IV. Turbulence in Three Dimensions

• A. Procedure
  • 1. Design of the Carts
  • 2. The Vscope
  • 3. Performing the Tests
• B. Results
• C. Analysis
  • 1. Relative order of shapes
  • 2. Sources of Error

A. Procedure

1. Design of the Carts

In general, drag is determined by an object's size, shape, and the speed at which it travels (in actuality, its terminal velocity). To ensure that the only variable in the experiment was the shape of the object, the cross-sectional area (size) was held constant. The speed or terminal velocity of each object is directly related to its shape; therefore, its shape was essentially the only thing being tested. The velocity reached by each design was dependent on the way the shapes factored into the equation mg(sin[theta]) = (1/2)CA[rho]v_T².
Each shape was built using copper wire as a frame and tracing paper as a covering (tracing paper was chosen for its smoothness to reduce the amount of drag due to surface friction) with electrical tape, craft glue, and scotch tape serving as adhesives. The shapes tested (Figure 11) included a hemisphere with a diameter of 45.2 centimeters and a mass of 127.62 grams, two cones of different lengths (base radii of 22.6 centimeters, lengths of 20.45 centimeters and 56.62 centimeters, and masses of 110.89 grams and 173.18 grams, respectively), and a hemisphere attached to each cone (forming teardrop-like shapes) of masses 238.51 grams and 300.80 grams). All of these were placed on a wooden cart with plastic wheels which were mounted on a hollow metal cylinder which rotated around a smaller solid metal cylinder. The cart had a mass of 588.69 grams. The shapes were attached to the cart's cardboard flooring using plastic stands which held metal rods in a vertical position. The tops of the rods were attached to a second piece of cardboard which was also glued to the base of the shape. In the case of the teardrop shapes, the piece of cardboard was glued to the base of the hemisphere and to the base of the larger cone.

2. The Vscope

The angle of the ramp, the mass of each object, and cross-sectional area of the shapes were measured. The density of air ([rho]) is a known value. In order to determine the drag coefficient for each shape, a value for terminal velocity was needed in the equation mg(sin[theta]) = (1/2)CA[rho]v_T². For this case, we used the velocity the cart reached when it traveled down the ramp.
Vscope is a computer interface program. One of its sensors was attached to the cart and a tower was placed at the end of the ramp. The tower emitted a series of infrared queries that were received by the button stimulating it to emit an ultrasonic response. The difference between the time the infrared query was emitted and the time that the ultrasonic response was received was recorded. Using this data, the program plotted a velocity graph as the cart moved down the ramp. By averaging the data obtained for each shape, the velocity was found and substituted for v_T in the equation to determine C, the drag coefficient.

3. Performing the Tests

A ramp with a length of 300 centimeters was used at an angle of 1.20 degrees (.021 radians) from the ground. Eight trials were done for each shape, four with it mounted on the cart facing forward and four with it facing backward.
Each shape was mounted on the cart and allowed to roll down the ramp facing both forwards and backwards. To ensure that the results were accurate, all the trials were conducted consistently, with the cart placed on the ramp in exactly the same position and the tower interface set up at the same location at the bottom of the ramp. The VScope program plotted the velocity of the cart in each trial.

B. Results

The force of air resistance was calculated using the equation listed with the formulas. In the equation, m represented the mass of the object; this included both the cart and the sail. The gravitational force constant that exists on Earth is represented by g and is equal to 9.80 m/s². The sine of [theta] is the sine of the angle opposite the cart at its starting point at the top of the ramp. The silhouette surface area of the objects is A, and in this experiment that number remained constant at 0.160 m². The density of air was represented by [rho], and it also is maintained as a constant equal to 1.226 kg/m³. The final variable in the equation, v_T, is the terminal velocity experienced by the different structures in the experiment.
This equation is used to calculate the drag coefficients listed in Table 2. The shape that had the least drag coefficient and therefore the greatest aerodynamic properties was the hemisphere in its forward orientation with a coefficient of 3.84. The large cone in its backward orientation had the second lowest drag coefficient of 4.16. The large cone in its forward orientation had the fourth lowest drag coefficient. While the hemisphere in its back orientation had the highest drag coefficient of any shape at 5.89. Other shapes having lower drag in their forward orientation were the large teardrop and the small tear drop. The large teardrop experienced the second largest difference between coefficients in different orientations. The forward version had a coefficient of 4.70 while the backward version had a coefficient of 5.85. The small cone is the only other shape that performed better in the backward orientation.

Figure 15: Terminal Velocity of Hemisphere

Figure 16: Terminal Velocity of Backward Hemisphere

C. Analysis

1. Relative order of shapes

Looking at all of the drag coefficients, it is difficult to find a clear correlation between the shape of the body and the coefficient calculated. However, if only the simple shapes are considered (ignoring the teardrop shapes), it can be observed that a round frontal shape is more aerodynamic than a pointed frontal shape which is in turn, more aerodynamic than a flat frontal surface. This is illustrated by the order of shapes of approximately the same length when listed according to increasing drag coefficients: the hemisphere had the least drag coefficient, then the backward (point-forward) large cone, and then the forward large cone (flat surface facing front). In addition, lengthening the structure seems to lessen the disturbance caused by the structure moving through the air and, therefore, lessen the drag on the object. When comparing the cones, the large cone was more aerodynamic in both forward and backward orientations than the small cone was. The teardrops did not fit this pattern, however. The theory seems to indicate that the large teardrop with its hemisphere facing front should have the least drag coefficient. Instead, the small teardrop had a lower drag coefficient in both orientations than the large teardrop.

2. Sources of Error

Because experiments involve the real world and not the ideal one for which laws and theories are formulated, there can be a considerable amount of error. In this case, the sources of possible error mostly had to do with the testing equipment involved. The ramp was a piece of wood propped up at one end by a small block of wood. The wood of the ramp was somewhat warped and the fact that it was bent would have altered the way the cart traveled and its velocity. This was the single greatest source of error in the experiment. Figure 17 illustrates the imperfection of the board. The graph appears to show a terminal velocity for the cart. The cart, because of its small silhouette area should not reach a terminal velocity in the time it takes to travel down the ramp. Thus, one can conclude that the graph actually shows an imperfection in the board.
Further, since the shapes were made with copper wire and tracing paper; all were not perfectly uniform. Calculation of the cross-sectional area was not very accurate, but it was limited by the amount of precision available because of shape irregularities. Another major source of error was friction. The board used as a ramp was rough and there was friction between that and the wheels of the cart. Additionally, the room in which the testing occurred was air conditioned and that ventilation may have affected the cart's velocity as it moved down the ramp. As a result of these sources of error, the numerical values of the drag coefficients that were found cannot be considered reliable; however, the relative listing of shapes which was based on these drag coefficients can be used to draw conclusions.