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AERONAUTICS

gave a standard anemometer which is easily maintained and reproduced and which is accepted throughout the world.

The essential parts of the anemometer are an open-ended tube facing the air current and a parallel walled tube with its axis along the wind, the walls of the tube being perforated by small holes. The open-ended tube is usually referred to as a " pitot " tube, the name being that of one of the early users, whilst the perforated tube is designed to give what is called " static pressure." If the perforations of the static pressure tube be some six diameters behind the closed end it appears that all such tubes give the same reading, independent- ly of size from a fraction of a millimetre upwards, and that the pres- sure inside the tube is the same as that on a body moving with the air stream. The pressure in the pitot tube is higher than that in the static-pressure tube and the difference, being due to the motion of the air and the stoppage of a central stream by the pitot tube, is usually referred to as " dynamic pressure " or " pitot head." The size of the pitot tube is unimportant and there is little difficulty in reproducing the standard tubes so that they agree with each other within a fraction of I %. This represents generally the order of accuracy of aerodynamic measurements, but for certain simple com- parisons of force and speed an accuracy of l /s % is attainable.

The experiments on the whirling arm at the National Physical Laboratory showed that the dynamic pressure of the anemometer was proportional to the square of the speed through the air. On physical grounds it is known that the dynamic pressure is also pro- portional to the density of the air. So long as the compressibility of the air does not enter into the effects of motion, the constant of proportionality is found to be equal to one-half, with a probable error of the order of Vio % The extreme range of speed was from a few in. per sec. to 50 ft. per second. On the principles of dynamical similarity, to be explained later, experiments at a speed of 20 ft. per sec. in water can be used to give information as to what happens at a speed of 250 ft. per sec. in air. Using the William Froude National Tank for the purpose, the dynamic pressure of the " pilot-static " tube anemometer has been calibrated to within I % up to speeds of 250 ft. per sec. in air.

Over the whole of this range the formula for dynamic pressure given by

-(I)

is an accurate representation of observations on the pitot tube anemometer. In this formula, p is the pressure in force per unit area, p the mass of unit volume, v the velocity of the air past the pitot and a the velocity of sound in undisturbed air at the place. So long as all the quantities are measured in a self-consistent set of dynamical units the equation is satisfied. The second term in the bracket will be seen to be small in comparison with the first up to speeds of 200 ft. per second. The velocity of sound being a little more than 1,000 ft. per second it will be seen that the second term is less than I % of the first within the range considered. This I % is a measure of the effect of the compressibility of air and illustrates a general rule that, for the purposes of aeronautics, air may be considered as an incompressible fluid. The statement is far from true as applied to the motion of a shell fired at usual velocities and may need modifica- tion in aeronautics when applied to airscrews. In ordinary practice the tip speed of an airscrew is upwards of 600 ft. per sec. and a few experimental forms have been made to reach tip speeds of 1 ,200 ft. per second. In the former case the effects of compressibility have not yet been disentangled from other effects, whilst in the latler some preliminary observations show marked changes of type of flow as a result of high speed and the introduction of modifications due to compressibility of the air.

Dynamical Similarity. The understanding of the laws of air motion in aeronautics and gunnery has been greatly assisted by the theory of dynamical similarity. An early formula was given by Lord Rayleigh 1 and had a marked influence on prog- ress, not only in Britain but abroad. In the later publications of the Advisory Committee for Aeronautics numerous references are made to aeronautical applications of the principle.

All the world is familiar with the idea of similarity in some form or other and there is little difficulty in appreciating the statement that human beings are similar to each other or, more accurately, are nearly similar; the horse would not be included so readily in the category of animals similar to man. The idea of dynamic similarity extends to motions what is more usually applied only to concrete bodies. Motions may be exactly similar, nearly similar, or very different, and in the case of an invisible fluid like air the eye is no guide to comparison. It is true that air may be coloured by smoke and the motion followed and that some work has been carried out on such basis. When it is found, however, that the fluid may be changed without loss of essential characteristics of the motion, a new line of attack is opened and

1 Advisory Committee for Aeronautics, 1909-10, p. 38.

the study of the motion of water or any other fluid will give the essential information. A striking experimental investigation of the reality of the law of equivalence in certain cases was made at the National Physical Laboratory. The motion of air past a square plate was observed and photographed. 2 Smoke ad- mitted to the current showed fluid impinging on the plate and spreading in the water. At a very low speed it was easy to detect a winding of the air round two axes roughly in the direction of the stream. A section of the stream across these axes would have shown particles moving in spirals winding inwards. This was a permanent state. At a higher speed a very noticeable change occurred in the type of motion. Instead of the spirals retaining a steady position, the smoke showed instability had occurred, and periodically loops formed across the two axes, broke away and travelled down stream. It is known by the principles of dynamical similarity that it is possible to produce similar flow in water. Exact conditions for the second experi- ment follow from those of the first. Further photographs 3 show that the comparison of types of flow is exact within the limits of observation. Neither of the motions described is calculable and the principle of dynamical similarity offers no assistance to understanding why an eddy occurs or what its type will be. It says, quite definitely, that if a given type of motion, eddying or otherwise, exists under certain circum- stances, there are sometimes a great number of other cir- cumstances in which the same type of motion must occur, and it lays down in precise terms the other circumstances in their relation to the given type. The instance given above re- lated to change of fluid; other changes might be those of velocity or size. Clearly the change of size covers the relation between model and full scale.

The applications of dynamics to similarity depend on fundamental theories. The common ground exists in Newton's laws of motion but superimposed on this common ground are a number of special cases. In investigating the motion of fluids at ordinary velocities, physi- cists have identified the property of viscosity; at high velocities compressibility matters and so on. The physical properties of fluids and the quantities involved in motion are expressed in terms of numerical factors and dimensions, e.g. 10 ft. per sec. means a velocity of a certain magnitude, the numerical factor 10 and the dimensions ft. and sec. being necessary to give full meaning to the idea of the particular velocity. If a complete dynamical equation be written down it must, if true, satisfy the condition that the numerical values of the two sides of the equation are equal and that, independently, the dimensions are equal. The latter point may be sufficient to give useful mathematical form to the physical ideas. For example, ima- gine an aeroplane to be gliding down through still air at some known speed. The resistance or drag will depend on its shape and size, its speed, the density of the air and the viscosity of the air. For the moment it will be assumed that the drag is dependent only on the quantities enumerated.

Force has the dimensions -- where M is the symbol for mass, L for length and T for time. Velocity, v , is represented by , density

by TJ and viscosity by -' (See footnote 4 )

Expressed in the form of an equation the assumptions so far made amount to

R =/(, l,v,*) (2)

where R is the resistance, I a typical linear dimension of the body and /a functional form which depends on the shape of the body. It is common to include in /the presentation of the body to the wind as well as its shape, but this can be excluded at will by introducing angular coordinates into the arguments of the function. The prin- ciple of dynamical similarity states that / may only have such a form as will make the dimensions of the two sides of (2) agree. For methods of finding the most general expression for/, consistent with dimensions, reference may be made to textbooks, etc. 6 ; it is found that (2) cannot have a more general form than

-(3)

1 Advisory Committee for Aeronautics, and Applied Aerody- namics, L. Bairstow.

3 Ibid.

4 The coefficient of viscosity used in dynamics is denoted by v and referred to as the " kinematic coefficient of viscosity." The other common coefficient M is related to v by the equation n = p y.

6 Applied Aerodynamics, L. Bairstow, p. 380.