LAST MONTH I described the basic mechanism that makes a boomerang return to the person who throws it. My account left some of the stranger aspects of the design and flight of boomerangs unexplained. This month I shall first deal with these aspects and then explain how you can experiment with the boomerangs you have made according to the techniques I described last month.

One interesting feature of certain older boomerangs is that they have a rough surface. Some of the ancient boomerangs found in Australia have this characteristic, as if the person who made the boomerang thought a rough surface would make it go farther. Although the idea seems implausible, it might be correct. One's initial reaction is that roughening would increase the friction between the surface and the passing air, so that the boomerang would lose forward velocity sooner and fall to the ground earlier. If a rough surface is desirable, the reason certainly has nothing to do with friction. Something else must be going on.

Many familiar objects have surfaces that have been made rough to increase their flight time. The dimpled golf ball is the best-known example. In the early days of golf all the balls were smooth, as intuition would suggest. Eventually someone noted that a ball scarred and pocked by use seemed to travel farther than a smooth one when both were struck in essentially the same way. Thereafter balls were pitted on purpose. Ascher H. Shapiro once did experiments comparing a dimpled golf ball with a smooth one. He found that the dimpled ball traveled more than four times farther than the smooth one, notwithstanding the difference in air friction.

Why would dimples aid an object in its flight? The answer lies in the behavior of the air flowing around a dimpled golf ball or a roughened boomerang. For the sake of simplicity first consider the flow pattern around a classic airfoil. Most of the air passing the airfoil is unaffected by the viscosity of the air, that is, by the friction between two layers of air or between a layer of air and the surface of the airfoil. Next to the surface, however, is a layer of air called the boundary layer that is affected by viscosity in an important way. The air closest to the surface does not move at all, being held in place by the friction between it and the surface. The next layer of air has a low velocity. As one considers more of these imaginary thin layers of air at greater distances from the surface one finally reaches layers moving with a speed that is unaffected by friction with the surface of the airfoil. The thickness of the boundary layer is defined as the distance from the surface to the level where the friction can be neglected. The clue to the behavior of dimpled golf balls and roughened boomerangs lies in the movement of the air in the boundary layer.

The air pressure around a classic airfoil is high in front of the airfoil and behind it, relatively low over the top of it and relatively high below it. Last month I explained that the pressure difference between the top and the bottom is-due to the Bernoulli principle. That pressure difference gives lift to the airfoil. This month I shall examine how the air moves from the front to the rear over the top of the airfoil.

Consider two small parcels of air. one parcel traveling inside the boundary layer and the other one just outside it. When they approach the airfoil, they are slowed to a stop by the high-pressure area just in front of the airfoil. When they reach the top of the airfoil (because it is moving), they must accelerate as they are pushed by the pressure difference between the front and the top. The parcel in the boundary layer must act against the viscous forces extending outward from the surface of the airfoil, however, and so it does not accelerate as much. When the two parcels pass over the top of the airfoil, the one outside the boundary layer is moving faster.

As the parcels then move toward the rear of the airfoil they encounter an increase in pressure and so slow down. The parcel outside the boundary layer will be brought to a stop by the time it reaches the rear of the airfoil and then will be accelerated farther rearward by the high pressure there until it regains the speed it had before it began to pass the airfoil. The parcel inside the boundary layer begins the trip to the rear with a lower speed, so that it may stop before it reaches the rear of the airfoil. If it does stop, the high pressure at the rear may even push the parcel back toward the top of the airfoil, forcing air emerging from below the airfoil up to replace it. The air of the boundary layer will be pushed away from the airfoil by this influx, an effect that is termed separation and is an important factor in the drag on the airfoil.

Part of the drag comes from the friction between the passing air and the surface of the airfoil. One can call this contribution skin-friction drag. Another contribution comes from the difference between the average pressure on the front and that on the rear of the airfoil. If the average pressures are about equal, this contribution is small and one need consider only the skin-friction drag. In some cases, however, the pressure difference is even larger than the skin-friction drag. Such a situation may arise when the boundary layer separates from the surface, leaving a relatively wide wake of turbulent air replacing what initially was high pressure. The pressure of the turbulence is intermediate between the low pressure on top of the airfoil and the high pressure in front of it. Thus the pressure difference between the front and the rear may be large, creating a strong drag on the airfoil. The earlier the separation develops in the passage of air over the top of the airfoil. the stronger the drag from the pressure difference will be. With an earlier separation the pressure in the wake is lower and the wake is wider; both effects reduce the average pressure on the rear of the airfoil.

