Why do Golf Balls have Dimples on Their Surfaces?

Transcription

1 Name: Partner(s): 1101 Section: Desk # Date: Why do Golf Balls have Dimples on Their Surfaces? Purpose: To study the drag force on objects ith different surfaces, ith the help of a ind tunnel. Overvie In this experiment, e ill use a ind tunnel to study ho the flo of air, over various balls, gives rise to a drag force. You sa in the Falling Cat lab that the air resistance, or the drag force, increases as the speed increases. We mathematically modelled this drag force as: F D bv Here, b is a coefficient that takes the shape of the object into account, along ith other factors like hat the object is falling through. If e further break don b into its parts, e can see this drag force in even more detail.

2 Separating out the cross sectional area A of the object, and the density of the fluid ρ, the equation becomes F D 1 Av D C (1) here CD is a dimensionless quantity knon as the drag coefficient. It is affected by the object s features, like the shape, the material etc. At lo speeds, it is approximately constant, hich means the drag force is proportional to v. Hoever, at high speeds, the drag coefficient changes, and the situation becomes rather more complicated. Figure 1 shos the drag coefficient of a ball moving in air. Drag coefficient CD Speed v Figure 1 For aerodynamics, it is important to kno the drag coefficient at lo and high speeds. There are no theoretical equations for it, so it can only be measured experimentally. That is hat e are doing today. The steps e ill follo are: Measure the drag force FD; Measure the cross section area A; Measure the speed of air flo v in the ind tunnel; Once the above measurements are done and the density of the fluid (air) is knon, Equation (1) allos us to calculate the drag coefficient CD. If e do that for different speeds, e can plot CD vs. the speed and get a curve similar to Figure 1.

3 Theory: Bernoulli equation Bernoulli s equation states that for an incompressible fluid, along any streamline of flo, the quantity 1 P v gh is a constant, or 1 1 P1 v1 gh1 P v gh here P is the pressure, ρ is the density of the fluid, v is the flo speed and h is the height relative to an arbitrary zero. Streamlines can be thought of as the paths folloed by the fluid particles. Wind tunnel A ind tunnel is a device that uses a fan to create a region in hich air moves ith constant velocity. The design is optimized to create a flo that is as smooth as possible in the central test section. Figure : Flo of air through a ind tunnel. A tennis ball is shon mounted in the test section. Finding the speed of air flo in the ind tunnel In order to use Bernoulli s equation to eventually find CD, e ill need to go step-by-step. The next couple of pages ill take you through the derivations. Choosing a streamline from far outside the ind tunnel (here the speed of air va= 0) to the test section, 1 Bernoulli s equation gives (ρa is the density of air): 1 P 0 Agh P Av Agh So v, the speed of air flo in the test section, can be found by measuring the pressure in the test section.

4 Question 1: In the space belo, derive the relation beteen v and the pressure difference P P, assuming the density of air ρa is knon (symbols only). Measure the pressure difference ith a manometer: To measure pressure difference, e ill use a manometer. A simple manometer is just a U tube ith ater (density ρ) inside: P P P P 1 h h height=0 1 height=0 Bernoulli Equation: P 0 0 P 0 gh Or: P P gh Bernoulli Equation: P 0 gh P 0 0 Or: P P gh The pressure of the test section in the ind tunnel is loer than ospheric pressure (the situation on the right). So P P ghd here hd is the reading of the Dyer manometer attached to the ind tunnel. Question : In the space belo, derive the equation to calculate the speed of air flo v from the reading of the Dyer manometer (hd), assuming the density of air ρa and the density of ater ρ are knon.

