THE FLOW of air around a cylindrical object in a direction perpendicular to the axis of the cylinder tends to generate vibrations. The ancients applied the effect in a musical instrument known as the aeolian harp, which consists of a set of strings stretched across a sounding box. The strings emit a series of harmonic tones that vary with the strength of the breeze.

The aeolian effect was investigated in some depth by experimenters of the l9th century, although the aerodynamic mechanism that is responsible for the vibrations is not fully understood even today. Interest in the vibrations has risen sharply in recent years. They can generate destructive forces in cylindrical structures ranging from smokestacks and water towers to rockets on launching pads. Recently Bonnie Jean Luessen, a high school student in Huntsville, Ala., built a small wind tunnel for investigating the aeolian effect quantitatively. The project placed her among the finalists in the 31st annual Science Talent Search. She describes her experiments as follows:

"When a stream of air flows around a slender cylinder at relatively low velocity, two stable, symmetrical vortexes form in the wake. At a somewhat higher velocity one of the vortexes becomes unstable, detaches from the cylinder and moves downstream. The action transmits a force through the air that disturbs the remaining vortex, which thereupon becomes unstable in its turn, detaches and drifts downstream. A new vortex forms at the site of the first disturbance. Thereafter vortexes form and detach alternately as long as the critical velocity persists.

"The swirling pattern of flow in the wake of the cylinder is known as von Kármán's vortex street in honor of the Hungarian aerodynamics Theodor von Kármán, who explained the effect mathematically some years ago. The periodic shedding of vortexes of opposite rotation generates alternating forces that act on the cylinder at right angles to the flow of air. The influence of the oscillating forces increases as the length of the cylinder increases and as the diameter decreases. The amplitude of vibration also increases as the frequency of the alternating forces approaches the natural period at which the cylinder vibrates.

"Wind speeds of less than 30 knots induced significant vibrations in the first Redstone rocket to be erected on its pad. The problem was solved by sheltering the vehicle from winds of more than 25 knots. Wind-tunnel tests subsequently indicated that the Saturn V vehicle would be susceptible to wind-induced oscillations at velocities occasionally observed at Cape Kennedy. To prevent the buildup of destructive forces the upper end of the rocket assembly was clamped to a bracing structure through a pair of hydraulic damping cylinders similar in principle to the shock absorbers of automobiles. The cylinders dissipate the wind-induced energy as heat.

"My experiment was designed for observing the aeolian effect in a homemade wind tunnel. In particular I set out to investigate von Kármán's vortex street by measuring the dynamic responses of the cylinder instead of injecting smoke in the wake to make the vortexes visible a procedure that is often followed. Experiments done during the l9th century by Lord Rayleigh and by Vincenz Strouhal of Czechoslovakia demonstrated that the frequency at which a cylinder sheds vortexes varies in proportion to the velocity of the wind.

"My tests were made with cylinder of two diameters. The cylinders were supported in the wind tunnel by a hinged fixture that allowed the model to move only in the vertical plane, at right angles to the axis of the cylinder an to the direction of the airstream. The vertical motion of the model was limited by a set of helical springs. The tension of the springs determined the model's natural period of vibration. The natural period could be altered by substituting springs of different stiffness. The velocity of the airstream could be controlled by regulating the speed of the fan that sucked air through the wind tunnel.

"The alternating forces generated b the vortexes are relatively weak in small wind tunnel. The model cylinder for my experiments were accordingly constructed of light material. Care was also taken to minimize the damping effect of the hinged fixture to which the models were mounted.

"The wind tunnel consists of four demoumtable elements [see illustration at right]. The rectangular entrance section curves inward, roughly in the form of an exponential horn. It is made of poster board mounted on a wood frame at the upstream end of the test section. The frame maintains the rectangular shape of the assembly. The frame can be detached from the test section for transportation.

"The open end of the entrance section is stiffened by strips of corrugated cardboard glued to the poster board. The corners of the board were similarly glued together. All joints were sealed with masking tape. The structure is surprisingly stiff and strong.

"The test section, a straight rectangular tube, is made of plywood. One exterior side supports a rectangular frame to which the fixture that supports the models is hinged [see illustration at left]. The motion of the fixture is limited in the vertical plane by helical springs. Motion in the horizontal plane is prevented by a cord between the fixture and the frame on the upstream side. The cord is maintained in tension by a helical spring on the downstream side.

"Air pressure in the test section is measured by a manometer, a transparent tube in the form of a U that is partly filled with alcohol. The two arms of the manometer are connected to glass tubes with a bore of approximately six millimeters. One tube, which measures static air pressure, terminates inside the test section about four inches above the model and two inches upstream from it. The end of the tube is flush with the inner wall of the test section. The axis of the tube makes a right angle with respect to the direction of the airstream.

