that each would carry the same load as if the other were absent. Trials with the planes two inches apart showed a falling-off of about 15 per cent, of the total load. Fig. 101.
It would appear that when the inclination of the plane exceeds five or six degrees, some interference is felt even at four inches separation. This is doubtless to be attributed to some change in the type of fluid motion, such as that suggested in Fig. 98; on the other hand, no increase in the velocity, and therefore diminution of the angle, is found to prevent the interference taking place when the distance apart is reduced to two inches.
Referring to these experiments Langley says:—“The most general, and perhaps the most important, conclusion to be drawn from them, appears to be that the air is sensibly disturbed under the advancing plane for only a very slight depth; so that for the ​planes four inches apart, at the average speeds, the stratum of air disturbed during its passage over it is, at any rate, less than four inches thick. In other words, the plane is sustained by the compression and elasticity of an air layer not deeper than this, which we may treat for all our present purposes as resting on a solid support less than four inches below the plane.” “(The reader is again reminded that this sustenance is also partly due to the action of the air above the plane.)” Fig. 102.
In the author's opinion the whole of this inference is unsound. Professor Langley appears to have overlooked the possibility of a superposition of the two systems of flow, such as is plotted for the Eulerian fluid in Fig. 73, Chap. IV. It is evident that such a superposition is not only possible but highly probable, each plane affecting the other profoundly so far as the actual stream-line system is concerned; but the combined supporting power of the planes, that is, the sum of the two systems, being substantially unaffected. Fig. 73 illustrates, from the case of the perfect fluid, the manner in which the two cyclic systems proper to the two planes may react on one another.
Beyond this it is evident that the speed of propagation of the compression and rarefaction within the fluid will be equal, or approximately equal, to that of sound, so that if this were really the determining factor as affecting the layer of fluid involved, ​the latter would be much greater than the observed four inches; it would, in fact, at the average velocity employed, amount to something like eight feet in either direction, that is to say, some sixteen feet in all.
The employment of a series of superposed members for the support of a load in flight was not new at the date of Langley's experiments. This system appears to have been well known to, if not actually employed by, Horatio Phillips, being foreshadowed in his specification of 1884, and further in 20,435 of 1890, and very thoroughly developed in his captive machine at Harrow about the same date. The supporting members adopted by Phillips were rightly of curvilinear section (see Fig. 60), but the critical distance of separation is evidently much the same for such a form as for a plane; at least Phillips appears to have independently adopted for his aerofoil spacing substantially the proportions subsequently proved by Langley to be admissible for the aeroplane.
§ 155. The Centre of Pressure as affected by Aspect.—The general behaviour of the centre of pressure as a function of the angle has been discussed in respect of the square plane in § 148. It remains for us to examine the subject in its relation to aspect.
So far as the author is aware, the only experimental determinations other than for the square plane are those of Kummer (Berlin, Akad. Abhandlungen, 1875–6), from which it appears that the displacement of the centre of pressure from its normal position is less in planes in apteroid aspect than in the square plane, and is greater in planes in pterygoid aspect. This is substantially what might be anticipated, for in the case of the infinite lamina in apteroid aspect the pressure distribution along its length is uniform, so that the centre of pressure for a very long plane will be sensibly undisturbed by its change of angle. On the other hand, in planes in pterygoid aspect the cyclic motion results in an increased pressure region under the leading edge, and in a partial vacuum in the region above. If the cyclic ​motion were perfect, as in Figs. 71, 72, 73, &c, (Chap. IV.), the motion of the fluid would be symmetrical, and the centre of pressure would not suffer displacement; owing, however, to the imperfection of real fluids, the pressure on the region of the following edge is not materialised, the motion becoming discontinuous, as depicted in Fig. 98, so that the centre of pressure is situated towards the forward edge of the plane.
Langley has observed that when the angle of flight exceeds a critical value, the displacement of the centre of pressure is greater for planes in apteroid than in pterygoid aspect, a reversal taking place similar to that discovered by him in the case of the total pressure reaction.
