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

i

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:

ii

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.

iii

CONTENTS

Page

Declaration i

Abstract ii

Contents iii

List of Figures v

1 Introduction 1

1.1 Literature Review 1

1.1.1 Wind Tunnel Operation and Instrumentation 1

Wind Tunnels 1

Measurement Systems 1

1.1.2 Airfoil Theory 2

Terminology of Airfoils 2

Forces on an Airfoil 2

Circulation 3

Kutta-Joukowsky Theorem 3

1.1.3 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

1.2 Objectives 5

2 Experimentation 5

2.1 Apparatus 5

2.1.1 Equipment 5

iv

2.1.2 Instrumentation 6

2.2 Procedures And Precautions 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

v

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 8

1

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

2

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.

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.

Figure 1 - Basic Airfoil Terminology (8)

3

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.

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.

(2)

(3)

(4)

Where

, = Force, in Newtons

= Moments, in N·m

, , = Coefficient parameters

A = Surface area, in m2

c = Chord length, in meters

4

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.

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.

(5)

Where

μ = Fluid viscosity, in kg/m·s

5

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.

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

Figure 3 - Wind Tunnel Schematic (9)

Figure 2 - Data for NACA 0012 Airfoil (8)

6

the airfoil. It could also change the angle of attack of the model, through the servo

motor.

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.

Figure 4 - Test Rig Schematic (9)

7

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.

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.

(6)

Where

φ = Pot value

8

2.5 Results

The results were plotted, in Figure 5 - Drag Coefficients against Angle of Attack, using

the values calculated in Appendix II.

3 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

Figure 5 - Drag Coefficients against Angle of Attack

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.500

-10.0 0.0 10.0 20.0 30.0

Coef

fici

ent

of

Dra

g

Angle of Attack (°)

Drag Coefficients for varying Angles of Attack

Re = 197 000

Re = 273 000

9

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.

10

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. 0-13-064633-

4

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.

11

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

12

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.

13

APPENDIX II – MEAN DRAG FORCE RESULTS

Angle

of

Attack

Mean

Drag

Force

Standard

Error

Drag

Coefficient

Drag

Coefficient

Error

For Re = 197 000

23.5 10.5283 0.0005 0.461 0.006

21.4 9.5030 0.0033 0.416 0.006

15.3 7.6680 0.0024 0.336 0.004

13.1 5.3827 0.0007 0.236 0.003

7.4 4.8419 0.0007 0.212 0.003

3.8 3.9232 0.0001 0.172 0.002

-0.7 3.9035 0.0003 0.171 0.002

-4.8 4.4277 0.0007 0.194 0.003

-1.2 3.8612 0.0002 0.169 0.002

7.3 4.3961 0.0004 0.193 0.003

15.9 8.1469 0.0007 0.357 0.005

23.6 10.4829 0.0006 0.459 0.006

For Re = 273 000

23.9 18.8434 0.0013 0.432 0.005

19.3 15.3940 0.0016 0.353 0.004

16.2 13.5693 0.0016 0.311 0.004

10.9 7.9995 0.0011 0.183 0.002

8.2 7.0660 0.0003 0.162 0.002

4.1 5.9702 0.0011 0.137 0.002

-1.1 5.8297 0.0004 0.134 0.002

-4.5 6.6874 0.0012 0.153 0.002

-0.9 5.6938 0.0003 0.131 0.002

7.4 6.9120 0.0005 0.158 0.002

15.9 13.7202 0.0008 0.315 0.004

26.0 20.2353 0.0020 0.464 0.005

14

APPENDIX III – RISK ASSESMENT FORM

Hazard Identification Risk Assessment Risk Control Review

No. What harm can it

cause?

Risk Score Control Measures

already implemented

Harm

Reduction

Whose

responsible

By when Controls

effective

Date finalised

1 Noise of Wind

tunnel

Low None PPE User Near future Unknown

2 Pressure within

wind tunnel

Low None Isolation School Near future Yes

3 Electricity supplied

to motor

Low Electrical cables well

shielded

Isolation School Near future Yes

4 Clutter around wind

tunnel

Moderate Cabinets used for

storage

Administrative School Next operation

of wind tunnel

Yes

5 Accidental start up

of wind tunnel

during maintenance

High Safety switch and

administrative control

Engineering Technician Immediate Unknown