It is possible that a correcting factor will be introduced into the
design of airscrews to allow for compressibility of the air. In such
a case, resistance coefficients based on (7) would provide the first
approximation to a rational formula.
Tests of the Water Resistance of Flying-Boat Hulls. Applications of dynamical similarity extend over the whole range of physics and an exhaustive discussion would lead far away from aeronautics. One other illustration is required to show the origin of the law of corre- sponding speeds applied in naval architecture to surface-moving craft. Experimentally it has been found that the resistance of sur- face craft at high speeds depends greatly on the generation of waves. If attention be concentrated on this new aspect of resistance it will be found by methods already indicated to give the law of corre- sponding speeds associated with the name of Froude.
At any point of a wetted surface the pressure is proportional to the head of water above that point and will be increased if a wave crest exists in the neighbourhood. The pressure depends on the head and on the weight of unit volume of the water; alternatively the weight may be expressed as the product of the density of the water and the acceleration due to gravity. Now consider the problem of similar motions between a ship and a model of it. The scale of the model must apply to the scale of the waves if similarity is to exist. It can be said therefore that the resistance depends on a linear dimen- sion I, velocity of test v, density of water p and the acceleration due to gravity g. The appropriate formula then follows and proves to be
-(8).
The law of corresponding speeds is therefore -j = constant.
When dealing with comparisons of motion on the earth's surface, g is constant and the law states that the speed of test for the model varies as the square root of the scale. This condition ensures that the waves in model and full-scale trials shall be similar. Equation (8) may apply in other cases, such as the disturbed motion of model and actual aeroplanes in free flight, the governing factor being the de- pendence of the motion on gravitational attraction.
Summary of the Aeronautical Uses of Dynamical Similarity. In measurements of resistance to the motion of a body through viscous
fluid the correct law of corresponding speeds is that = constant;
this is applicable so long as the velocity of motion is not more than about one-quarter that of sound. At higher velocities, compressibility of the fluid modifies the flow, the changes depending on a further
factor , i.e. on the ratio of the velocity of the body through the
a fluid to that of sound in the fluid.
If the wave-making resistance alone be considered the law of cor-
v 1 responding speeds for terrestrial surface craft is y = constant ;
where resistance depends partly on wave-making and partly on viscosity it is generally assumed that the two can be treated by special assumptions. A very accurate method of treatment of the complex problem does not lead to practicable formulae.
The Resistances of Bodies of Various Shapes. A somewhat sharp division exists between the resistances of wings and aerofoils and those other bodies with which aeronautics is concerned. In the latter cases the resulting air force is either directly opposed to the motion or is little inclined to it. In the case of wings at the most efficient angle of presentation the resultant force is almost normal to the direction of motion. Since there is always a real drag the direction of the resultant force must fall behind the normal but the amount may be less than three degrees.
It has been found experimentally that all aeroplane wings whatever their variations of shape have certain common charac- teristics. The best ratio of lift to drag is obtained only at a particular angle of attack of the wing to the air and a considerable loss of effi- ciency is incurred if, as is usual in aeroplanes, departure from this
FIG. 173. Flow past wing.
8. Below critical angle.
angle to the extent of 5 or 6 be permitted. At the highest speed of flight of the aeroplane of 1921 it is improbable that the lift exceeds 12 times the drag, whilst the maximum ratio exceeds twenty.
Apart from efficiency there is a limit to the greatest force which can be obtained at a given speed by a wing of finite area. Omitting very special complex wings for the moment, the limiting force at any given speed is obtained when the wing is inclined at 15 or 20 to the wind. One of the most efficient types of wing form for high-speed
FIG. i?b. Flow past wing.
20. Below critical angle.
flight has a limiting lift of about 7 Ib. per sq. ft. at a speed of 50 m. per hour. Other forms of fixed section are known which give 12 Ib. per sq. ft. at the same speed. The general experience of all experi- menters with aerofoils has been that, so long as the shape of the sec- tion is invariable, high loading at the angle of maximum lift cannot be obtained at the same time as high efficiency for maximum speed.
Much attention has been paid therefore to flexible and variable wings; if it were possible to vary the area of a wing at will without introducing unreliable mechanism or adding greatly to the weight of the wing structure that solution would offer the maximum aero- dynamic advantages. It should be pointed out here, that the addi- tion of weight to an aeroplane in such a place as not to add directly to the resistance leads to an immediate and calculable indirect in- crease of resistance at a given angle of incidence; the amount may be estimated as about one-eighth of the weight under favourable condi- tions. So far no satisfactory proposals exist for the mechanical variation of the area of the wings of an aeroplane. More practical success has met the endeavours to vary the section of a wing of given size so as to obtain the advantages of high lift and consequent low speed for alighting and high efficiency at flying speeds. It has already been shown that either condition may be obtained by a wing of fixed section. A further general observation is that the high-speed wing is thin and flat whilst the high-lift wing is thick and greatly curved. Means of constructing flexible ribs for wings to admit of continuous change from one shape to another have been developed and the me- chanical difficulties do not appear to be insuperable. A less obvious method of attack has shown greater promise. Mr. Handley Page 1 found by experiments in a wind tunnel that the properties of high lift could be obtained by allowing air to pass through the front part of a wing from the lower to the upper side. By dividing the wing of an aeroplane into a small aerofoil hinged at its leading edge and a large main wing the device becomes both mechanically and aero- dynamically effective.
The motion of an aeroplane is now realized to be dominated by other considerations than those of lift and drag and it may be that a particular high-lift wing would be useless because it led to failure of lateral control at low speeds. This point is of growing importance and aeroplane design can no longer ignore the complex interactions of aerodynamic properties. For this reason it may be anticipated that the full advantages from variable wings will not be obtained im- mediately but that the processes of evolution will be followed. Past history has been simpler; early experiments by Langley (1896) covered the properties of flat plates used as aerofoils and laid the general foundation of practical aviation. Lilienthal later showed that curved surfaces were more efficient than flat ones and attention was given to sections suggested by bird wings, a subject of interest still occupying the minds of designers. With little guidance as to good forms, the early pioneers of night, Wilbur and Orville Wright, Far- man, Bleriot and others, introduced wing sections in the period 1906- 10 and on these Eiffel based his first series of experiments. 2 Design then began to be regularized. One of the more promising wing sec- tions examined by Eiffel in his wind tunnel at the Champs de Mars, designated "Bleriot II bis," was adopted by the Royal Aircraft Factory for the BE2A. In 1911, thje National Physical Laboratory adopted this form as the starting-point for systematic variation of wing form. In the series of experiments which followed, 3 the thick- ness of the wing was changed, also its shape on upper and lower sur- faces and the bluntness of the nose, and in each case measurements of lift and drag were made. From this series it was possible to make a rational choice of wing section to fit the conditions of the day. The absolute maximum of aerodynamic efficiency demanded a wing too
1 Jour. Royal Aeronautical Society, 1920.
2 Resistance de I'atr el I' Aviation, 1910-1. 8 A.C.A., 1911-2, pp. 73-77.
