Dynamics

DYNAMICS.) The equations of motion of a slightly viscous fluid such as air can be formulated, but have not in general been solved, and in no instance has the complete motion of the air passing a body, such as a wing, been calculated directly from the laws of motion. Aero nautics is thus dependent for its data mainly upon direct experi ment, and since quantitative experiments upon the full scale are difficult and costly, the bulk of the data available has been ob tained upon small models of the parts concerned suspended in ar tificially induced currents of air. The technique of the measure ment of air reactions upon models suspended in "wind tunnels," (see AERONAUTICS) as the tunnels containing these air currents are called, has reached a high development and many tunnels of great size and power have been built in all countries seriously concerned with aeronautics. The largest wind tunnel in the world is at Langley Field, Virginia; it has a cross section of 3o feet by 6o feet and is operated by two fans 35 feet 5 inches in diam eter, driven by a 4,000 horse power slip-ring induction motor. The tunnel can accommodate a moderate sized aeroplane, one, for example, that would carry two or three passengers. New tunnels in England and Germany approach the size of the one at Langley Field.

It is clearly of importance to determine how nearly similar is the manner in which the air flows around such models to the man ner in which it flows around the corresponding full scale aeroplane. In many aeronautical problems air can be treated as though it were an incompressible and inviscid fluid. In these cases it can be shown that similar bodies moving in a similar manner will generate similar air flows, no matter what the size of the body, the density of the fluid, or the speed of the motion. Even the nature of the fluid, whether liquid or gas, is of no consequence, and ex periments intended to relate to air may equally well be carried out in water, or in any other fluid which, for the purpose in hand, can be treated as incompressible and inviscid. The simple laws of aeronautics, such as that the pressure varies in proportion to the density of the air and to the square of the speed, are based upon this assumption.

Effects of Compressibility and Viscosity.

In aeronautics air can generally be treated as though it were incompressible, un less velocities exceeding some 300 or 400 miles per hour are in volved. When velocities higher than this occur, compressions and expansions of the air as it passes the solid body may seriously in fluence the flow ; the rule for similarity of flow becomes less simple. It is then necessary that corresponding velocities in the motions to be compared should all bear the same ratio to the velocity of sound in the fluid in which the motion takes place. For example, if high speed experiments in air and water are to be strictly comparable, the experiments in water must be carried out at speeds four times as great as the corresponding speeds in air, because sound travels some four times as fast in water as in air; but when both the model and the full scale body move through air the actual speed of the model must be the same as that of the full scale body. Such high speeds as 30o to 400 miles per hour are not yet commonly met in aeronautics, except in racing aeroplanes and at the tips of the blades of high speed airscrews, and hence it is unnecessary to obey this law of similarity, except in connec tion with the tests of such screws.

For fluid flows which are appreciably influenced by viscosity yet another condition must be fulfilled to ensure similarity between two motions. Letµ be the viscosity of the fluid, p its density, V the velocity of the body through the fluid, and let I be some length which defines the size of the body—for instance the diam eter, if the body be a sphere, or the length, if it be an airship.

Then the quantity must have the same value in each of the motions to be compared, otherwise viscosity will play a different part in each motion, and strict similarity will be impossible. This quantity AV lhas zero dimensions and is therefore a pure number itt or ratio, independent of the units in which its separate factors are expressed, provided that those units form a dynamically consistent system, i.e., a system of units which allows Newton's laws of mo tion to be expressed in the form :—Force = mass X accelera tion, the pound-poundal-foot-second and the gram-dyne-centi metre-second systems being examples. It is universally known as the Reynolds number, after the celebrated scientist Osborne Rey nolds. When the Reynolds number of a flow is small, viscosity will play a predominating part in the motion, which will be of the kind usually associated with the flow of oils or treacle. When the number is large, the flow will be of the kind usually associated with water, and will approximate to that of an inviscid fluid. Thus the same fluid may act in a highly viscous manner when flowing slowly past small objects, and yet behave almost as though inviscid when flowing rapidly past large bodies. The Reynolds numbers which are of interest in aeronautics are of the order of a million ; in these circumstances the air behaves almost as though it were inviscid, and the simple laws of comparison between model and full scale, to which reference has been made, can generally be applied without serious error. Certain flows, which are of great interest in aeronautics (e.g., the flow about a wing at the stalling angle which is defined later) are, however, highly critical, in the sense that small disturbances may cause an entire change in the type of flow. These critical flows may be sensitive to changes in the Reynolds number, even though the value of the number is very large, and hence the greatest caution must be used in apply ing the results of small model experiments on these critical phe nomena. The highly practical importance of some of these critical flows has led to great efforts being made to devise laboratory ex periments in which the Reynolds number will be the same as that of the full scale flow to be represented. One way in which this has been successfully accomplished, though at great cost, is to en close the whole wind tunnel in an air-tight chamber pumped up to a pressure of some 20 atmospheres; this increases p some 20 times, which compensates for the reduction in V and 1 necessary to bring the experiment within the scope of laboratory methods.