University of the Witwatersrand, Johannesburg School of Mechanical, Industrial, and Aeronautical Engineering
Title:
Variation of the drag coefficient with Reynolds number
Subject:
MECN3007 - Mechanical Engineering Labs II
Author:
Brandon Heukelman (555597)
Due date:
Thursday 18 April 2013
DECLARATION I, Brandon Heukelman (555597), declare that this laboratory report is my own, unaided work, except where otherwise acknowledged. It is being submitted for the degree of Bachelor of Science in Mechanical Engineering in the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination at any other university. I further declare that I am aware that plagiarism (the use of someone else’s work without their permission and/ or without acknowledging the original source) is wrong. I understand that the University of the Witwatersrand, Johannesburg may take disciplinary action against me if it can be shown that this task is not my own unaided work, or that I have failed to acknowledge the source of the ideas or words in my writing in this task.
Name: Brandon Heukelman Student Number: 555597 Group Number: 25 Due Date: 18th April 2013
Signature: i
ABSTRACT The objective of this experiment was to calibrate the angle of attack of the model, to collect accurate data on the drag forces at different angles of attack, and lastly to compare drag coefficients against the angle of attack for different Reynolds numbers. This was done by using a closed-circuit wind tunnel and an external balance. It was found that drag is a minimum when no lift is produced, and this drag, called parasitic drag, is inversely proportional to the Reynolds number. When the Reynolds number is increased the drag coefficient is decreased throughout the range of angles tested. At low angles of attack, the drag coefficient varies parabolically. When the stall angle is reached, the drag coefficient varies linearly. The gradient of this linear trend is directly proportional to the Reynolds number. The stall angle was also found to decrease with an increase in Reynolds number.
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CONTENTS Page Declaration
i
Abstract
ii
Contents
iii
List of Figures
v
1
1
Introduction 1.1
Literature Review
1.1.1
1
Measurement Systems
1
Airfoil Theory
2
Forces on an Airfoil
2
Circulation
3
Kutta-Joukowsky Theorem
3
Dimensionless Parameters
4
Geometric and dynamic similarity
4
Reynolds Number
4
Coefficients of Lift, Drag and Pitching Moments
4
1.1.4
Previous Studies
4
Joukowski Airfoil
4
NACA0012 Airfoil
4
Objectives
Experimentation 2.1
2
Terminology of Airfoils
1.1.3
2
1
Wind Tunnels
1.1.2
1.2
Wind Tunnel Operation and Instrumentation
1
Apparatus
2.1.1
Equipment
5 5 5 5 iii
2.1.2 2.2
Instrumentation
Procedures And Precautions
6 6
2.2.1
Procedure
6
2.2.2
Precautions
7
2.3
Observations
7
2.4
Data Processing
7
2.5
Results
8
3
Discussion
8
4
Conclusion
9
List of References
10
Appendix I – Uncertianty Analysis
11
Angle of Attack
11
Plan Form Area of Airfoil
11
Density
11
Drag Coefficient
11
Appendix II – Mean Drag Force Results
13
Appendix III – Risk Assesment Form
14
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LIST OF FIGURES Figure 1 - Basic Airfoil Terminology (8)
2
Figure 2 - Data for NACA 0012 Airfoil (8)
5
Figure 3 - Wind Tunnel Schematic (9)
5
Figure 4 - Test Rig Schematic (9)
6
Figure 5 - Drag Coefficients against Angle of Attack
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v
1
INTRODUCTION
1.1
Literature Review
1.1.1 Wind Tunnel Operation and Instrumentation Wind Tunnels A low-speed wind tunnel is, in essence, a large venturi where airflow is driven by a fan connected to some motor. The fan draws air through the venturi, with the model placed inside. The nozzle of the venturi increases the velocity (hence decreasing the pressure) of the airflow, once the airflow passes the test section, the diffuser returns the airflow to the previous velocity and pressure as efficient as possible (1). There are two general wind tunnel types: open or closed circuit type. The closed circuit type reuses the airflow from the exhaust forming a loop, thus reducing operating costs but the extra ducting requires more space. The open circuit type draws air directly from the atmosphere and is exhausted out the back. Wind tunnel interference or turbulence is defined as the relative magnitude of velocity fluctuations in the three planes (2). To