5 Finding the drag force on a ball If air had zero viscosity, then the flo of air around a sphere ith a smooth surface ould be as shon on the left. Notice that the flo is steady and that streamlines both upstream and donstream of the sphere have the same shape. The pressure ould thus also the same on both sides, according to Bernoulli s equation. The drag force on the ball ould be zero in this - not very realistic - case. The actual flo around a sphere only resembles this for very lo flo speeds. As the speed ith hich air passes over the surface of the ball increases, the streamlines eventually are unable to follo the contour of the ball, and separate from its surface. For a smooth ball, this separation of flo occurs near the idest extent of the ball, as shon on the right hand side of the figure above. Donstream of this the air forms a ake. A ake is the region of relatively sloly moving air behind the ball. Comparisons of a velocity profile of air upstream of the ball and in the ake behind it are shon in the figure belo. Because of the pressure difference beteen the front and back of the ball, there is a net force, hich is the drag force, FD. In order to find the drag force, 16 little manometers are installed on the donstream side of the ball to measure the pressure difference. The height in each manometer, hi, measures a pressure difference: P P P gh here i = 1,,, 16 i i i Roughly speaking, the average height of the manometer, havg, ill help to measure the average pressure difference ΔPavg. Multiplied by the cross sectional area A, e can get the drag force.

6 Question 3: Calculate the drag force FD from the average height of 16 manometers havg and the cross sectional area A. (You are still orking in symbols only.) Finally, e use Equation (1) from page, and the ansers derived in Questions and 3 to calculate the drag coefficient. Question 4: Derive the equation to calculate the drag coefficient CD in the space belo. The derivations are complete. No e ill move on to data and numerical calculations. Numbers you need in your calculations: Gravitational acceleration g = (9.81 ± 0.01) m/s The density of air ρ A (room temperature) = 1.17 kg/m 3 (±%) The density of ater ρ = kg/m 3 (±0.%) The diameter of the golf ball and the rounded golf ball are both: (4.5 ± 0.05) cm 1 inch =.54 cm = m (Uncertainty calculations are not required in this lab. Make all your results to the correct number of significant digits, but remember to not do any rounding during calculations.) Sample data (assigned A, B or C): There are 3 sample-data pages available, and one ill be given to you in class. In the lab, you ill be producing a data set that ill look like these, so hile you are aiting for your turn at the ind tunnel, you can practice measuring the drag coefficient. You should have one that is different from your partner, so calculate the folloing: 1. Calculate the flo speed of ind tunnel.

7 . Measure / calculate the average height of 16 manometers in inches. 3. Calculate the cross sectional area A. 4. Calculate the average pressure difference across the ball and then the drag force on the ball. 5. Using Equation (1), calculate the drag coefficient CD. 6. Calculate the drag coefficient using your simplified equation from Question 4.

8 Wind Tunnel data taking and analysis: 1. Change the speed of the air flo and observe the change of the manometer readings.. You ill be assigned an airflo speed and a ball to analyze. For your given case, finish a data sheet like the ones in the sample data. Everyone in your group ill do their on data sheet. 3. Calculate the drag coefficient CD using your simplified equation from Question 4 and enter it in the 1101 Collective Data sheet. Do the calculation on the back of the data sheet for your case. 4. On the back of the data sheet for your case, do all of the complicated calculations (ind speed, pressure difference, etc.) like you did for the sample data. The drag coefficient should come out the same as the simplified version. 5. Once all of the data for the balls has been collected, the 1101 Collective Data sheet ill be ed to you. Dra the CD vs v curves for both types of ball. You can do this on graph paper or by computer, ith both curves on one graph, on the same scale. 6. Attach your data sheet and the CD vs v curves to this handout. Revie and Discussion 1. Summarize hat you can measure ith a ind tunnel. What is the purpose of a ind tunnel? (Do not just say calculate pressure differences or hatever. Talk about hat the ind tunnel can do for any kind of research or mechanical design.)

9 . Why do golf balls have dimples on their surfaces? Justify your anser by referring to your CD vs v curves from the measurements made in this lab. (Compare the curves for the dimpled and smooth balls. Also refer to the purpose/goal of golf.) 3. What other balls or animals have special surfaces that can strongly affect ho they move through fluid? Describe ho they ork, referring to hat you learned in this lab. Note that this is referring to surface texture, not body shape.