Aerodynamics

“ If, for example, we suppose a plane of infinite lateral breadth gliding edgewise through the fluid to have a force applied at right angles to the direction of motion, this force is borne immediately by the fluid, and the conditions necessary to the development of the cyclic system are fulfilled. In a perfect fluid, however, a plane can move without resistance in any aspect, and thus it is not possible to generate a difference of pressure between its two sides except for the period whilst the normal component of its velocity is undergoing acceleration. ”

“ Taking first the case of a normal plane travelling at a velocity greater than that of the particles, we have the resistance proportional to the energy per unit volume (§ 131) , the energy being reckoned only in respect of motion in the direction of flight, of either plus or minus sign. This energy is made up of two parts, the corpuscular energy of the medium, of which one third only counts as being in the direction of the axis of flight, and the energy of translation. ”

“ The kinetic energy in the field of flow round a body in motion is imparted to the fluid when the body is started from rest, and is given up when the motion is arrested. The effect of the fluid motion is thus to add to the apparent inertia of the body, so that a given force requires to act through a greater distance to impart a given velocity than for the same body in vacuo. Not only has a force to act for a greater distance, but also for a longer time, which means that the body possesses in effect a greater store of momentum for a given velocity. ”

“ When, however, the principle is applied to bodies aerodynamically supported in the free atmosphere the matter is not so self-evident; here, for example, we find that the weight of a parachutist is borne by the earth's surface almost from the moment he leaves the car, and his presence overhead, or the presence of a passing flight of birds, could be detected barometrically if we possessed an instrument of sufficient delicacy. ”

“ The deficiency in the present experiments is chiefly that of apparatus and opportunity. The launching of free flight models requires a suitable apparatus to be designed, by which the initial velocity shall be placed under definite control; beyond this it must be considered quite essential, if reliable results are required, that experiments should be conducted inside a building; the absolute calm necessary for aerodynamic determinations is so rare a phenomenon as to render outdoor experiment almost impossible. ”

“ 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. ”

“ 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. ”

“ The aerodynamic balance, Fig. 153, consists of a horizontal arm or beam pivoted about a vertical axis the amplitude of motion permitted being regulated by the screws which also form electrical contacts.
For the determination of a normal plane is attached to one end of the beam and a friction plane to the other, the areas of the two being adjusted until they exactly balance. The instrument is used either by being exposed to the wind and held stationary, or fitted in front of an automobile vehicle in still air. ”

“ It may be noticed that this result is in appearance out of harmony with prop, i., for there it was shown that the most favourable velocity at which to run an aerodrome in order to cover the greatest distance on a given quantity of energy is that at which the aerodynamic resistance is equal to the total direct resistance, that is whereas according to the present proposition the most economical conditions are met with when which is only a portion of the total. ”

“ It is probable that the difference in the behaviour of air or any other gas, and the medium, will be least at very high and very low velocities; at intermediate velocities the present mode of treatment is unlikely to be of any utility. ”

“ Author's Experiments. Method (continued) .—In addition to the free flight experiments enumerated above, an attempt has been made to effect the direct measurement of by means of a new instrument, which may be appropriately termed an aerodynamic balance. [6]
The magnitude of as determined by the method of free flight suggested that, in spite of the failure of previous experimenters, it should be quite possible to effect a direct measurement of this quantity by the aid of a suitably designed appliance. ”

“ Dines says:—“It had been assumed in the above that the wind pressure varies as the square of the velocity. The experiments have proved this to be the case, for when upon a calm day equilibrium for any plate is once attained, it has been found impossible to disturb it by any alteration of the velocity of rotation, and since the centrifugal force varies as the square of the velocity the wind pressure must do so also. For the smaller planes the maximum velocity of which the machine is capable is about seventy miles per hour.” ”

“ A physical conception of velocity potential under these circumstances is somewhat difficult, but if we revert to the dynamical hypothesis and regard the velocity potential system as the pressure system by which the motion is generated, we encounter at once the same difficulty in another form. Before we are able to interfere either to start or to stop the fluid in cyclic motion, we must introduce some imaginary barrier in its path. ”

“ 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. ”

“ If, in the case of a body propelled at a constant velocity, the entire run be removed, as in § 23, the consequence is a surface of discontinuity emanating from the periphery of section. Under these circumstances, if we neglect the influence of viscosity and the consequent loss of wake pressure, the work done appears wholly in the counterwake current, on the production of which energy is being continuously expended. ”