important factors to be measured is the intensity and scale of turbulence. The significant effect of turbulence is important in the boundary layer region. However, the stability of simple shapes, such as airfoils, are highly insensitive to turbulence in the wind tunnel, as seen in the work by Volluz (2). Measurement Systems The pitot-static probe (more commonly known as a pitot tube) is used to measure the velocity of a fluid stream at a point. The probe causes the fluid stream to stagnate; this pressure is then measured and is known as the total pressure (po). The static pressure (p) is also measured through a static pressure orifice. The difference in these pressures is known as the dynamic pressure, from Bernoulli’s equation. Once the dynamic pressure is known, the velocity can be calculated from equation 1. (1) Where = dynamic pressure, in Pascal = fluid stream velocity, in m/s po = total pressure, in Pascal p = static pressure, in Pascal ρ = fluid density, in kg/m3
1
A multi-tube manometer is created when one limb of the U-tube has a cross section sufficiently large that the level of the fluid does not appreciably change. This limb or reservoir can then be connected to a bank of tubes measuring different pressures. At least one of these tubes is required for a reference. This reference level can easily be changed by raising or lowering the reservoir. (2) Internal or external balances measure forces on the model within the wind tunnel. Internal balances are fixed to the model, and only indirectly give values of forces. External balances are placed outside the wind tunnel, and require ample space. However, external balances have three advantages over internal balances. Firstly, it can measure large forces with a high accuracy, secondly the model may be mounted anywhere with respect to the moment center of the balance and lastly less space is used within the wind tunnel. (2) 1.1.2 Airfoil Theory Terminology of Airfoils Airfoils are composed of a leading edge and trailing edge. The leading edge is usually rounded, while the trailing edge tapers off. This is seen in Figure 1 - Basic Airfoil Terminology. The chord is the length between these two edges and is named the chord line. The camber is the midpoint between the upper and lower edges. The angle of attack is the angle between the chord line and the oncoming fluid stream.
Figure 1 - Basic Airfoil Terminology (8) Forces on an Airfoil At small angles of attack, fluid flow around the airfoil remains attached, thus no turbulence is formed. There is also very little induced or pressure drag during this time; most drag is from viscous effects.
2
The lift produced by an airfoil is dependent mostly on the geometric factors of airfoil. When the flow is symmetrical, no lift is produced, this may occur at a negative angle for some airfoils. Lift can be calculated by equation 2. Drag is the force that resists the motion of fluid over the airfoil. It could be due to pressure on the leading edge or from viscous effects of the fluid. Drag can be calculated by equation 3. The pitching moments depend on where the moments are taken, which is calculated by equation 4. The point usually chosen is a quarter of the chord from the leading edge. The point where no moments are produced is called the aerodynamic center. (2) (3) (4) Where = Force, in Newtons , = Moments, in N·m = Coefficient parameters , , A = Surface area, in m2 c = Chord length, in meters Circulation Circulation is defined as the line integral of the velocity around any closed curve (3). When the airfoil is placed in steady flow two stagnation points are created, one at the leading edge and one at the trailing edge. Normal ideal calculations show the rearward stagnation point is slightly above the trailing edge. Although in reality the rearward stagnation point is at the trailing edge, this is known as the Kutta condition (3). The stagnation point is moved by adding circulation, this creates a pressure difference between the upper and lower surfaces. Kutta-Joukowsky Theorem This theorem states that any cross section that has circulation around it, within a fluid stream, produces a lifting force (3). This holds for any structure as long as the region with circulation is fully enclosed. The theorem assumes that there is smooth flow around the airfoil, i.e. for small angles of attack when flow is still laminar.