The position of the centre of pressure as a function of the inclination is of most interest in the case of planes of extreme proportion in pterygoid aspect. Under these conditions experiment is most difficult; no reliable data are at present available.
Lord Rayleigh has given the theoretical solution in the case of the infinite lamina in pterygoid aspect, on the Helmholtz hypothesis (§ 97). It is a curious fact that, when plotted, Rayleigh's curve is almost identical with that based on Langley's observations for the square plane, the departure only becoming noticeable at small angles; see Fig. 94 (L = Langley, R = Rayleigh).
§ 156. Resolution of Forces.—It is one of the advantages of the aeroplane as a medium of experiment that, if we neglect any tangential forces acting on its surfaces, the total pressure reaction, the resistance in the line of flight, and the reaction at right angles thereto, are correlated by an ordinary parallelogram of forces. Thus in Fig. 103, assuming the direction of flight to be horizontal, if $W$ be the weight supported, $R$ be the total normal reaction, and $S$ be the force of propulsion, the relative magnitude of these forces will be given by the resolution shown. Expressing $W$ and $S$ in terms ​of $R$ and the angle $\beta ,$ we have:—
$W=R\ \cos \beta \quad$and$\quad S=R\ \sin \beta .$
The quantity $S$ is in many cases very small and difficult to measure directly, so that it is usually, for small values of $\beta ,$ deduced from the normal reaction, values of which have been given in Figs. 93 and 96.
It is evident that if the form of aerofoil under investigation be other than plane no such simple relationship as the foregoing exists; the vertical and horizontal components require to be measured independently. Fig. 103.
The propriety of neglecting tangential forces has sometimes been questioned; such forces certainly cannot be neglected on the grounds of their negligibility (an error actually fallen into by Langley), and it is desirable to inquire into their exact nature in order that the consequences may be clearly understood and a correction provided.
If we had to deal with a plane devoid of thickness, so that its whole boundary surfaces might be said to lie in one plane, then there is only one possible kind of tangential force, i.e., that due to the viscosity of the fluid ; there must not only be viscosity within the fluid, but also physical continuity between the fluid and the plane itself capable of transmitting viscous stress. We could imagine this source of tangential force disposed of, either ​by making the fluid inviscid or by supposing the surfaces of the plane frictionless and not attached to the fluid in any way. Whichever be the assumption, the quantity that is being ignored is that known as “skin-friction,” the general principles relating to which have been discussed in Chap. II.
In actual planes it is impossible to do away with thickness, so that in addition to skin friction there must be the possibility of a longitudinal pressure component due to the shape of the plane. Thus, if the plane be of “fair” form, i.e., a stream-line solid based on an axis plane (Fig. 104), the pressure distribution, not being in any sense symmetrically disposed, may conceivably possess a longitudinal component of quite considerable value; or if the plane be of uniform thickness and square edges, as in the planes of Langley, Fig. 104. we have no means of computing the edge pressure resultant, for it is by no means certain that it can be represented by the resistance of the edge equivalent divested of its associations. There might, for example, in the types of motion illustrated in Fig. 98, be a region of negative pressure or suction on the front edge of the plane such as would entirely invalidate any ordinary computation.
§ 157. The Coefficient of Skin-Friction.—The hypothetical case of an aeroplane of zero thickness in edgewise motion offers the simplest possible case of skin-friction. The magnitude of the resistance due to this cause has been variously estimated, but at present is not known with any great degree of certainty. The value of skin-friction can be conveniently expressed as a coefficient, this coefficient being the resistance of a plane moving edgewise in terms of the resistance of the same plane when normal to the direction of motion. Reasoning from the facts ​known in connection with skin-friction in the case of water, we may infer that this form of resistance will vary approximately as the square of the velocity, but more accurately, proportionately to some power of the velocity rather less than the square, the index being lower than in the case of the normal plane. A consequence of this is that the “coefficient” will be greater for small planes at low velocities and less for larger planes at higher velocities.