A smooth golf ball is a blunt object from which the boundary layer separates early, perhaps even before the air has gone halfway to the rear. A normal golf ball is dimpled in order to delay the separation. (Streamlining would accomplish the same result, but a golf ball shaped like an airfoil would be unlikely to roll well on the green.) At first the effect of the dimples seems counterintuitive because the dimpling surely also increases the skin-friction drag. The reduction in the pressure-difference drag is so large, however, that the overall drag is decreased and the golf ball travels considerably farther.

The dimples are designed to create a turbulent boundary layer that will rapidly mix the air in the boundary layer and the air outside the layer. As a result the air in the boundary layer does not have a chance to slow down with respect to the outside air because it is always receiving momentum from the outside air. When boundary-layer air passes over the top of the ball and heads toward the rear, it does not slow to a stop before it reaches the rear and the high pressure at the rear does not push it away from the surface. The wake is therefore relatively narrow, and the pressure difference between the front and the rear of the ball is not as large as it would have been if the ball had been smooth.

For the past several years a golf ball with a new dimple design has been offered by Uniroyal. The regular arrangement of circular dimples is replaced with a random arrangement of hexagonal dimples. Based on the research of John Nicolaides of the University of Notre Dame, the ball reportedly travels an average of six yards farther than the conventional ball. I am not sure why, but the cause is likely to be the creation of a better turbulent boundary layer.

Recently I saw a demonstration of the effect of dimpled golf balls at the Ontario Science Centre in Toronto. One smooth ball was attached to a vertical rod hinged at the top so that the ball could move when an airstream was directed at it. A dimpled ball was mounted in the same way. When the airstream was turned on, the ball with the lesser drag would be displaced less and its rod would swing less from the vertical. When I turned on the air, I found that the dimpled ball was displaced the least. The dimpled ball has less drag.

Should a boomerang be roughened or dimpled? Maybe. One must bear in mind the likelihood of separation on an airfoil traveling at moderate speed, as a boomerang does. When investigators study the flow of a fluid past objects, they must be able to compare the likelihood of separation and turbulence even though they are using different fluids (having different densities and viscosities) and different objects (having different sizes). To make the comparison a dimensionless number called the Reynolds number is computed. Named for Osborne Reynolds, who contributed greatly to the analysis of fluid dynamics in the 1 9th century, the number is calculated from the density and speed of the fluid multiplied by a typical length of the object and divided by the viscosity of the fluid. When the Reynolds number is high, separation and turbulence are likely, the boundary layer is turbulent and the flow is said to be only slightly viscous. The fluid may actually have a high viscosity, but under the circumstances of the high Reynolds number the flow is almost as it would be if there were no viscosity. A given pressure difference propelling a parcel of the fluid will create an acceleration of the parcel that depends on the density of the fluid rather than on the viscosity.

When the Reynolds number is low, separation and turbulence are unlikely, the boundary layer is laminar and the flow is said to be viscous. Again the behavior does not depend directly on the actual value of the viscosity, which may in fact be low. Under the circumstances of the low Reynolds number the pressure difference attempting to propel a parcel of the fluid is almost matched by the opposing viscous forces, with the result that acceleration is minimal.

For a golf ball the Reynolds number is in a middle range, high enough to make separation and wake turbulence likely but not high enough to create a turbulent boundary layer. Dimpling the surface (even at the cost of increased skin-friction drag) is justified. What about a boomerang? Unfortunately the Reynolds number of a typical boomerang also lies in a middle range of values, making the effect of roughening unclear. If the performance of a particular boomerang is not much affected by early separation of the boundary layer, roughening the surface is inefficient because the overall drag is increased by the additional drag of skin friction. On the other hand, if the performance of a boomerang is considerably affected by early separation, a roughening of the surface may increase its flight time.

A related consideration in reducing the chance of separation is whether or not the object should be streamlined. Airplane wings are streamlined to reduce separation and the drag caused by pressure difference. Should the arms of a boomerang be streamlined too? For the sake of simplicity again consider a classic airfoil. When air in the boundary layer moves over the top and toward the rear, it meets a progressive increase in pressure that threatens to stop its progress and force a separation. If the path to the rear is short, the pressure increase is sudden and separation may be unavoidable. Hence an object such as an airplane wing has a tapered rear to make the pressure increase gradual. The tapering also increases the distance over which the passing air rubs against the surface, thereby increasing the skin-friction drag. Still, the avoidance of separation is worth the increased drag.