3
1.1.3 Dimensionless Parameters Geometric and dynamic similarity Geometric similarity depends on size and shape of the object in question. While dynamic similarity requires that dimensional parameters are equivalent between the model and prototype. Reynolds Number The Reynolds number, in equation 5, is the measure of the ratio of inertia to viscous forces. This parameter describes the type of flow around the object. (5) Where μ = Fluid viscosity, in kg/m·s Coefficients of Lift, Drag and Pitching Moments Coefficients are dimensionless quantities, which change when the angle of attack and Reynolds number changes. These coefficients allow the researcher to compare aerodynamic forces of different airfoils. 1.1.4
Previous Studies
Joukowski Airfoil The airfoil is created by using a conformal mapping, called the Joukowski transformation. This mapping takes an airfoil shape and converts to a simple geometry, namely a circle. This allows the researcher to establish theoretical flow coefficients of the complex geometry. NACA0012 Airfoil This is a symmetrical airfoil with a width of 12% of the chord. As seen in Figure 2 Data for NACA 0012 Airfoil.a, the theoretical coefficient of lift rises linearly with the angle of attack, until it reaches a maximum value then rapidly falls off. This occurs because of flow separation, and the airfoil stalls. In Figure 2 - Data for NACA 0012 Airfoil.b, how the theoretical drag coefficient rises much faster than the lift coefficient. The lift coefficient then reaches its maximum value then falls away while the drag coefficient continues to increase.
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Figure 2 - Data for NACA 0012 Airfoil (8) 1.2
Objectives
The objectives of this experiment were: 1.
To calibrate the angle of attack for the Joukowsky airfoil.
2.
To accurately collect data about the drag experienced on the Joukowsky airfoil.
3.
To compare the coefficient of drag at different Reynolds’s numbers.
2
EXPERIMENTATION
2.1
Apparatus
2.1.1 Equipment A closed-circuit wind tunnel was used for this experiment, as described in Figure 3 Wind Tunnel Schematic.
Figure 3 - Wind Tunnel Schematic (9) A test rig, as show in Figure 4 - Test Rig Schematic, was used in the experiment. It supported the model and included an external balance to measure the forces acting on 5
the airfoil. It could also change the angle of attack of the model, through the servo motor.
Figure 4 - Test Rig Schematic (9) 2.1.2 Instrumentation Bubble Inclinometer: Used to calibrate the angle of attack of the model. Thermometer:
Used to measure air temperature, with uncertainty of 0.5°C.
Data Logger:
Recorded values of the forces acting on model, at 2.5 Hz, against time.
Computer Station:
Allows control and display of data recorded, the angle of attack and temperature.
The pitot tube and manometer was not operational and values were obtained via the lab assistant. 2.2
Procedures And Precautions
2.2.1 Procedure 1. Calibrate the angle of attack with the bubble inclinometer, by: a. Aligning the inclinometer with the model, and reading the angle b. Change the angle of attack, and make another reading. c. Now a linear relationship can be assumed between the point’s measures. 2. Record the initial atmospheric conditions with thermometer and barometer. 3. Place the model securely within the test chamber, then seal the test chamber. 4. Start the wind tunnel, and increase motor speed until the correct airspeed is reached. 5. Insert pitot tube to find the pressure difference and then calculate airspeed. 6. Make any necessary adjustments to motor speed. 6
7. Adjust the angle of attack of the model until stall angle is reached. 8. Decrease angle of attack incrementally while taking necessary measurements. 9. Increase airspeed, and repeat steps 5 to 9. 2.2.2 Precautions Refer to Appendix III, for the Laboratory Risk Assessment. 1. Ensure that the Pitot tube is installed perpendicular to the airflow. 2. Make sure the bubble is not touched until the reading has been taken. 3. Once the motor speed has been changed, allow a steady state to form. 2.3
Observations
Atmospheric conditions were observed at 303 K and 83.3 kPa within the wind tunnel. Although the temperature was recorded with every measurement, there was little change in the temperature. Hence, it was decided to take the temperature as constant. The angle of attack was calibrated with the inclinometer, and it was found that the angle of attack could fit equation 6. (6) Where φ = Pot value Flow visualization was in the form of tufts attached to the airfoil. These tufts were placed in strategic locations, to allow estimates of the flow patterns produced. These locations are places such as the trailing edge, upper surface, and lower surface. 2.4
Data Processing
Results were handled in spreadsheets, using Microsoft Excel. It was quickly found that this program had very limited high-volume data handling. However, once the data had been broken into smaller sizes, the data could be handled easily. Data was recorded at 2.5 Hz throughout the experiment. The data was then dissected into sections for each angle of attack, where each section held more than 3000 data points. The mean value was taken in section and the standard error was calculated, as seen in appendix I. The mean results are in appendix II.