Langley in his Memoir (“Experiments in Aerodynamics,” pp. 9 and 25), and Hiram Maxim (Century Mag., xlii., 829 and 836, 1891) have both stated explicitly that the influence of skin-friction in its relation to flight is negligible. Langley gives this result as a deduction from certain of his experiments, also as a matter of calculation based on Clerk Maxwell's value of the viscosity of air. He concludes from the latter that the frictional resistance is “less than 1/50 of one per cent, of that of the same plane moving normally,” that is to say, he arrives at a coefficient of skin-friction of less than .0002.
The author finds that Langley' s deduction in this matter is not justified by the experiments upon which it is founded, and, further, that his calculation is based upon inadequate data and is in error. The author has further shown, in Chap. VII, that skin-friction is a dominating factor in the economics of flight.
The direct measurement of skin-friction is a matter of considerable difficulty, so much so that experiments specially devised merely to detect its presence (as in the disc experiment of Dines) have proved abortive. The author, by means of experiments (described in a subsequent chapter), has succeeded in measuring approximately the value of the coefficient of skin-friction $\xi$ the following conclusions may be stated:—
$\xi =$
(2) For larger planes, .5 to 1.5 square feet area, at higher velocities (about 20 to 30 feet per second), $\xi =$ .009 to .015.
(3) A plane of about ${\tfrac {1}{2}}$ square foot area, coated with No. 2${\tfrac {1}{2}}$ (Oakey's) glass paper gave, $\xi =$ .02 (approx.).
(4) For single surfaces (as the surface of a stream-line body) the half value of $\xi$ must be employed.
The experiments upon which the above results are based were made with planes of from 3 : 1 to 4 : 1 ratio in pterygoid aspect; the values are probably lower for square planes or planes in apteroid aspect. These experiments are still in progress.
§ 158. Edge Resistance in its Relation to Skin-Friction.—There is a subtle interaction between direct edge resistance and skin friction which merits discussion. Where the plane is bounded by square cut edges, or edges of bluff form, a certain amount of direct resistance is experienced. The work done from this cause is largely employed in setting in motion the air that impinges on the leading edge of the plane, and which afterwards “washes” its two surfaces. This has for a consequence the lessening of the skin- or surface- friction, for the air in contact with the plane, having already a velocity imparted to it, does not exercise so great a viscous drag. The influence of this edge effect is comparable to the diminution of the coefficient, as the distance from the “cut-water” is increased (discovered by Froude in the case of water); here the fluid, having been set in motion by the first part of the plane, does not exercise so great a drag on the part that follows. In a plane such as we are considering the total resistance will not be the sum of the edge resistance and skin-friction separately assessed, but will be less than this amount, and may be very little greater than the one or the other of the resistances measured separately.
It is probable that for planes of less than a certain proportionate thickness the augmentation due to the edge area is imperceptible, and that for such thin planes edge effect can be ignored. Equally it is probable that for planes of rectangular section of ​more than a certain proportionate thickness the skin-friction disappears and the total resistance may be assessed as edge effect.
Amongst the former may be classified laminae of mica (such as used by the author) of an inch or a few inches breadth and 1/1000 inch or 3/1000 inch thickness. To the latter might be said to belong a plane of the proportions of a common floor-board. Probably the planes employed by Langley, about one square foot area, and of various proportions, by 1/10 inch thickness, would be intermediate, where edge effect and skin-friction give a total greater than either, but far less than their separately computed sum.
§ 159. Planes at Small Angles.—It commonly happens in physical problems that the conditions are greatly simplified when limited to the case of some particular angle being small, that is to say, within the range for which the angle (in circular measure), its sine, and its tangent, are sensibly equal to one another. In a case such as the present, where a high degree of accuracy is not important, and not attainable, such a range may be said to extend to as much as ten or fifteen degrees, and thus include practically the whole range of angle that can be usefully employed in the application of the aeroplane to aviation. It is consequently of importance to examine the extent to which simplification is possible under these restricted conditions.
