THE PROBLEM OF SHAPING A SURFACE to some desired contour occupies amateurs in many fields. Where the basic design is firmly established, as in telescope making, the problem is mainly one of craftsmanship. It may not be easy to make a telescope mirror of high precision, but the maker knows in advance exactly what shape he must try to achieve. In what follows we are going to consider a type of project in which the object is to discover the best or most efficient shape.

This problem applies whenever one sets out to build a racing sloop, an airplane or even a kite. Any boy can make a kite that will climb a hundred feet or so. But to design one which will fly higher than all competitors with a given length of string is a challenge of astonishing sophistication. The performance of a kite, like that of a glider, an airplane or any other vessel in a fluid medium, depends critically on the shapes of its working parts.

Two types of force are of prime interest to those who must design hydrodynamic and aerodynamic shapes. One is summed up in the aerodynamic term "lift." Here the major effect acts at right angles to the motion of the fluid. It accounts for the rise of a kite or an airplane and for the ability of a sailboat to tack against the wind. Generally shapes which create forces perpendicular to the streamlines manage in one way or another to cause the fluid (air, steam or water) to flow at greater velocities on side of the object than on the other. This differential motion causes a drop in pressure on the side of higher velocity, and the object tends to move in that direction. If you hold a sheet of paper horizontally and blow over its upper surface, the sheet will not bend down but will jump up momentarily, because of the reduction of air pressure on top. The effect explains why gusts of wind sometimes lift the roofs off houses, why ships traveling close together may be sucked together and why it is dangerous to stand near the edge of a railway form when a train speeds by.

The second interesting force, exerted in the direction of the flow, is called "drag." The air drag on the tail of an ordinary diamond-shaped kite is partly responsible for keeping it heading into the wind. A box kite will rise higher both because it is lighter and because there is less drag on its surfaces. The force of drag is due to friction between adjacent air layers of different speeds around the airfoil. Anything that promotes turbulence in the air layers increases drag. At a speed of 200 miles per hour an airplane's wings with protruding rivet heads and lap joints show almost 50 per cent more drag than when the wings are perfectly smooth.

An impressive array of devices, ranging from mile-long towing basins to yard-long shock tubes, has been devised investigating the forces of fluid flow. Most of them are beyond an amateur's resources. But Francis W. Niedenfuhr, an instructor in engineering mechanics at Ohio State University, describes for this department an ingeniously simple apparatus which anyone can make to study fluid dynamics. He writes:

"The letter from J. J. Cornish in your May issue about his aerodynamic smoke tunnel brought to mind another series of experiments which can be performed by amateurs interested in either hydrodynamics or aerodynamics. I refer to the Hele-Shaw apparatus, which enables you to see fluid flow and to measure the resulting forces in two dimensions.

"Sir George Stokes first proved that when a thin layer of viscous fluid such as water is made to flow between a pair of parallel plates, the streamlines approximate those of a theoretical nonviscous liquid. Ludwig Prandtl subsequently pointed out that air behaves as a nonviscous fluid outside the boundary layer. H. J. S. Hele-Shaw constructed an apparatus based on Stokes's observations and made useful measurements concerning the flow around ship hulls. Although the proofs presented by Stokes apply to flow between two parallel plates, Hele-Shaw found that good results can be obtained by letting water flow in a thin sheet over one plate, leaving the top surface of the water free. The device is useful for investigating the performance of airfoils, because the air flow over a long narrow wing is essentially two-dimensional except near the tips.

"In the Hele-Shaw apparatus a thin sheet of water flows very slowly over the surface of a glass plate around a cross-section model of an airfoil, a set of sails or a ship's hull [see Figure 1]. The streamlines are made visible by means of a dye, such as potassium permanganate. They can easily be photographed for close study, and the flow lines will yield quantitative results concerning the pressure distribution over the model.

