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Contents
About the AE1111-I project ... 2
Grading ... 8
2D Pressure distributions on aerofoils ... 9
LI Lift and weight ... 21
DR Drag and Longitudinal stability ... 30
TH Thrust ... 37
PW Power ... 42
3D Execution of 3D Wind Tunnel Experiment on a Swept Wing ... 47
AD Aerodynamic Design ... 57
RE Range and Endurance ... 62
FE Flight envelope, climb rate and glide ... 65
BT Building and testing the aerodynamic model... 67
M1 Mars Mission Design I ... 70
M2 Mars Mission Design II ... 72
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Change sheet
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About the AE1111-I project
The AE1111-I project is a first exploration of the aerospace engineering field, which means it aims to enhance knowledge taught in parallel courses by using the exploration theme.
Aerospace engineers create machines, from airplanes that weigh over a half a million pounds to spacecraft that travel over 17,000 miles an hour. They design, develop and test aircraft, spacecraft and missiles and supervise the manufacturing of these products. Aerospace engineers develop new technologies for use in aviation, defence systems and space exploration, often specializing in areas such as structural design, guidance, navigation and control, instrumentation and communication, or production methods. They often use computer-aided design (CAD) software, robotics, and lasers and advanced electronic optics. They also may specialize in a particular type of aerospace product, such as commercial transports, military fighter jets, helicopters, spacecraft, or missiles and rockets and can be experts in aerodynamics, thermodynamics, celestial mechanics, propulsion, acoustics, or guidance and control systems. Almost all jobs in engineering require some sort of interaction with co-workers. Whether they are working in a team situation, or just asking for advice, most engineers must have the ability to communicate and work with other people. Engineers should be creative, inquisitive, analytical, and detail-oriented. They should be able to work as part of a team and to communicate well, both orally and in writing. Communication abilities are important because engineers often interact with specialists in a wide range of fields outside engineering.
This project is focused on flying wings, which have long been a dream of many designers. The biggest problem found when building a flying wing aircraft is that such designs are inherently unstable and they do not easily stay in level flight. Yet such an all-wing aircraft would have excellent payload and range capabilities because it produces less drag than a conventional aircraft as the tail and the fuselage of a conventional aircraft are responsible for a significant amount of drag. Eliminate the tail and fuselage and you might be able to eliminate a great deal of drag, enhance performance, reduce the amount of fuel required and improved the handling capabilities of the airplane; an attractive prospect in the age of fuel running short and an increase in air traffic transportation.
This manual is the general guideline for the different parts of the AE1111-I project and tells you, the student, what has to be done for this project. Most of the exercises and lab sessions will aid you in designing and building your own actual flying wing except for the part of the project called ‘Mars Mission Design’. In that part of the project you will explore, with your team, the design of a solar-powered aircraft which can map part of the surface on Mars and present the design on a poster. You will find that some of the exercises throughout the student manual also refer to Mars. They can be treated as stand-alone exercises that are not related to the Mars Mission Design.
The schedule for this project can be found on the AE1111-I BrightSpace site, along with other important information for this project. It should be noted that attendance to all project parts is mandatory, according to the exam regulations, and will be registered. You can be absent for no more than two project parts per period, which must be compensated for in consultation with your tutor. If you are absent more than twice or do not compensate for being absent, you can be expelled from the project.
It is recommended that you bring your own laptop with you during the project, since there are only two computers available per project group in the project rooms.
As every part of this project requires a report answering all questions stated, a short description is given below about the things required in a report:
Frontpage: Frontpage, including student names, student number, group number and title.
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Introduction: shortly introduce the topic and the outline of the report
Bibliography: All websites, books or information retrieved elsewhere, must be clearly and properly documented at the end of the report. The reference list at the end of this chapter can be seen as a good example.
Table with contributions of each student, to keep track of what you and your colleagues have been doing during project, but it also important for the teaching assistants to keep track that work is distributed equally.
Although the course AE1110-I Introduction to Aerospace Engineering is running parallel with this course, it might occur that the project is running ahead. Nothing can be done about this unfortunately, and you, the students, are asked to do some discovery yourself.