Shapiro gives an example of streamlining in his book Shape and Flow: The Fluid Dynamics of Drag. He compared the overall drag on a cylindrical wire and a streamlined airfoil when air passed them at 210 miles per hour. The wire had the same drag as the airfoil, which was almost 10 times the width of the wire. Intuition suggests that the much thicker airfoil should have the greater drag, but intuition does not always take into account the subtle effects of boundary-layer separation and pressure-difference drag. At high Reynolds numbers it is better to streamline the object.

Suppose the fluid passes the object at a Reynolds number that is low enough to make separation unlikely. Streamlining the object (at the cost of increased skin-friction drag) would then be wrong. Instead the object should be blunt, presenting as little surface as possible for the air to pass.

Should the arms of a boomerang be streamlined? They probably should be, but I have not found conclusive evidence to verify my guess. Once again the Reynolds number for a typical boomerang is in the awkward middle region, and I do not know if separation is sufficiently common to justify the increased skin-friction drag the boomerang would have if the arms were streamlined.

You may wonder why none of the boomerang shapes is just a straight stick. Even if a straight stick is suitably curved on one side and flattened on the other so that it presents a classic airfoil to the passing air, it will not serve as a boomerang. The reason is a bit subtle, because it involves the stability of a rotating object against the small perturbations it encounters as it rotates.

Suppose you bind a book with a strong rubber band so that the book remains closed and flip it into the air. As is shown in the illustration above, the book can rotate about three principal axes. Rotation about two of them (either A or B) is stable, but around the third axis (C) the book wobbles vigorously.

The axes are characterized by the distribution of the book's mass with respect to an axis. With rotation about axis A the book has its mass distributed as close to the axis of rotation as possible. The book's moment of inertia is the lowest for this axis. The moment of inertia is the highest for axis B because then the book has its mass distributed as far as possible from the axis. About axis C the moment of inertia is intermediate and the rotation of the book is unstable.

The stability depends on what the angular momentum of the book does when it deviates slightly from being exactly parallel to the principal axis about which the book is rotating. (The book's angular momentum is a vector with a magnitude that is the spin rate multiplied by the moment of inertia and with a vector direction that is perpendicular to the plane in which the book is spinning.) For two choices of a principal axis (A and B) a wandering angular-momentum vector is quickly rotated (precessed) back to being parallel to the principal axis. For axis C a wandering angular-momentum vector is rotated away from being parallel. If you could carefully spin the book around the wobbly axis with the angular-momentum vector exactly parallel to the axis, the book would spin stably. Any deviation from that parallel condition (a likely development) will increase quickly during the flight and the book will be unstable. If the greatest moment of inertia and the intermediate one are almost equal, as they would be for a square book, the spins around both of the axes associated with them will be unstable.

The standard banana-shaped boomerang spins around the axis about which it has the greatest moment of inertia and so its spin and flight are stable. With a straight boomerang, however, the greatest moment of inertia and the intermediate one would be almost equal, and the boomerang would be unstable. The wobbling would prevent the boomerang from meeting the air at the proper angle of attack, and so the device would get no lift. In short, a straight boomerang does not boomerang.

A curious feature of a regular boomerang is the tendency for it to "lie down," that is, for its spin plane to rotate from being almost vertical to being horizontal. At the beginning of a flight the boomerang's forward velocity is great enough so that the spin plane needs only a slight tilt from the vertical to have enough upward lift to counter the weight of the boomerang. As air drag slows the forward velocity, progressively more of the lift has to be upward to hold the boomerang up. By the time the boomerang returns to the thrower the spin plane is almost horizontal.

Two possible causes for lying down can be identified. One, which is true for all types of boomerang, involves the air deflected by the arm that is turning forward of the boomerang's center of mass. Whenever an arm rotates through the forward position, it deflects the passing airstream, which (slightly later) flows around the other arm as the boomerang travels forward. Hence the trailing arm does not meet undisturbed air and does not experience the same lift as the leading arm.

For example just after a throw by a right-handed thrower the boomerang has a horizontal lift to the left of the thrower. When the air leaves the leading arm, it must be deflected to the right of the thrower. (According to Newton's laws of motion, every action has an equal and opposite reaction. If the boomerang is forced to the left by the air. the air must be forced to the right by an equal amount.) As the trailing arm turns into air that is already flowing to the right it has a bit less lift than the leading arm because the air does not flow past the leading arm at the best attack angle. (The attack angle may even be negative.) The result is that as a boomerang spins, the leading arm always has a bit more lift than the trailing arm.

The difference in forces attempts to tilt the spin plane so that the rear of the boomerang would swing to the right (as viewed by the thrower) and the front would swing to the left. As I argued last month, this simple tilting does not appear because the boomerang is spinning and so has angular momentum. Instead what happens is that the torque resulting from the forces causes the spin plane to precess, that is, to rotate about a horizontal axis until the spin plane is horizontal by the end of the flight.