"It is obvious that when an obstruction reduces the area through which a given amount of fluid flows, the velocity of the flow must increase. Where the streamlines flow around the model, they will be crowded into a narrower area. By measuring the distances between streamlines (marked by filaments of dye) at various points along the flow channel, we can get an idea of the relative velocities and pressure distributions at these points.

"The construction of the Hele-Shaw apparatus is not difficult. The plate over which the water flows can be a piece of plate glass about 30 inches long and 20 inches wide. To enclose the sides of the channel a metal angle along each side, sealed to the glass with aquarium cement, will serve. The plate is leveled by means of leveling screws in the frame which supports it. A film of water is fed onto the plate from a settling trough fitted with a smoothing screen [Figure 1]. "When the plate has been set up and leveled, water is made to flow very slowly over the surface in a film of uniform thickness. The plate should be thoroughly washed with soap and rinsed with strong ammonia water so that its whole surface will be wetted to make the film uniform. A small squeegee or an automobile windshield wiper will help in the cleaning job.

"The model to be placed in the stream can be made from a sheet of balsa wood a quarter of an inch thick. If the model is to represent a sail, it will need to be thinned to a section one sixteenth of an inch thick. After cutting the model, sand its lower surface as flat as possible and waterproof it with two coats of dope. The model can be held in place on the plate either with cement or by a weight on it. If you put a large sheet of graph paper beneath the glass plate it will be easier to make measurements of the distances between streamlines.

"When the film of water has assumed a uniform flow, place a crystal of dye directly upstream from the model so that the filament of color will flow around it. At what is called the 'stagnation' point, just in front of the model, the streamline splits in two, one half flowing around each side of the obstruction. The two halves of the filament come together again at the trailing edge and re-form a single filament. The split streamline defines the boundary layer of flow around the model.

"Now if you place other crystals of dye at uniform intervals (say every quarter inch) in a row along the starting edge of the plate, they will generate filaments which will flow in parallel lines downstream until they come the region where the model diverts the flow. There the parallel filaments will curve and be pushed closer together or farther apart, depending upon the shape of the model.

"Now a couple of simple formulas enable us to calculate the comparative velocity of flow and the pressure at various points around the model. These quantities can be computed simply from the distance between the filaments, or streamlines, at those points. Where the stream is flowing freely and uniformly, before it reaches the model, the average velocity between one streamline and the next, multiplied by a certain constant, is equal to the reciprocal of the distance between the streamlines: e.g., 1/ 1/4 when the distance is 1/4 of an inch. The general formula is VC =1/D, with C standing for the constant and D for the distance between streamlines. At a point upstream from the model the distance between streamlines may be expressed by a [see upper drawing on page 125]. The equation then reads VC = 1/a. At a point A on the model, where the distance between the surface of the model and the first streamline is b, the equation is V_AC = 1/b. Now we can eliminate the C by division and get the ratio V_AV = a/b. In other words, the ratio of the original distance between filaments to the distance between filaments at a given point on the model is a measure of the change of velocity of flow at the model. When the distance between filaments narrows, the flow speeds up in proportion to the reduction distance.

"Similarly we can calculate the change in pressure at a given point on the model by another formula. The difference between the pressure in the free stream and that at a point on the model is equal to k [(V_A/V)² - 1], the letter k standing for a constant.

"Let us illustrate the method with a specific problem. The model is a circular obstruction [see lower diagram on page 125]. At the starting line of the flow the streamlines are a quarter of an inch apart. At point B on the leading side of the circle the distance has widened so that the first streamline is .482 of an inch from the surface of the model. Thus V_B/V = .25/.482, or .518. This means that the velocity of flow between the model and the first streamline at point B is .518 of the original velocity. The change of pressure there is k(.518² - 1), or -.732k. The minus sign corresponds to the fact that the pressure is inward.

"At point A on the model the distance to the first streamline narrows to an eighth of an inch, and V_A/V=.25/.125, or 2, meaning that the velocity of flow is double that in the free stream. The pressure here falls by 3k.