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2.5
Results
The results were plotted, in Figure 5 - Drag Coefficients against Angle of Attack, using the values calculated in Appendix II. Drag Coefficients for varying Angles of Attack
0.500 0.450
Coefficient of Drag
0.400 0.350 0.300
Re = 197 000
0.250 0.200
Re = 273 000
0.150 0.100 0.050
0.000 -10.0
3
0.0
10.0 20.0 30.0 Angle of Attack (°) Figure 5 - Drag Coefficients against Angle of Attack
DISCUSSION
The angle of attack was found, in degrees, by calibrating the sting. This relationship was assumed linear. Hence, a straight-line equation could be fitted to the data. The drag caused by the sting was assumed to be constant, through all the angles of attack experienced. Thus it took no part in the calculations. The coefficient of drag has a parabolic nature at low angles of attack (-10° to 14°). At larger angles of attack (> 14°), a linear relationship forms. The minimum drag coefficient is due to the parasitic drag only, thus no lift force is produced at these angles of attack (-1.1° for Re = 197000 and -2.4° for Re = 273 000). The parasitic drag is composed of form, skin friction and interference drag. Only the shape of the airfoil, which is kept constant throughout the experiment, produces form drag. The surface finish of the airfoil, which is also kept constant throughout the experiment, produces skin friction drag. When the drag coefficient is greater than the minimum, induced drag is formed. This occurs because of the pressure gradient between the upper and lower surfaces of the 8
airfoil, which generates a lifting force. This shows that no system can be 100% efficient. When the stall angle (± 14°) is reached, the relationship becomes linear. The gradient of this relationship is sharply increased. This occurs because of the separation of the boundary layer from the airfoil. Boundary layer separation occurs because of particles reaching a pressure gradient that they cannot overcome. It had been observed that the airfoil with a greater Reynolds number has a lower drag coefficient throughout the range of angles tested. Parasitic drag is lower (0.132 < 0.173) and the angle (-2.4° < -1.1°) that this minimum occurs is lower for a higher Reynolds number. The stall angle is lower for the high Reynolds number test (11° < 14°). This shows that boundary layer separation occurs earlier at higher Reynolds numbers. The gradient of the linear region is also higher for the large Reynolds number test. This shows that there is a larger pressure gradient, for the air particles, to overcome. Without dimensionless parameters to compare, extracting trends from results would be much more difficult. This shows that dimensionless parameters are vital when comparing data. 4
CONCLUSION
The experiment was a success and the following conclusions could be drawn:
Calibration of the model’s angle of attack is necessary for accurate interpolation of the angle in degrees.
The induced drag coefficients, at low angles of attack, increase parabolically.
Above the stall angle, induced drag forces increase linearly, because of the separation of the boundary layer.
The drag coefficients, at low angles of attack, are inversely proportional to Reynolds number.
Parasitic drag coefficients (no lift drag coefficient) are inversely proportional to Reynolds number.
The stall angle of the airfoil is inversely proportional to Reynolds number.