"The other arm of the manometer connects to a second glass tube of the same size. This tube extends through the wall of the test section approximately one inch from the static tube, but the glass is bent at a right angle and installed with the open end facing squarely into the wind. The tube measures total pressure. From the difference between the static pressure and the total pressure it is possible to calculate the velocity of the airstream, as I shall explain.

"The downstream end of the test section connects to the diffuser section, a trapezoidal tube that allows the air to expand gradually as it approaches the fan. The diffuser section creates maximum flow velocity within the test section. Maximum velocity is achieved by maintaining laminar flow between the test section and the fan, which should be larger in area than the test section.

"My first diffuser failed. Its sharp taper allowed the air to expand so quickly that separation occurred: the airstream pulled away from the walls, resulting in loss of energy and relatively low velocity. I replaced the section with a successful diffuser that tapered at an angle of seven degrees. The character of the flow was observed by attaching tufts of wool yarn to the end of a wire rod and supporting them in the airstream.

"A single 21-inch floor fan, which is coupled to the downstream end of the diffuser, can suck air through the test section at a maximum velocity of 32.6 feet per second. The velocity can be increased to 40 feet per second by using two fans back to back. The speed of the fans can be adjusted within reasonable limits by a variable transformer. The velocity is calculated from manometer measurements by using Bernoulli's law: in which is the static pressure in the test section, is the density of air, V is the velocity and is the total pressure in the test section. The manometer indicates the difference in the pressure of and in inches of alcohol: , where D is the density of alcohol and h is the difference in the height of the alcohol in the two arms of the manometer. Combining the equations and solving them for velocity gives .

"I wanted to express velocity in feet per second. The density of alcohol in corresponding British units is 49 (pounds per cubic foot). The density of air is .002378 (slug per cubic foot). The difference in the height of the alcohol in the two arms of the manometer was measured in inches. The measurements were converted to feet (divided by 12) for use in the equation. For example, a manometer reading of .21 inch (.21/12 = .0175 foot) indicates a wind velocity in the tunnel of feet per second.

"Since the forces developed by the vortexes are not large, the mass of the test cylinders was minimized by making the models of balsa wood. The fixture that supports the models in the test section was built of soft pine and poster board. Friction was minimized by coupling the distant end of the fixture to the frame with a spring hinge.

"Ideally the motion of the model should be pure translation. I approximated the ideal motion by making the fixture twice as long as the models. The models are attached to the fixture by a pair of metal strips that enter the test section through half-inch holes in the wall. The static pressure at this point in the test section is somewhat below atmospheric pressure. Some air doubtless enters the tunnel through the holes. I did not want to hamper the free movement of the models by sealing the openings with flexible membranes. Subsequent experiment demonstrated that the leakage was not sufficient to interfere with the vortexes.

"The free end of the model, which is visible through a window in the opposite wall of the test section, carries a triangle that functions as a vibrometer for measuring the amplitude at which the model vibrates. The altitude of the triangle is approximately one inch. The base measures .3 inch. The triangle is divided into three equal parts [see illustration at right]. When the model vibrates in the vertical plane, persistence of vision causes the eye to perceive two relatively light overlapping triangles plus a smaller dark triangle in the region where the two light triangles overlap. The altitude of the small dark triangle varies inversely with the amplitude at which the model vibrates. It is possible to estimate the altitude of the dark triangle to within 1/15 inch and therefore to estimate the amplitude of vibration to within .02 inch.

"The Langley Research Center of the National Aeronautics and Space Administration made a similar study of von Kárman's vortex street in 1966 as part of the Saturn rocket project. These tests were made with a cylinder three feet in diameter in a 16-foot wind tunnel. Since my tunnel could accept a model eight inches long, I scaled the cylinder in proportion, making one of my models 2.25 inches in diameter. A second model was made with a diameter of 1.75 inches for observing the influence of reduced diameter on the performance of the apparatus.

"At approximately what frequency should these cylinders vibrate in my tunnel? Strouhal demonstrated that the frequency with which a cylinder sheds vortexes varies directly with the velocity o the airstream and inversely with the diameter of the cylinder. The relation is described in terms of a constant, which is known as the Strouhal number. It is the product of the vortex frequency times the diameter of the cylinder divided by the velocity of the airstream , in which S is the Strouhal number, f the frequency, d the diameter of the cylinder in inches and V the velocity of the airstream in inches per second). The value of the constant is .2. The frequency at which the cylinder should vibrate is therefore equal to .

"With the maximum velocity of my wind tunnel, running on one fan, being about 32 feet per second, if one assumes a flow of 25 feet (300 inches) per second a model 2.25 inches in diameter should oscillate at the rate of .2 x 300/2.25 = 26.6 vibrations per second. This frequency seemed reasonable for a small cylinder of balsa wood. The problem of suspending the model with springs of the stiffness required for a natural period of about 26 vibrations per second was solved experimentally. I merely suspended the model with a set of springs, plucked it with my finger, measured the rate of the resulting vibration with a stroboscope and then substituted springs of greater or lesser stiffness as required to generate the desired frequency of about 26 vibrations per second.