“ III.—The constancy of demonstrated in the present proposition has for an immediate consequence the constancy of the gliding angle (if be ignored) , that is to say, the thrust required to maintain an aerodrome in flight will be constant for a given value of and if this thrust be supplied by a component of gravity (Fig. 110) , and if be the angle of descent, we have:—
that is to say, is constant. If we take account of the body resistance , we shall find that the value of will increase the higher the velocity. ”

“ In reality, however, such increase of momentum is only apparent; the momentum of the body and fluid system combined is actually less than that of the body at the same velocity in vacuo by the amount due to its fluid displacement. That is to say, if the body be of the same specific gravity as the fluid the total dynamic system possesses no momentum at all, whatever the velocity. ”

“ If, when the velocity V increases, we suppose that the layer of fluid affected to any given degree becomes thinner, and vice versa, it is clear that the viscous forces will rise in a greater ratio than directly as V, for the velocity gradient will be steeper, also the inertia forces will be less, for the mass of fluid to be set in motion will be less. It is, therefore, evident that we may suppose the thickness of the affected strata varies with the velocity in the degree necessary to preserve a balance between the viscous and inertia forces. ”

“ 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. ”

“ The Hydrodynamic Standpoint.—Let us revert to the Hydrodynamic aspect of the subject as expounded in Chap. IV. We have seen that the supporting reaction is due to a cyclic motion in the fluid which is maintained by, and is in equilibrium with, the load on the aerofoil; it is of course understood that there is a superposed motion of translation, i.e., the motion of flight. Now in the case of an aerofoil of infinite lateral breadth we have seen that this equilibrium is permanent, and we have in
Fig. ”

“ The leaping mode is, in fact, a means of adjustment, by means of which the conditions of least resistance can be approximated under considerable variations of velocity. If one of the larger birds, with its limited relative area, were to force its velocity up to the point at which leaping flight would pay, it would require an amount of energy per second far beyond its actual horse-power capacity. ”

“ Another standpoint from which we may view the present problem is that the turbulence, by effecting a rapid transference of momentum from one part of the fluid to another, acts in effect to augment the apparent viscosity, and in this way adds to the pressure reaction. In any case experiments made on a fixed plane or other body in moving air, cannot be regarded as valid when the conditions are reversed. ”

“ 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
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 becomes negligible. ”

“ 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. ”

“ It is only those who have watched and waited for a really calm day who can fully appreciate its rarity. In repeating these experiments it would be well to arrange for the use of a large hall, well secured against draughts; the equilibrium of low velocity models, such as it is necessary to employ, is very sensitive; even the previous motion of a person across the line of flight will affect the gliding path. High velocity models, although possessing greater stability, are not well suited to the determination of aerodynamic data. ”

“ That is to say, that a model aerodrome, made to a th scale and adapted for motion under water, will, at a velocity proportioned to the square root of its linear dimension, that is th the full scale velocity, give rise to a geometrically similar disturbance in the fluid, and will itself undergo geometrically similar disturbance, and density for density the resistance will be proportional to the cube of the linear dimension—that is to say, in the ratio of of the full scale model ”

“ The validity of the above reasoning is unquestionable if we are dealing with a fluid whose properties are those of the Newtonian medium; the energy of turbulence being represented by a variation in the individual velocities of the particles such as will not affect their mean velocity. ”

“ Under these conditions the greater the energy of the vortex system set up in the fluid the lower will become the pressure in the internal part of the region, so that the plus and minus momentum of the equal and opposite flow taking place across any imaginary barrier plane is accounted for by the ordinary static pressure on the confines of the region, and does not give rise to any added pressure. ”

“ As a maximum estimate of the range of flight conceivably possible without some fundamental discovery in fuel and prime movers we may take the following supposititious case. Using liquid hydrogen as fuel, and carrying 25 per cent, of the total mean weight, and assuming a yet-unheard-of thermal efficiency of 50 per cent., a total mechanical efficiency of 90 per cent., and a propeller efficiency of 70 per cent., with the minimum angle of given in the Table (= 6°) , the exhaustion of fuel will be complete after a flight of 6,800 miles distance. ”

“ If the turbulence contain only one harmonic component, or if, which amounts to the same thing, the bird is only able to take advantage of the harmonic component in the line of flight, the available energy for a given maximum velocity will be only half that on the basis of circular motion ”

“ In the case of a body of streamline form, the motion can be observed much closer to the axis of flight than is the case for a sphere or other bluff form; also when the movement is complete nothing further happens. In the case of a sphere, the looked-for movement duly takes place; but immediately after the whole of the fluid under observation is involved in a state of seething turbulence, where the wake and counterwake currents are mingling. ”