If you have any questions or suggestions about this project, please pass these on to your mentor, so he or she can further take that into consideration. We hope you enjoy taking part in this thematic project!
Aerodynamics of flying wings
Designing an aircraft, such as a flying wing, consists of many steps going from preliminary to detailed design in which relevant disciplines like aerodynamics, propulsion, flight dynamics and structures are considered.
With respect to aerodynamics, we will limit ourselves to the main aspects of aerodynamics of flying wings, where complicating elements like the fuselage and the tail planes (denoted as the empennage) are left out of the analysis. The wing is the main element of the aircraft that exhibits all physical aerodynamic phenomena that occur, which makes a flying wing attractive as a study object to elucidate the discussions on the basic laws of aerodynamics as discussed in chapters 1 to 5 of Introduction to Flight (Anderson, 2008). Moreover, flying wings may have some advantages when it comes to aerodynamic or performance efficiency when they are used as a simple platform to carry equipment, such as cameras, either in the Earth’s atmosphere or at remote planets.
NASA Mars Explorer (1999) Northrop YB-49 (1948)
Blended Wing Body (BWB) design Helios, high altitude research plane (2001) Figure 1: Examples of “flying wing” aircraft with different missions.
The flying wing shows remarkable similarities with the blended wing body (BWB) (Figure 1) which is being studied word-wide. In this sort of design the entire airplane (including the
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fuselage) generates lift and is streamlined in such a way to minimize drag and produce a high lift-to-drag ratio. On conventional transport aircraft, the cylindrical fuselage is a source of drag and generates close to no lift. Therefore, the design of a flying wing can have significantly better fuel characteristics than aircraft with a traditional layout. There are however some serious drawbacks to flying wings which need to be overcome, such as the inherent aerodynamic longitudinal instability and the limited space for cargo if the wing thickness is a limiting factor (drag). You may find a lot more information that is relevant for this typical aircraft design in open literature.
As the aerodynamics of wings is discussed extensively in Anderson’s book we will frequently refer to this work for further reading. Furthermore, use will be made of the sample problems at the end of the chapters to practice solving simple problems in the same manner as will be done in subsequent exams.
In subsequent chapters the lift production of wings will be discussed, starting with the typical behaviour of aerofoils. This will be done through a dedicated wind tunnel test in which the pressure distribution is analysed. The results may be assumed to be typical for aerofoils in the low speed regime, i.e. the effects of compressibility are neglected (M<0.3). In reality compressible flow is found in most instances on larger transport aircraft and you are encouraged to carefully go through the particular sections of Anderson’s book.
On Mars, the compressible regime is reached for even lower velocities. This is because of the low temperature which lowers the speed of sound. Several wind tunnels exist to simulate the Mars environment. The university of Aarhus (Denmark) operates a wind tunnel that is mainly used for instrument testing and properties of the dust in the Martian atmosphere. NASA has a wind tunnel for the same purpose. The University of Oxford uses their wind tunnel to test wind meters.
Aarhus: http://marslab.au.dk/en/windfacilities/
NASA: http://www.nasa.gov/centers/ames/business/planetary_aeolian_facilities.html Oxford: http://www.atm.ox.ac.uk/user/wilson/matacf.html
The way how lift, drag and pitching moment are created for 3-dimensional wing situations will be considered as well as this is necessary to design the wing in a later stage of the project. Here again the characteristics and aerodynamic phenomena are studied using wind tunnel tests. Also, simplified theory as discussed in Anderson, 2008 is used. One should realize that more accurate and sophisticated techniques are required to arrive at an acceptable design. However, this extension is left to forthcoming lectures in the later years in the BSc and MSc curriculum. In this project, the analysis of the aerodynamic characteristics is merely used to get acquainted with the basic laws of aerodynamics.
Flight Mechanics and Space Systems
The term flight mechanics is commonly used for the combination of aircraft performance, stability and control (Torenbeek and Wittenberg 2009). The field of aircraft stability and control is also known as flight dynamics. In this first-year course on flight mechanics, the main emphasis is on aircraft performance whilst the subject of flight dynamics will be treated in the second and third year of the BSc curriculum.