It has been shown in the case of the square plane that Duchemin's formula: $P_{\beta }=P_{90}\ {\frac {2\ \sin \beta }{1+sin^{2}\beta }}$ does not greatly differ from the results of direct experiment, and we know that for small values of $\beta$ the quantity $\sin ^{2}\beta$ may be neglected, so that the expression becomes: $P_{\beta }=P_{90}\times 2\ \sin \beta ,$ or neglecting the difference between $\beta$ and $\sin \beta$ (which for 10 degrees is less than 2 per cent.), we have: $P_{\beta }=P_{90}\times 2\ \beta ,$ where $\beta$ is expressed in circular measure.
The same form of expression is found to apply to planes ​generally for small angles, though the departure from the law with increase of angle would appear to be less in the case of the square plane than for planes of elongate form, whether in pterygoid or apteroid aspect. This is rendered evident by reference to Figs. 95, 96, 99. Fig. 105. If the above form of expression held good each curve would be represented by a straight line passing through the origin. The actual curves are, in the vicinity of the origin, sensibly straight, but the departure from the straight is more marked in the case of planes of extreme proportion, and the applicability of the straight line law in this case is consequently more restricted, or subject to greater error.
Let us write the general expression for small angles in the form $P_{\beta }/P_{90}=c\ \beta ,$ where $c$ is a constant. Then the value of $c$ which determines the slope of the line when plotted, depends upon the shape and aspect of the plane, or, in the case of a rectangular plane, its aspect and aspect ratio. ​ Fig. 106.
Fig. 105 illustrates the manner in which $c$ may be plotted as a function of the aspect ratio $n$. The values of $c$ are at present not known with any pretence to accuracy; $c$ is probably different in the case of an aeroplane from what it is in the case of a pterygoid aerofoil. For the former Langley found that variations in $n$ gave rise to very considerable variations in $c$ (Fig. 99); Dines failed to discover any variation at all (Fig. 101).
The values given in Fig. 105 are “plausible values” (see Chap. VIII.) for a pterygoid aerofoil. The same data have been laid out in Fig. 106, where abscissae give angle and ordinates pressure reaction.
In addition to the equation,
${\frac {P_{\beta }}{P_{90}}}=c\ \beta$
we may also formulate as a direct consequence of the small angle hypothesis,
${\frac {W}{A}}=P_{\beta }$
and from the resolution of forces we have,
$y=\beta \ W$
where $y=$ aerodynamic resistance.
Consequently,
${\frac {W}{A}}=c\ \beta \quad P_{90}=c\ C\ \rho \ \beta \ V^{2}$
$W\propto \beta \ A\ V^{2}\quad \beta \propto {\frac {W}{A\ V^{2}}}$
and by (3) and (4),
$y=\beta \ W={\frac {W^{2}}{c\ C\ \rho \ A\ V^{2}}}$
$y\propto {\frac {W^{2}}{A\ V^{2}}}$
or for given value of $W,$
$y\propto {\frac {1}{A\ V^{2}}}$
| Work done aerodynamically per second $=y\ V,$ or $\ \propto {\frac {W^{2}}{A\ V}},$ or for given values of $W$ and $A$, Power (h.p.) $\propto {\frac {1}{V}}.$ | (6) |
We may interpret and summarise the above as follows:—
(1) The normal reaction of any given plane is proportional to its angle. The constant connecting the quantities depends upon the aspect ratio, and increases with the aspect ratio according to a law not at present known.
(2) The weight supported is sensibly equal to the normal reaction.
(3) Neglecting skin-friction and edge effect, the resistance in the line of flight varies as the angle multiplied by the weight sustained.
(4) Other things being equal, the weight supported varies as the square of the velocity.
(5) Neglecting skin-friction and edge effect, the resistance in the line of flight is directly as the square of the weight sustained and inversely as the area and the square of the velocity.