"By plotting the pressure distribution at various points over the surface of a model, you can compute the lift that will be exerted on a given shape at a given angle to the direction of the stream flow.

"Another project which may interest amateurs is the study of the flight dynamics of flexible model aircraft. Small wooden gliders are adaptable to this study.

"When an airplane flies through a gust of wind, it may nose up and down slightly with an oscillation of a certain frequency. The airplane's fuselage, being elastic, also bends at a characteristic frequency. When the two modes of vibration chance to coincide, the airplane begins to gallop through the air, bending and pitching in an oscillation of growing amplitude.

"The phenomenon can easily be demonstrated with a model glider To make the usual stiff model sufficiently flexible, you can introduce springs consisting of piano wire into the structure of the frame. With a little ingenuity may also discover how to produce wing flutter in the craft by inserting elastic hinges at appropriate places in the wings. [see Figure 3]. To demonstrate the classic flutter effects the wing should be allowed to bend upward and twist about its center at the same time.

"As airplanes have gone to higher speeds and lighter structure, the problems of flutter and aeroelasticity have become very important to the aircraft industry. While an amateur has little hope of solving these complex problems, experience gained with models will prove valuable if he ever turns professional."

Dushan Mitrovich, a high-school student of Chestnut Hill, Mass., submits the design of a smoke tunnel which he built at home [see Figure 4]. Photographing the stream-lines of smoke around a model, he can calculate pressure distributions by the same method employed with the Hele-Shaw apparatus. An advantage of Mitrovich's smoke tunnel is that it is somewhat more convenient in a workshop which lacks running water and a drain.

"The tunnel," writes Mitrovich, "has a bellmouthed air inlet made of four pieces of balsa. The observation chamber measures six inches high, eight inches long and an inch wide. The entering air flow is straightened by a grid of 23 thin brass vanes, each two inches long and spaced at quarter-inch intervals. The diffuser, made of quarter-inch plywood, starts as an oblong rectangle and gradually becomes an octagon at the exhaust end.

"The air is drawn in by a pair of crossed, nine-inch propellers notched front and back so they fit into each other and rotate in the same plane [Figure 5]. They are housed in the octagon at the exhaust end and driven by a 15-volt, direct-current motor of the kind used in model trains. The propeller-motor assembly is supported in a brass tube, two inches long, which is held in the center of the octagon by four streamlined struts. A spinner on the propeller and faring aft of the motor streamline the whole power plant.

"The smoke generator, located below the air inlet, consists of a cylinder four inches in diameter and two and one half inches long. The funnel-shaped top feeds smoke to a half-inch manifold fitted with 24 smoke nozzles The nozzles are made from eighth-inch tubing flattened slightly into ovals at the point of discharge. They. enter the inlet at an angle of 45 degrees and end flush with the inner wall halfway between each of the smoothing vanes.

"Construction details of the smoke generator are obvious in the drawing. I burn joss sticks (available in Chinese stores) to generate the smoke; they work better in this tunnel than cigarettes, which produce too much heat, or titanium tetrachloride, which is corrosive and clogs the nozzles. The joss sticks, cut to a length of about one inch and impregnated with a few drops of lighter fluid, are placed on the wire mesh, which serves as a grate. To provide enough air to keep them burning, the propeller must be running when the generator is put in place. The pile is ignited, permitted to burn for a few seconds and then blown out. It continues to deliver a dense smoke.

"Air speed in the experimental chamber can be varied between 4 and 15 feet per second by adjusting the input voltage to the motor. The rear wall of the chamber is painted flat black. Illumination for photography is supplied by two 25-watt tubular bulbs equipped with cardboard reflectors. They are placed about three inches from each side of the chamber. To counteract reflection from the glass front of the chamber, I place a sheet of flat-black cardboard in front of the tunnel and photograph through a hole cut in the center for the lens. My models are made and tested by the procedures recommended by J. J. Cornish."

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