Aircraft performance can be defined as the ability of an aircraft to effectively fulfil its mission. Consequently, performance includes the study of the forces acting on an aircraft and the effects that these forces have on the resulting translational motion. Key elements in the field of aircraft performance are how far, high and fast a certain aircraft can fly. The elementary forces acting on the aircraft are aerodynamic forces, propulsive (thrust) forces and the gravitational force. Aircraft performance is therefore a topic where the results from several disciplines must be integrated.
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A typical feature of an aircraft is its type of propulsion system. The main engine types for manned aircraft are the piston engine, the turbofan and the turboprop. The turbofan is mainly used for high-subsonic, long-haul aircraft. The conversion of the power delivered by piston engines and turboprops into thrust is accomplished by means of a propeller.
A distinction can be made between point performance and path performance. Point performance capabilities concern the performance of an aircraft at a given point in time, that is, at a given point on the flight path. This means that point performance values refer to instantaneous quantities, such as maximum speed in level flight or maximum rate of climb. Path performance is related to a duration of the flight, such as time to climb and range and may often be established by integration of quasi-steady point performance capabilities. The study of performance problems in which the exchange between kinetic and potential energy is of significance may fall within the category of flight dynamics.
Design performance concerns the prediction of the capabilities of a new aircraft type and extends to include the validation of the estimates by executing flight tests. To prove compliance with the airworthiness requirements experimental data is required. This is also the case for the construction of the aircraft flight manual (AFM), specifying the conditions under which the aircraft can be used safely. Operational performance governs the operation of an aircraft during its entire lifetime.
Results from the field of aircraft performance can be used to explain the morphology of aircraft. For example, an aircraft that is designed to fly at high speeds will have a very different shape than an aircraft that is designed for low flight speed. An example is given in Figure 2, where a supersonic aircraft is shown with very short wings on the left and a slow human powered aircraft with relatively long and slender wings is shown on the right.
The concepts introduced above (flight mechanics, flight dynamics, point- and path performance) are valid for an aircraft moving in the Earth’s atmosphere as well as other planets’ atmospheres.
For the motion of rockets and satellites as a result of gravitational forces and propulsion (mostly outside the atmosphere), the terms orbital mechanics and astrodynamics are used more frequently.
Contrary to a jet engine, a chemical rocket engine requires the vehicle to take both the fuel and the oxidizer along (Cornelisse et al. 1979) for the complete journey. Depending on the state of the fuel, chemical propulsion systems can be further subdivided into solid, liquid or hybrid systems. For systems that require lower thrust, other propulsion systems can be used, such as electrical propulsion (Wertz and Larson 1991). Generally, such low-thrust systems are more efficient. Curious minds can consult references 10-12 or the references listed at the bottom of the orbital mechanics Wikipedia page.
References
1. Torenbeek, E., Wittenberg, H., Flight Physics – essentials of aeronautical disciplines and technology with historical notes, Springer, 2009.
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2. B.W. McCormick, Aerodynamics, Aeronautics, and Flight Mechanics, Wiley, New York, 1979. 3. R.S. Shevell, Fundamentals of Flight, Prentice-Hall, Inc., Englewood Cliffs, New Jersey,
1983.
4. F.J. Hale, Aircraft Performance, Selection, and Design, Wiley, New York, 1984.
5. C.E. Padilla, Optimizing Jet Transport Efficiency, Performance, Operations & Economics, McGraw-Hill, New York, 1996.
6. J.D. Anderson, Aircraft Performance and Design, McGraw-Hill, New York, 1999. 7. M.E. Eshelby, Aircraft Performance - Theory and Practice, Arnold, London, 2000. 8. M. Saarlas, Aircraft Performance, Wiley, New York, 2007.
9. G.J.J. Ruijgrok, Elements of airplane performance, VSSD, Delft, The Netherlands, 2009. 10. J.W. Cornelisse, H.F.R. Schöyer, K.F. Wakker, Rocket Propulsion and Spaceflight Dynamics,
Pitman Publishing, London, 1979.