(6) Neglecting skin-friction and edge effect, the work done per unit time, i.e., the power required for a given weight sustained and a given area, varies inversely as the velocity of flight.
§ 160. The Newtonian Theory Modified; the Hypothesis of Constant “Sweep.”—In the theory of the Newtonian medium, for a given velocity the mass of fluid dealt with is proportional to the sine of the angle </math>\beta ;</math> in a real fluid it is evident that the particles cannot cross each other's paths as depicted in Fig. 92, but will be ​constrained to move in a congruent manner. Thus if one layer be supposed to strike the plane and follow its surface, the next layer will be in turn deflected and move parallel to the first, and so on. If the particles of fluid were artificially constrained so as to be unable to undergo any change of velocity along the axis of flight, or to spread laterally, this influence would be transmitted from layer to layer with undiminished amplitude, or in the case of an elastic fluid until the initial displacement had been absorbed by compression. Fig. 107. If we suppose the artificial constraint to be removed, then the amplitude rapidly diminishes as we get further from the plane owing to the longitudinal motions of the fluid particles; this may be regarded as a leakage of the fluid round the plane from the compression to the rarefaction side. (Compare Chap. IV., § 109.)
Now the facility with which the air or fluid can escape round the plane from one side to the other is evidently, for small angles at any rate, independent of the angle and dependent only on the size and shape of the plane, and for planes of elongate form it evidently depends largely upon the smaller dimension and to a less extent upon the greater. Thus in the case of a plane in pterygoid aspect the thickness of the layer affected by the passage of the plane will depend upon the dimensions of the ​latter and not upon its angle, and for a given plane the thickness of the layer will be constant.
In the foregoing paragraph the term “thickness” is used somewhat loosely. It is evident that there is no definite point at which the influence ceases altogether; and this brings us to a convention which it is found advantageous to adopt.
Let us suppose that the plane be supported by a definite stratum of air to which a uniform downward motion is imparted (Fig. 107); let us term the vertical cross-section of this stream or stratum the “sweep” of the plane and denote its downward velocity by $v$.
Then it is clear that for similar planes the sweep will bear a definite constant relation to the area $A;$ let us, as in § 109, denote the sweep by the symbol $\kappa \ A$ where $\kappa$ is a constant proper to the shape of the plane; in the case of rectangular planes a given value of $\kappa$ will correspond to some definite aspect ratio.
Now the mass of air handled per second will be $=\rho \ \kappa \ A\ V,$ and the momentum $=\rho \ \kappa \ A\ V\ v=\rho \ \kappa \ A\ V^{2}\ \sin \beta ,$ which for small angles $=\rho \ \kappa \ A\ V^{2}\ \beta ,$ where $\beta$ is in circular measure. We therefore have $P_{\beta }=\rho \ \kappa \ V^{2}\ \beta ,$ that is, ${\frac {P_{\beta }}{P_{90}}}={\frac {\rho \ \kappa \ V^{2}\ \beta }{C\ \rho \ V^{2}}}={\frac {\kappa }{C}}\ \beta$ under the conditions of the present hypothesis.
But by § 159 we know from experiment that for small angles (such as under discussion) ${\frac {P_{\beta }}{P_{90}}}=c\ \beta$ where $c$ is a constant depending upon the plane form and aspect; thus our hypothesis leads us to an expression of the correct form.
If we endeavour to deduce the constant $\kappa$ from $c$ and $C$ (constants experimentally determined and known) from the resulting equation, $\kappa =c\ C$, we obtain a value far in excess of ​that indicated by the experiment of the superposed plane (§ 154), hence it is evident that the hypothesis is insufficient.
§ 161. Extension of Hypothesis.—According to the principles laid down in Chaps. III. and IV., the neighbourhood of a plane or other aerofoil sustaining a load becomes the seat of a cyclic disturbance, and the air in advance of the aerofoil is in a state of upward motion; it has been shown that this up-current con- tributes to the supporting power of the plane or aerofoil, that is to say, its momentum contributes to the total load carried.