"A model so tuned might tend to oscillate in response to mechanical vibrations developed by the fan or some other source. On the other hand, if the amplitude of vibration approaches maximum as the velocity of the airstream approaches the critical value predicted by the equation and thereafter declmes as the velocity increases, one could conclude that the oscillations are induced by von Kármán's vortex street even though the presence of the vortexes is not confirmed by a different technique, such as injecting smoke into the wake of the cylinder. In order to prove the existence of the vortexes without resort to such alternative techniques I made up four model configurations.

"Two cylinders of 2.25-mch diameter were tuned to resonate respectively at 26.4 and 30.4 cycles per second. Two 1.75-inch cylinders were tuned to 27 and 34.5 cycles per second. The amplitude at which each cylinder vibrated was observed at 10 discrete velocities as the air speed was increased in increments through the range from approximately 16 to 32 feet per second. At each velocity I tabulated the air pressure as indicated by the manometer and the amplitude of vibration as indicated by the vibrometer.

"From these data I calculated the corresponding air speed and Strouhal number. The accompanying table [at left] lists a typical set of data. The results of the four tests were displayed as graphs made by plotting amplitude versus velocity [see illustration lower right]. During each test the amplitude of vibration increased to maximum and then decreased as the velocity of the airstream increased from minimum to maximum. The frequency of vibration remained constant, as expected at the rate to which the models were tuned. The measured value of the Strouhal number at which maximum amplitude was observed ranged from .178 to .192. The Strouhal numbers averaged within 8.3 percent of the theoretical value of .2.

"Finally, the data of all four tests were combined in a single graph [see illustration lower left]. The amplitude at which each model vibrated at each increment of air speed was divided by the maximum amplitude of that model. The quotients were plotted against Strouhal numbers ranging from .12 to .24. The resulting graph peaks at a Strouhal number of .19, within 5 percent of the theoretical value, indicating that the vibrations were indeed generated by the aeolian effect. Moreover, the results have assured me that I can use the small wind tunnel with some confidence for doing other aerodynamic experiments, the outcome of which may not be so easily verified by theory."

"Fill a bottle with water to within an inch or so of the top and insert the diver. Place a snug-fitting stopper in the neck of the bottle. Press the stopper gently. The increased air pressure forces water up into the ornament. The diver sinks. Releasing the pressure on the cork causes the diver to rise. If the bottle is filled with water completely to the stopper, pressure applied to the stopper will be transmitted directly to the air in the diver. In this state the device is inconveniently sensitive and difficult to control.

"It is easy to demonstrate that the diver is unstable at any depth. If the diver starts to sink at any position of the stopper, the pressure of the water increases, more water enters the bulb and the bulb loses buoyancy and sinks deeper. Conversely, when the diver starts to rise, the pressure of water at the neck of the bulb decreases, the trapped air expands, water is ejected and the rate of ascent increases.

"The diver can be stabilized, however, by altering the density of the fluid. Fill the bottle to about half of its capacity with fresh water and insert a funnel with a stem long enough to reach the bottom [see illustration at right]. (A long-stemmed funnel can be improvised by taping a soda straw to the stem of a conventional funnel.)

"In a separate container make up a brine solution by dissolving a few table spoons of salt in fresh water. Chill the brine with a cube of ice. By means of the funnel slowly add chilled brine to the bottom of the bottle until the upper surface of the fresh water floats to within an inch or so of the top of the bottle. By looking through the bottle toward a source of light you can see the interface between the brine and the fresh water as a result of the effects of optical refraction.

"Make a stirring rod by taping a dime across the end of a soda straw. Insert the rod in the water and gently lower the dime to the level of the interface. Stroke the rod slowly up and down about a dozen times, beginning with quarter-inch excursions and ending with two-inch excursions. Remove the rod. This action, when it is carefully performed, creates a zone of increasing density that is approximately five inches thick.

"Observe that the diver is now stable at any depth within the five-inch zone at which you place it by manipulating the stopper. When a slight pressure is exerted on the cork, the diver first sinks and then rises and oscillates about a new position of equilibrium. Increased pressure forces solution into the diver, causing it to lose buoyancy and sink, as in the first experiment. This time, however, the diver sinks into solution of increasing density, where it gains disproportionate buoyancy. The diver rises. After passing through the zone of equilibrium it again loses fluid, but the surrounding solution is less dense, so that the motion again reverses. After a few oscillations of diminishing amplitude the diver comes to rest at the new level of equilibrium. The density gradient will usually persist in the solution for a week or more before it is destroyed by diffusion."

Bibliography

WIND TUNNEL TESTING Alan Pope. John Wiley & Sons, Inc., 1947.

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