Frederick William Lanchester

“ The aeroplane, thanks to its power of evading a considerable portion of the resistance due to skin friction, is capable of being utilised for the support of the load without any very great loss of efficiency. Considered thus, and compared to an aerofoil of pterygoid form, it is found to give results that are really remark- able. Experimenting on a small scale, it is difficult to construct a model with a pterygoid aerofoil that, so far as gliding angle is concerned, will perform better than a ballasted aeroplane of the most crude description. ”

“ Now the cyclic motion is in equilibrium with the reaction to which it gives rise, so that if we suppose an aerofoil supported entirely by the cyclic reaction and in equilibrium at any instant with the cyclic reaction, then if, firstly, it be supposed of infinite extent it will be in equilibrium at every other instant of time ”

“ V. of § 166) , the gliding angle is constant, neglecting body resistance, so long as the aerofoil is properly designed, and does not depend upon the velocity; we may therefore regard the gliding angle as made up of two parts: the part which is constant in respect of velocity, and the part due to body resistance, which varies as the velocity squared, The resistances, and therefore the gliding angles, may be presented in the form of a diagram (Fig. 111) , in which abscissae represent velocity and ordinates the gliding angle ”

“ It is probable that the relative advantage of the pterygoid form becomes greater when the size of the aerodrome is increased, owing to the relations of weight and area discussed in § 196, and the relatively less importance of skin friction. If this should prove to be the case the present theory would account for the remarkable difference between the flight and wing form of birds and insects, showing in detail that which was anticipated in Chap. II. (Compare § 196.) In general the wings of flies, dragon flies, moths, etc., are approximately flat—they are in fact aeroplanes ”

“ In the Eulerian fluid there is no reaction on the aerofoil possible except that due to the cyclic motion, but in a real fluid this is not the case; a reaction may always be generated and always is generated when the motion gives rise to discontinuity, whether kinetic or physical. ”

“ When we have to deal with a case of finite lateral extent we have seen that there must be a continual dissipation of the cyclic motion, which vanishes in the manufacture of the trailing vortex filaments which the aerofoil is continually shedding on either hand. ”

“ In the case of smaller birds or insects, such as the dragon-fly cited, it is an open question to what extent the problem is modified by the exigencies of active flight, and so the evidence, as confirming or otherwise the present theory, is at the best inconclusive. ”

“ If the view put forward in § 106 is correct, as to the cause of the noise made by bodies in rapid motion, then the " whirring " noise made by the wing of a bird in flight is a direct proof of the existence of discontinuity. ”

“ The question of aerofoil weight as affecting the phenomenon of flight as presented by Nature takes its place in the later portion of the work. It is only necessary here to point out that the larger birds must, in general, be more influenced than the smaller ones, just as in the case of other forms of aerodrome ”

“ Now since the angle of least resistance is the minimum gliding angle, the line will be a tangent to the curve at the point and since the least power expenditure corresponds to the slowest rate of fall, the tangent to the curve at the point will be horizontal ”

“ In Tables IV. and V. the angle is given both in circular measure and in degrees; in the case of the gliding angle Table VI. the equivalent is given in the inverse form, i.e., as a gradient. In all cases the assumption is that of the small angle as already stated.
In actual aerodrome models, owing to the necessity for organs of equilibrium the resistance is greater than that due to the considerations taken into account in the foregoing Table; there is additional resistance due to the added surface, or body resistance.
Table VII. (Aeroplane) .
Least Resistance. ”

“ It is probable that for an aerodrome the size of that chosen as an example in the preceding section the value of , is in excess of that necessary, but it is questionable whether it would be possible to construct a machine of moderate velocity capable of bearing a man, without the aerofoil considerably exceeding one-fifth of the total weight. ”

“ It is scarcely necessary to remark that the values given in the preceding Tables are not based on sufficiently reliable data to justify their being carried to so many places of decimals; the figures as tabulated have a probable error of 10 per cent, or even 20 per cent, one way or the other, and the employment of the third significant figure is only justified as a means of showing the relation of any one value to those adjacent to it in the Table.
It has not been thought necessary to re-tabulate in metric
Table XII.—Aeroplane Data. ”

“ When an aerofoil is designed according to the foregoing specification, whether as a solid as in the case in point, or as a lamina of the same mean section, the equivalent area for uniform values of and will evidently be that of a plane whose plan ordinates are those of a segment, that is, proportional to the thickness ordinates. ”

“ This method of grading, independently of the plan-form to which it is applied, ensures the proper tailing off of the load towards the extremities as set forth in § 120. It is evident that if the thickness of the stratum at different points along the aerofoil is in constant relation to its width at each point, the mass dealt with per unit length will be everywhere as the width. But for a given height of the arched section the values of and will be inversely as the width, and consequently the load sustained for any particular element of the length will be constant in respect of width ”

“ Owing to this and other causes which will be explained later, the gliding angle is never found to be as low as the theoretical value, and in the most carefully made model is usually at least 50% greater than theory would indicate. ”

“ On the value of we are entirely without information so far as direct experiment is concerned. If the value of were known for an aerofoil of given aspect ratio, the value of can be obtained from the equation given in the preceding article, i.e.,
We may provisionally assume that is a function of and constant in respect of other variables. ”

“ It is true that we have taken no account of the influence of plan-form, but we may legalise our position in this respect by specifying some standard form such as a rectangle, and leave the onus of drawing up tables of equivalent proportions in any other form to future experimenters. ”

“ Aeroplane Loads for Least Resistance.—The pressure, or load, per unit area that an aeroplane will economically sustain is considerably less than that tabulated in the preceding section for aerofoils of pterygoid form. ”

“ In the case of the aeroplane, in spite of the fact that the same values of are employed in the calculation the reverse is the case, and investigation shows that in effect the value of is considerably less in the case of an aeroplane than its ordinary value, and may amount to no more than half the coefficient as ordinarily found applicable. ”

“ Now if is a function of alone, is also a function of alone, and if we can by experiment or theory establish a form of expression in the one case, the other follows from the equation.
It is evident that the circumstances determining are foreign to our present hypothesis, and we shall require to temporarily take our stand outside this hypothesis in order to investigate the question.
Fig. 113.§ 179. An Auxiliary Hypothesis.—Let us suppose an aerofoil represented in plan in Fig. 113 supported in a continuous medium ”

“ Of the two explanations of the genesis of the vortex continuations offered here and in Chap. IV., it is not necessary that either should be in error. The previous explanation also opens out a possibility that must not be lost sight of as bearing on the phenomenon now under discussion. ”

“ Now (3) is a superadded resistance with which for the moment we will not concern ourselves, so that we have—
But we know that for Least Resistance these terms are equal, consequently under the conditions of Least Gliding Angle () we have that is to say, the least gliding angle will be
If be expressed in degrees this will require to be multiplied by
§ 175. Taking Account of Body Resistance.—The foregoing investigation has included the temporary assumption that the whole direct resistance is constituted by the skin friction of the aerofoil, as in prop. i. ”