To make sense of the lying down imagine an overhead view of a right-handed boomerang that you have just launched with its spin plane vertical and therefore with its angular-momentum vector to your left. On the average the lift in front of the center of mass is more than the lift behind the center of mass because of the deflected air met by the rear of the boomerang. For the sake of simplicity consider the lift on the rear to be zero. (The results of my argument are the same even if you give that lift its proper value.) The change in angular momentum is due to the torque caused by the average lift on the forward section. To locate the direction of the change you should employ a rule I gave last month: Point the index finger of your right hand from the center of the boomerang toward the place where lift is applied on the boomerang's arm, point your middle finger parallel to the lift and then stretch your thumb perpendicularly to both fingers. Your thumb shows you the direction of the change in angular momentum: it is upward, perpendicular to the existing angular momentum.

Since the initial angular momentum is to the left and the change is upward, the angular-momentum vector rotates to be more upward, with the result that the spin plane of the boomerang is also rotated, turning from the vertical toward the horizontal. The boomerang has begun to lie down. This rotation, called precession because it is a rotation of the angular-momentum vector, continues throughout the flight. By the time the boomerang has returned to the thrower the spin plane is almost horizontal and the lift on the boomerang is almost entirely upward.

The second possible cause of lying down is similar. Perhaps one arm has a shape that gives more lift than the other arm. (In the illustration on the opposite page the arm with more lift is labeled A and the other arm is labeled B.) Both arms have the most lift when they move into the upward position (as I noted last month) because the air speed past the arm is highest when the arm is turning into the oncoming airstream. When A is in its best position, it happens to lie in front of the center of mass. When B is in its best position, it is behind the center of mass. If the greatest lift exerted on A is always greater than that exerted on B, as will be determined by the cross-sectional shape of the arms, the lift on the average will be greater in front of the center of mass than it is behind it. Once again a difference between the force in front of the center of mass and the force behind it creates a torque that tilts the spin plane. Just as in the argument about the deflected air, the torque rotates the angular-momentum vector of the boomerang from being initially horizontal to being finally upward, causing the boomerang to lie down.

If the boomerang did not lie down, its flight would be shorter and far less interesting, because at some point during the flight the forward velocity would be too low to provide enough upward lift to support the boomerang. If a boomerang lies down so quickly that it is horizontal about halfway through the flight, it might continue to precess, swinging its angular-momentum vector out of the vertical and thereby dipping its spin plane below the horizontal. The boomerang would then begin to curve in the opposite direction and might even follow a figure-eight path on its return to the thrower.

It takes as much effort to build a boomerang that flies straight as it does to build one that returns. The horizontal lift must be almost eliminated, but enough vertical lift must remain to hold the boomerang up against its own weight. The horizontal lift is removed by twisting the arms to give a negative angle of attack near the tips of the arms and a positive angle of attack near the center. The overall lift is greatly reduced, but not quite to zero. The boomerang is thrown with its spin plane nearly horizontal, so that the net lift is almost vertical.

If the boomerang is launched with its spin plane about 20 degrees above the horizontal, a small horizontal lift causes it to fly in a straight line toward the thrower's left. The small amount of lying down makes the spin plane rotate until it is horizontal and then makes it dip below the horizontal. Afterward the boomerang veers somewhat to the thrower's right. Since the path is not perfectly straight, the thrower must know the boomerang well if a distant target is to be hit.

Explanations of the flight and return of boomerangs have been put forward for many years, but much of the early work was erroneous and lacking in experimental verification. Except for a modern understanding of airfoil theory, the basic mechanics of boomerang flight were known as early as 1837, but apparently the information was ignored or misunderstood by later writers. A classic theoretical explanation was given by G. T. Walker in his article "On Boomerangs" in the Philosophical Transactions of the Royal Society of London in 1897.

The most exhaustive modern treatment of the subject of boomerangs is the doctoral dissertation by Felix Hess listed in the bibliography for this month [below]. Hess reviews virtually all the older information and then examines boomerangs himself both experimentally and theoretically. Some of his earlier information was published in this magazine [see "The Aerodynamics of Boomerangs," by Felix Hess; SCIENTIFIC AMERICAN, November, 1968]. Much more information is available in the dissertation including several interesting three-dimensional illustrations of computer-simulated flights that the reader examines through a stereoscope viewer provided in the back of the book.

