Aeroplane Construction and Operation

by John B. Rathbun

There are almost an infinite number of different forms, but for the present the above examples will fill our purpose. As an example in showing how greatly the form of an object influences its resistance, we will work out the resistance of a flat plate and a spherical ended cone, both having the same presented diameter. The cone is placed so that the spherical end will face the air stream. The area A of both objects will be: 0.7854 × 2 × 2 = 3.1416 square feet. With an assumed wind velocity of 100 miles per hour, the resistance of the circular flat disc will be: R= KAV²=000282 × 3.1416 × (100×100) = 87.96 lbs. For the cone, R = KAV²=0.000222 × 3.1416 × (100 × 100) = 6.97 lbs. From this calculation it will be seen that it is advisable to surround the object with a spherical cone shaped body rather than to present the flat surface to the wind. In the above table the value of K is given for two positions of the spherical based cone, the first is with the apex toward the wind, and the second condition gives the value with the base to the wind. With the blunt end forward, the resistance is about one-half that when the pointed apex enters the air stream. This is due to the taper closing up the stream without causing turbulence.

Figs. 4a-5-6-7-8-9-10-11-12. The Values of the Resistance Co-efficient K for Different Forms and Positions of Solid Objects. Arrows Indicate the Direction of the Relative Wind. (Eiffel.)

With the apex forward there is nothing to fill up the vacuous space when the air passes over the large diameter of the base as the curve of the spherical end is too short to accomplish much in this direction.

Skin Friction. The air in rubbing over a surface experiences a frictional resistance similar to water. At the present time the accepted experiments are those of Dr. Zahm but these are still in some question as to accuracy. It was found in these experiments that there was practically no difference caused by the material of the surfaces, as long as they were equally smooth. Linen or cotton gave the same results as smooth wood or zinc as long as there was no nap or lint upon the surface. With a fuzzy surface the friction increased rapidly. This is undoubtedly due to a minute turbulence caused by the uneven surface, and hence the increase was not purely frictional, but also due to turbulence. In the tests, the air current was led parallel to the surface in such a way that only the friction could move the surface. The surface was freely suspended, and as the wind moved it edgewise, the movement was measured by a sharp pointer. End shields prevented impact of the air on the end of the test piece so that there was no error from this source. The complete formula given by Dr. Zahm is rather complicated for ordinary use, especially for those not used to mathematical computations. If Rf = resistance due to friction on one side of surface, L= length in direction of wind in feet, b = width of surface in feet, and V= velocity in feet per second, then

Rf = 0.00000778L⁰.⁹³V¹.⁸⁶b.

It will be noted that the resistance increases at a lower rate than the velocity squared, and at a less rate than the area. That is to say, that doubling the area will not double the resistance, but will be less than twice the amount. Giving the formula in terms of area and miles per hour units, we have: Rf = 0.0000167A⁰.⁹³V¹.⁸⁶. Where A = area in square feet and V = miles per hour. The area is for one side of the surface only. A rough approximation to Zahm's equation has been proposed by a writer in "Flight," the intention being to avoid the complicated formula and yet come close enough to the original for practical purposes. The latter formula reads: Rf = 0.000009V² where Rf and V are as above. Up to 40 miles per hour the results are very close to Zahm's formula, and are fairly close from 60 to 90 miles per hour. This approximation is only justified when the length in the direction of the wind is nearly equal to the length. If the length is much greater, there is a serious error introduced.

This formula is applied to surfaces parallel to the wind such as the sides of the body, rudder, stabilizer, and elevator surfaces (when in neutral). A second important feature of the friction formula is that it illustrates the law of "similitude" or the results of a change in scale and velocity, hence it outlines what we must expect when we compute a full size aeroplane from the results of a model test.

The Inclined Plane. When a flat plate is inclined with the wind, the resistance or drag will be broken up into two components, one at right angles to the air stream, and one parallel to it. If the plate is properly inclined, the right angled component can be utilized in obtaining lift as with an aeroplane wing. This is shown in Fig. 13 where L is the vertical lift force at right angles to the air stream and D is the horizontal drag acting in the direction of the wind. As in the case of the plate placed normal to the wind, there is pressure at the front of the plate and a partial vacuum behind. The resultant force will be determined by the difference in pressure between the front and the back of the plate. The forces will vary as V² since the reaction is caused by turbulent flow. Both the lift and drag will vary with the angle made with the stream, and there will be a different value for the co-efficient K for each change in the angle. The angle made with the air stream is known as the "Angle of incidence" or the "Angle of attack." The change of drag and lift does not vary at a regular rate with the angle.