11. J.R. Wertz, W.J. Larson, Space mission analysis and design, Kluwer Academic Publishing, Dordrecht, the Netherlands, 1991.
12. D.A. Vallado, Fundamentals of Astrodynamics and Applications, 2nd Edition, Microcosm Inc and Kluwer Academic Publishing, Dordrecht, the Netherlands, 2001.
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Grading
To pass this project you will need a grade six or higher and you may have only one grade lower than 6.0. To get a grade, you will also need to fulfil the presence requirements. If you are absent for too many parts and/or did not compensate for your absence, you will not get a grade and will thus fail the project.
The final individual grade for this project consists of three main parts: Group poster (20%)
Individual test (40%)
Individual assessment (40%)
Group poster
Each group will make a poster at the end of the project to present their final design. This poster will then be presented at the competition day and a grade will be determined by a group of staff members, including teachers from the course Introduction to Aerospace Engineering.
Individual test
A multiple-choice computer test will take place at the end of the project, in January, where the knowledge you should have gained during the project will be tested. This test will be completed individually. Your grade for this test needs to be 5.0 or higher, and may only be lower than 6.0 if your other grades for this project are above a 6.0.
Individual Assessment
The last part of your grade is determined by your tutor together with the staff members of this project. Your grade for this assessment needs to be 5.0 of higher. For this grade six different aspects of your individual performance throughout the project will be looked at:
Technical quality (50%) Commitment (10%) Attitude (10%) Initiative (10%) Management of Resources (10%) Communication (10%)
During the project, all students will complete a peer evaluation in which you will be asked to comment on your own performance and the performance of the members of your group. An elaborate description of the tips and tops is for your own benefit and will allow you and your fellow students improve in this and future projects. Please fill it in before the given deadline!
Structures and materials sessions
The first sessions of this project will be dedicated to a construction exercise and materials testing, performed in the Hal. These sessions are not described in this reader, but are firmly explained on BrightSpace. They can be found under Assignments > S1+S2. Please read the practicum instructions beforehand.
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2D Pressure distributions on aerofoils
This first part of the project consists of two parts, the preparation of 2D wind tunnel experiment and the wind tunnel experiment itself. As there are ‘only’ four 2D tunnels and up to eight project groups per one half day, the tasks which do not require the wind tunnels need to be completed around the measurements at the project tables.
A flying wing needs lift for which you need the velocity relative to the air, also known as airflow. This airflow behaves in a certain way for which a 2D analysis provides insight.
Although it is possible to calculate the flow around entire airplanes using high-performance computers, wind tunnels are still very important. Besides measurements on aircraft aerodynamics, fundamental work on specific flow characteristics is done using modern measurement methods. Experimental research and numerical research in aerodynamics are complementary as the numerical research can give more insight in flow phenomena and the experimental research is used to validate numerical calculations. In this respect there seems to be a growing demand for dedicated wind tunnels. Apart from the necessity of experimental research, wind tunnels also provide a relatively simple means to show the basic flow phenomena and offer hands on experience of aerodynamics.
The wind tunnel experiment discussed in this and the following chapter is meant to provide a better inside in the airflow around a wing profile and the associated pressure distribution, which are also treated in the aerodynamics lectures (AE1110-I). The experiment will be performed on a TecQuipment AF10 Airflow Bench, a small wind tunnel equipped with a wing profile in the airflow. This profile has pressure holes to measure the air pressure and its angle of attack ( ) is adjustable.
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Learning
objectives
The student should:
become familiar with total, static and dynamic pressure. understand the basic working principle of wind tunnels,
pressure holes and manometers.
obtain insight in the behaviour of 2-dimensional aerofoils. understand the working of a simple wind tunnel and the
manometer measuring device.
have the ability to execute a 2D wind tunnel experiment. be able to convert measurements from one quantity to the
other.