A certain amount of research on boomerangs is carried out in wind tunnels in which investigators study the paths of airflow around a stationary boomerang. Much can still be done by amateurs throwing carefully constructed boomerangs. If you pursue this research, your first problem will be to determine what your boomerang does during the flight, because you will be at only one end of the flight and can easily misjudge distance and height. To determine the distance you can station several observers along the flight path. To determine the height you can measure the angle the path makes in your field of view. By knowing both the distance and the angle you can calculate the height the boomerang attains.

Some investigators have chosen to attach a light to the boomerang and throw it in darkness while leaving the aperture of a camera open. The result is a trail of light on the photograph marking the boomerang's path. Examples of the technique were shown in Hess's article in this magazine. With a single camera, however, the three-dimensional nature of the path is difficult to appreciate. Even if you position two cameras to photograph the path from different perspectives, you will still have trouble combining their separate information to get the three-dimensional trajectory. Better results might be obtained if you photograph the lighted boomerang with the stereoscopic technique I described in this department last December. When the resulting two photographs are put in a stereoscope, one sees a combined image that has depth.

Hess and others have lighted boomerangs by attaching a small circuit composed of batteries, an electric oscillator and a small light. The arrangement is convenient, but unless you sink the objects into the wood (as Hess did) it has the disadvantage of distorting the airflow around the boomerang. The batteries and the oscillator are fastened to the center of the boomerang with only the small light bulb out on an arm, so that the boomerang's moment of inertia is altered as little as possible. A sparkler of the kind that celebrators light on the Fourth of July would also show the flight path.

Many variables stand ready for investigation if you would like to experiment with boomerangs. Although Hess has done a marvelous job in simulating boomerang flight with his computer programs, even he has not ascertained the variables to the point of designing the best boomerang or turning a bad one into a good one by some small change in construction. The major variables are those of the launching and those of the construction. A boomerang you bought would serve for a study of the variables in launching, but you would have to build one to examine the variables in construction.

To investigate launching try to find a large, open space with little wind. Choose one variable and then keep all the others constant as closely as you can. I know that holding the launching conditions constant is difficult because you will never make two successive tosses in exactly the same way. Photographing the launching with a motion-picture camera would enable you to objectively compare the launch angle, the spin and the forward velocity from one trial to another. How do the flight time, the total distance traveled and the maximum height depend on the direction of launching (normally you throw toward the horizon), the angle between the spin plane and the vertical (the greater the angle, the steeper the climb), the spin rate and the forward velocity?

With your homemade boomerangs how does the flight depend on the cross-sectional shape of the arms, the amount of twisting and the weight? I would be particularly interested to know if streamlining the arms is always advantageous. Should the tips of the arms be tapered or rounded? Should the boomerang be narrow in the center and wider at the tips of the arms? Does the flight change if the leading edge is sharp instead of blunt?

Since the effect of surface roughness is unresolved, you might want to explore it. Instead of cutting holes or grooves in a wood boomerang, try attaching cellophane tape that is sticky on each side. The tape itself might create a turbulent boundary layer, or you could create turbulence by sprinkling sand on the sticky stuff. Some researchers have investigated the effect of turbulence by mounting a thin wire just in front of the leading edge of each arm. Hess's results in such an experiment were inconclusive.

Some boomerang enthusiasts have discovered that by adding ballast to the arms near the tips they can greatly increase the distance of the outward flight of a returning boomerang. In 1972 Herb A. Smith, whose "Gem" design for a boomerang I described last month, threw such a boomerang for an outward distance of 108 yards. According to the Guinness Book of World Records, that is the longest outward flight ever recorded. Smith suggests adding ballast amounting to as much as a third of the boomerang's weight and countersinking the ballast (usually pieces of lead) in holes drilled about an inch from the tip of an arm. Weight added at the center will probably keep the boomerang in a more level flight, but more distance might be achieved with the ballast near the tips because then the boomerang's moment of inertia is also increased. You might see how much effect ballast has on your homemade boomerangs. If world records interest you, you might try to beat Smith's throw.

Bibliography

SHAPE AND FLOW: THE FLUID DYNAMICS OF DRAG. Ascher H. Shapiro. Anchor Books, Doubleday & Company, Inc., 1961.

MANY HAPPY RETURNS. Peter Musgrove in New Scientist, Vol. 61, No. 882, pages 186-189; January 24, 1974.

BOOMERANGS, AERODYNAMICS, AND MMTION. Felix Hess. 1975 dissertation available from the author, c/o Dr. H. Rollema, Eikenlaan 51, Peize, Netherlands.

PROJECT BOOMERANG. Allen L. King in American Journal of Physics, Vol. 43, No. 9, pages 770-773; September, 1975.

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