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Chapter II

Elements of Aerodynamical Theory

Liquid, Fluid, and Perfect Fluid

Both liquids and fluids may be defined as substances which flow or are capable of flowing. A liquid is incompressible and therefore of constant density, a fluid is compressible and of varying density. Thus water is eommonly spoken of as a liquid, air as a fluid, yet the hard and fast distinction is un- fair, since water itself is slightly compressible.

In the transportation speeds employed in aeronanties, the variations in pressure of the air, and the consequent varia- tions in density are so slight, that the air may also be regarded as incompressible. Thus for a dirigible at a speed of 100 miles per hour the increase in pressure at the nose is only about one per cent. It is only at the tips of fast moving propeller blades that the compressibility of air assumes any importance.

The motion of fluids is so complex that no complete mathe- matical theory lias yet been evolved for it. In hydrodynamics the mathematicians have stipulated a perfect fluid possessing no viscosity. In such a fluid all bodies may move without encountering resistance. Although the conception of a perfeet flnid may seem of no practical importance, yet hydrodynamical theory serves as a guide in the theory of aeronauties and we shall have to make occasional reference to this idea.

Density of Air

In setting forth data from the laboratories the air will be assumed as having a temperature of 15° C. and a density of .07608 lbs. per cubic foot at sea level.

Variation of Density of Air with Height

Height (ft.)

Principle of Relative Motion

We shall assume throughout without further reference that the same resistances will be brought into action whether a body is moving through a fluid or a fluid is streaming past a body, provided the relative motion is the same.

This is an idea which often presents difficulties and is very difficult of theoretical demonstration, yet it is merely a matter of common sense. In La Technique Aéronautique of May 15th, 1913, M. Lecorun has given a very sound disenssion of this point. We will venture a rough illustration. Imagine a boat propelled through a river at rest at a speed of 5 miles per hour. The oars will exert a certain force of propulsion. Now if the river has a contrary current of 5 miles an hour, the boat will remain at rest relative to the banks, yet exactly the same force will be exercised by the oarsman. There is really nothing more to be grasped underlying the principle of relative motion.

Bernouilli's Theorem for Fluid Motion

In the steady flow of a fluid the current at any point is always in the same direction and magnitude and may be rep- resented by a series of stream lines, or by tubes of flow. The energy of a fluid consists of three parts: (1) The po- tential energy, or the energy due to its position of height through which it may fall, (2) The pressure energy, (3) The kinetic energy due to its motion, negleeting the effects of vis- cosity or friction. Bernouilli's theorem states that along any stream line, the sum of these energies is a constant, and if

In considering air flow in aeronautics where we deal with a fluid ocean of immense depth, the variations in height are negligible, and the theorem becomes:-

The theorem is of fundamental importance in aeronauties; its proof will be found in any text-book on hydrostaties. This equation may also be written in the following useful form, by multiplying both sides of the equation by g:

Total Energy of a Fluid Applied to the Theory of the Pitot Tube

The Pitot tube, so frequently employed in aeronautics to measure the speed of a machine in aetual flight, furnishes an excellent illustration of the principles just set forth. In Fig. 1 is given a diagram of such a tube.

Its main function is to measure the velocity of flow for

  • (p) is used for Density to prevent confusion with D for Drag

and to conform with standard usage.23