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LIST OF REFERENCES 1. D, Anderson J. Fundamentals of Aerodynamics. 1st. New York : Macgraw Hill, 1984. 0-07-001656-9. 2. J, Volluz R. Handbook of Supersonic Aerodynamics: Wind Tunnel Instrumentation and Operation. Ordnaice Aerophysics Laboratory. Daingerfield : The John Hopkins University, 1961. 20. 3. J, Bertin J. Aerodynamics for Engineers. 4th. s.l. : Prentice Hall, 2002. 0-13-0646334 4. Baals D. D, Corliss W. R. Windtunnels of NASA. Scientific and Technical Information Branch, NASA. Washington : NASA, 1981. NASA SP-440. 5. G, Davanport A. Wind Tunnel Testing: A General Outline. Faculty of Engineering Science. Ontario : The University of Western Ontario, 2007. 6. M, White F. Fluid Mechanics. 4th. Rhode Island : Macgraw Hill, 2007. 7. Scott, Jeff. Angle of Attack and Pitch Angle. [Web] [ed.] http://www.aerospaceweb.org. s.l. : http://www.aerospaceweb.org, 29 Febraury 2004. 8. T. Cebeci, E. Besnard, H. H. Chen. An interactive boundary-layer method for multielement airfoils. Long Beach : California State University, 1988. 9. Naidoo, Prinal. Wind tunnel testing. School of Mechanical, Industrial, and Aeronautical Engineering, University of Witwatersrand, Johannesburg. s.l. : University of Witwatersrand, Johannesburg. student short report.
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APPENDIX I – UNCERTIANTY ANALYSIS Angle of Attack
Plan Form Area of Airfoil
Density
Drag Coefficient Drag force error (
) is the standard error calculated by the standard deviation (σ) and
the number of data points (n). The standard deviation is different for each angle of
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attack, because of the change in the drag force and number of elements. Hence, each data point has a different standard error, which can be seen in Appendix II.
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APPENDIX II – MEAN DRAG FORCE RESULTS
Angle of Attack
Mean Drag Force
23.5 10.5283 21.4 9.5030 15.3 7.6680 13.1 5.3827 7.4 4.8419 3.8 3.9232 -0.7 3.9035 -4.8 4.4277 -1.2 3.8612 7.3 4.3961 15.9 8.1469 23.6 10.4829 23.9 19.3 16.2 10.9 8.2 4.1 -1.1 -4.5 -0.9 7.4 15.9 26.0
18.8434 15.3940 13.5693 7.9995 7.0660 5.9702 5.8297 6.6874 5.6938 6.9120 13.7202 20.2353
Drag Standard Drag Coefficient Error Coefficient Error For Re = 197 000 0.0005 0.461 0.006 0.0033 0.416 0.006 0.0024 0.336 0.004 0.0007 0.236 0.003 0.0007 0.212 0.003 0.0001 0.172 0.002 0.0003 0.171 0.002 0.0007 0.194 0.003 0.0002 0.169 0.002 0.0004 0.193 0.003 0.0007 0.357 0.005 0.0006 0.459 0.006 For Re = 273 000 0.0013 0.432 0.005 0.0016 0.353 0.004 0.0016 0.311 0.004 0.0011 0.183 0.002 0.0003 0.162 0.002 0.0011 0.137 0.002 0.0004 0.134 0.002 0.0012 0.153 0.002 0.0003 0.131 0.002 0.0005 0.158 0.002 0.0008 0.315 0.004 0.0020 0.464 0.005
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APPENDIX III – RISK ASSESMENT FORM Hazard Identification
Risk Assessment
Risk Control
Review
No. What harm can it Risk Score cause?
Control Measures Harm already implemented Reduction
Whose responsible
By when
Controls effective
1
Noise tunnel
of
Wind Low
None
PPE
User
Near future
Unknown
2
Pressure wind tunnel
within Low
None
Isolation
School
Near future
Yes
3
Electricity supplied Low to motor
Electrical cables well Isolation shielded
School
Near future
Yes
4
Clutter around wind Moderate tunnel
Cabinets storage
5
Accidental start up High of wind tunnel during maintenance
Safety switch and Engineering administrative control
used
for Administrative School Technician
Date finalised
Next operation Yes of wind tunnel Immediate
Unknown
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