Schedule
(40 min.) Preparation and familiarizing with the theory, including reading the related chapters in Anderson (2008). (15 min.) Divide the tasks over the group members. (60 min.) Working out of the exercises. (60 min.) Preparation and execution of wind tunnel experiment for six different angles of attack (all group members need to be present).
(30 min.) Working out the results of the wind tunnel experiment.
(30 min.) Presentation of results to tutor.
Deliverables
Make sure that you write clearly on everything you hand in who has worked on it. Worked out exercises [Task 1: 1.1-1.7] Measurement report of the wind tunnel exercise [Task 1 &2]
Worked out exercises and produced graphs [Task 3]
The combined wind tunnel measurements for all six angles of attack delivered to the tutor as well as to all members of the group (needed for LI).
Clear table containing work and hour distributions of all team-members
Theoretical background
The theory used in this chapter and during the wind tunnel experiment will also be treated in the aerodynamics lectures (AE1110-I) but for this project it is important that you know the working principle of wind tunnels, the different pressures and manometers before you start working out the results of the wind tunnels. Therefore, you should read chapter 4.10 until 4.11.1 in Anderson (2008). If after reading there are still things on these concepts unclear, please feel free to do your own investigation to find the required information. Below you will find a short summary of the most important equations, but be aware, these are only the equations in their final form. For the derivations, please read the related chapters in Anderson (2008).
Continuity equation
One of the basic principles from physics is that mass can neither be created nor destroyed. If this should be applied on a flowing gas, the following equation must hold: m1m2. This means that the mass flow at point 1 in a stream tube must be equal to the mass flow at point 2 (see Figure 1). This can be written as:
1 2 1 1 1 2 2 2
m m A V A V (1)
This can be further simplified if we assume that the flow is incompressible, which means that the density does not change within the stream tube and Equation (1) becomes
1 1 2 2
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Bernoulli’s equation
The derivation given in Anderson (2008) for Bernoulli’s equation assumes a steady, incompressible, irrotational, and inviscid flow and can be written as:
2 2
1 1
1 2 1 2 2 2
p V p V (3)
Using Bernoulli’s equation with a Pitot-static tube (information on the Pitot-static tube can be found in chapter 4.11 in Anderson (2008), the following equation can be formed:
2 1
1 0
2
p V p (4)
The first term is the static pressure, the second term describes the dynamic pressure and the last term states the total pressure. The total pressure is sometimes also written aspt.
Manometer
During the wind tunnel experiment you will measure the pressure differences along the aerofoil using a manometer. In this section, the basic principles behind manometer measurements will be discussed. A simple manometer is a tube in an ‘U’ shape, filled with a fluid, see Figure 2. In this situation one side of the U tube is connected to a certain pressurep1. The other side is connected to a certain pressurep2. A difference in fluid height, h , in the U tube is a measure for the difference in pressure, p2p1. In equation form this looks like:
1 2
p A p A w A h (5)
Here, A is the cross-sectional area of the tube and w is the specific weight of the fluid. w is defined as the density of the fluid times the acceleration of gravity. Equation (5) can be written as:
1 2
p p g h w h (6)
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If one of the pressures is known, say the ambient pressure; and the height difference is measured, Equation (6) can be used to calculate the unknown pressure on the aerofoil. Please note that the width of the tube, A, on either side is not important, if it is equal!
Speed of sound
The speed of sound, as the words already suggest, is the speed at which sound propagates through a medium. In many aerodynamic problems, the speed of sound plays an important role. How the speed of sound fulfils this role becomes clear in the coming aerodynamics lectures and during the rest of your studies. The principle is introduced in this section, but the complete derivation of the formula for the speed of sound in a gas can be found in Anderson, 2008. The most important result of this derivation is that the speed of sound in a perfect gas depends only on the temperature of that gas:
a R T (7)
The parameters and R are constants (look up their values). Next to the speed of sound, there is another widely used quantity, the so-called Mach number. It is defined as:
V M
a
(8)
See chapter 4.11 of Anderson (2008) for further details.
Execution of 2D Wind Tunnel Experiment
For a wing to be able to fly, it must consist of a number or wing profiles which are extruded over the span. How the flow over these profiles causes changes in pressure can best be demonstrated in a wind tunnel.