Let us represent this cyclic disturbance by supposing that in Fig. 107 the air stratum, instead of meeting the plane horizontally, has an upward component so that its motion (plotted relatively to the plane) be inclined at an angle $a$ (Fig. 108), so that its upward velocity will be $V\ \sin a,$ or for small angles $V\ a.$
Then the mass per second will be $\rho \ \kappa \ A\ V$ as before, and the momentum $=\rho \ \kappa \ A\ V^{2}(\alpha +\beta )$ or ${\frac {P_{\beta }}{P_{90}}}={\frac {k}{C}}(\alpha +\beta ).$
But we know by § 159 that ${\frac {P_{\beta }}{P_{90}}}=c\ \beta ,$ so that we now have the equation—
${\frac {k}{C}}(\alpha +\beta )=c\ \beta ,$ whence ${\frac {\alpha +\beta }{\beta }}={\frac {k}{C}}$
${\frac {\alpha }{\beta }}={\frac {c\ C}{k}}{\boldsymbol {-}}1.$
Thus for any given plane, $C$ and $c$ being known experimentally, and $\kappa$ being estimated from trials of superposed planes, we can calculate the equivalent up-current due to the cyclic disturbance, within the limits of the present hypothesis. This qualifying phrase is necessary because the supposed motion of the fluid, as depicted in Fig. 108, is conventional, and it is only on this conventional basis that we have effected a solution. The theory on the present lines is more fully developed in Chap. VIII., where it is made to perform useful work. The author, however, does not regard it as by any means final; the theory of the future should be based on a more comprehensive ​treatment of the whole motion of the fluid, in which the pressure reaction should appear as an integration; the present theory may be said to be based on the assumption that this integration of the whole motion of the fluid may be fairly represented on the hypothesis of a finite layer uniformly acted upon.
While pointing out the imperfect nature of the hypothesis at present adopted, it is perhaps fair to say that its defects are comparable to those of the Rankine-Froude method of dealing with the problem of propulsion, and in common with that method it may be found to perform all that is practically required. Fig. 108.
At present there are some difficulties, as will appear when the method is more fully discussed; these difficulties relate principally to the application of the somewhat unreliable data at present available—in particular, estimates of the value of $\kappa$ from existing data can be little more than guess-work, and it is questionable whether experiments conducted with merely a pair of planes are sufficient; in all probability the true value can only be obtained when a veritable screen of planes is employed.
It would appear highly probable that a separation that might be sufficient to prevent loss of pressure where two planes only are superposed would prove quite insufficient if a greater number of planes were involved, for, according to § 122 (Fig. 73), the ​individual systems of flow fuse into one greater system, and are not, as Langley supposed, independent, consequently they will each and all react on one another, and the more numerous they become the wider they will require to be separated.
A limiting width will evidently be approached asymptotically when the number of planes becomes very great, and the limiting condition is that which most nearly resembles that of our hypothesis, for the whole depth of the fluid is then acted on with approximate uniformity, and the sweep of each plane will be fairly represented by the area included between any two adjacent planes. Hence the value of $k$ deduced from experiments with pairs of superposed planes may he less than its true value according to the requirements of hypothesis.
Let an aeroplane (Fig. 109) be loaded in the manner shown so as to bring its centre of gravity to a position from a quarter to one-third of its width from one of its edges (we may assume it to be a plane of about 4 : 1 ratio in pterygoid aspect), and let it be launched in free flight with the ballasted edge leading to its line of flight. It will then be found that, provided the air is sufficiently calm, the plane will glide after the manner of a bird in passive flight, and will show itself to be possessed of complete stability.
The ballasted aeroplane in free flight may be employed for the determination of aerodynamic data as follows:—
(1) The value of c for planes of different aspect ratio in the expression, $P_{\beta }=c\ \beta \ P_{90}.$
(2) The determination and plotting of the position of the centre of pressure as a function of the angle of inclination for small angles.
(3) The determination of the value of the coefficient of skin-friction, $\xi$.
The most satisfactory results can be obtained by employing planes of mica, of only a few thousandths of an inch in thickness, the ballasting being effected by a split lead shot, as shown in the figure. Such planes show a perfection of equilibrium that appears to be unattainable with any other material than mica; it is also important that the ballast should be applied in a compact mass centrally and not distributed along the front edge. In order to improve the “sense of direction” it is found to be advantageous to “dog-ear” the front corners, slightly turning them upwards. It may be further noted that the rectangular form is very advantageous; in general other forms give inferior stability.
The theory of the equilibrium of the ballasted aeroplane belongs more correctly to the domain of aerodonetics, but the ​importance of the matter warrants its premature introduction as touching the aerodynamic aspect of the subject.
Let us suppose that the position of the centre of gravity be such as will coincide with the centre of pressure when the plane makes an angle $=\beta _{1}$ with its direction of motion. Now we know (§ 184) that the position of the centre of pressure varies as a function of $\beta$ and that its distance from the front edge of the plane diminishes the less the angle; if then the angle from any accidental cause becomes less than $\beta _{1}$ the centre of pressure will move forward in advance of the centre of gravity so that the forces acting on the plane will form a couple tending to increase the angle and so restore the condition of equilibrium. Likewise if the angle become too great the centre of pressure will recede and the resulting couple will tend to diminish the angle, and again the equilibrium is restored; thus the conditions are those of stable equilibrium, the plane tends to maintain its proper inclination to its line of flight.
There is not only equilibrium between the angle of the plane and its direction of motion as above demonstrated, but also between the gliding angle and the velocity of flight; thus if the velocity is deficient, so that the weight is insufficiently sustained, the gliding angle and the component of gravity in the line of flight automatically increase and the aerodrome undergoes acceleration. Conversely, if the velocity is excessive, the gliding angle (and so the propulsive component) diminishes, and the velocity is thereby reduced.
* ↑ A word due, in its present usage, to Langley. By aspect is meant the arrangement of the plan-form of an aeroplane, or other aerofoil, in relation to the direction of flight.
* ↑ A term used in the present work to denote the lateral dimension of an aeroplane, or other aerofoil, in terms of its fore and aft dimension, denoted by the symbol n.
* ↑ With the greater dimension arranged in the direction of flight, in contradistinction to pterygoid.
* ↑ With the lesser dimension in the direction of flight, as in the wing planform of birds.
* ↑ See also § 97.
* ↑ Compare Chap. X.
* ↑ Dines, “On Wind Pressure upon an Inclined Surface” (Proc. Royal Soc., XLVIII., p. 243.
* ↑ See foot-note, § 172.
* ↑ The curved section shown in Fig. 107 relates to the subsequent discussion (Chap. VIII.); it is perhaps easier to represent the conception on which the hypothesis of constant sweep is based by showing a curved section than a plane; in the latter case the flow has to be shown of angular path. The difference is otherwise unimportant.
* ↑ An erratum published in Volume 2 has been applied: "P. 231, line 4 from foot (also in footnote), for 'Moulliard' read 'Mouillard.'" (Wikisource contributor note)
* ↑ That so keen an observer as M. Mouillard should have fallen into so extraordinary an error is almost incredible; the following passage, however, occurs in his work: "Avant d'aller plus loin, je suis forcé d'énoncer une propriété de l'attraction sur les corps en mouvement: propriété qui est connue ou inconnue, je ne sais; mais qui en tout cas existe, c'est celle-ci: Quand un corps se meut, son centre de gravité se deplace, et se transporte en arrière du sens du mouvement." These words leave no loophole for a second interpretation, and even if they did so, the subsequent argument leaves no vestige of doubt.
* ↑ See account of author's experiments, Chap. X.
* ↑ The above explanation of the automatic stability of an aerodrome is, in a condensed form, that given by the author in his paper to the Birmingham Natural History and Philosophical Society in 1894.