This chapter will cover one of the more practical sessions during the aerodynamics part of your first-year project. You will do measurements on a NACA0020 aerofoil in a small vertical wind tunnel. All preparation steps for the measurements and the measurements themselves need to be done by you and your project group. First you will get some theoretical background to be able to operate the wind tunnel. After this you will do the measurements at different angles of attack and you will have to record the measurement results. Finally, you must process the results and do some exercises related to the wind tunnel experiment.
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Theoretical background
The wind tunnelThe wind tunnel we use for the experiments is the TecQuipment AF10 Airflow Bench which is a simple vertical wind tunnel (see Figure 1). The wind tunnel generates a controlled airflow for various experiments with standard equipment.
Principle of the wind tunnel
The wind tunnel has a fan that draws air from the atmosphere and generates a flow along a pipe into a settling chamber (or air box) above the test area. In this settling chamber the flow is conditioned to produce a uniform laminar flow. Any unsteadiness or unevenness of the flow is further reduced by the increase of velocity towards the test section (the air is accelerated through a contraction). The velocity of the flow can be regulated by a valve that is positioned in the pipe. The rectangular exit at the end of the contraction is provided with quick release joints to which various test sections can be coupled. In the following experiment, such a test section is used.
Operating the wind tunnel
All complementary assets of the wind tunnel are breakable and very expensive. So treat everything with utmost care!
The wind tunnel has two important buttons:
Start-button: the black button on the right side below the wind tunnel
Stop-button: the red button on the right side below the wind tunnel next to the black one The valve of the wind tunnel can be set from 0 to 1. The setting 0 means the valve is near to close (low wind velocity), while setting 1 means the valve is completely open (highest wind velocity). Neither of these settings is very good when starting up the wind tunnel, therefore the
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setting of the valve is fixed and should not be changed! When you are ready to perform the tests, just press the start button and begin with the measurements.
The multi-tube manometer
The multi-tube manometer (Figure 2) is a manometer with fourteen tubes that can be connected to pressure tubes. Figure 2 is only meant to explain the principle of reading measurements from a manometer, it is not the actual set-up you will be using. The manometer liquid is water coloured by a dye, this to improve the visualization of the water height. The density of the water is not significantly altered by addition of the dye. The manometer can be used inclined to a suitable angle to increase sensitivity, however this is not necessary for the experiments we are performing. The reservoir of manometer liquid can be set to different heights to adjust to the experimental situation. Know the liquid height does not tell anything about individual pressures you want to measure! The only thing that you will be able to measure is the pressure difference that can be obtained by reading the shift in liquid height in the manometer tube. The reservoir connection must be left open to atmospheric pressure during the measurement. The pressures are not read directly from the level of the manometer. It depends on what quantity is desired. First the level (in mm) is read and then the pressure difference desired is calculated e.g. hphpt p pt[mm H2O)] (Figure 2). After that the pressure difference can be calculated using the conversion factor; pressure readings taken in terms of mm of water may be converted to units of N/m2 with the relationship:
1 mm water = 9.80665 N/m2 Description of the equipment
For this experiment a module that houses a tapped aerofoil profile, as shown in Figure 3, is coupled to the wind tunnel exit. The layout of the module is indicated in Figure 4, Table 1 and Table 2. The aerofoil has 6 pressure holes on the upper surface and 6 on the lower surface. All pressure holes are connected to the manifold next to the test section, the manometer is connected to this manifold. The angle of attack of the aerofoil can be adjusted by simply turning the aerofoil (NOTE: turning the aerofoil clockwise represents a positive angle of attack).
Figure 2: Multi-tube manometer and the principle of reading pressures [Markland (2009)]
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Table 1: Aerofoil model details
Table 2: Tapping positions on the NACA0020 model Tapping
number Tapping position from the leading edge [mm] 1 2.2 2 3.9 3 6.1 4 8.7 5 11.8 6 14.8 7 20.0 8 25.6 9 31.4 10 37.3 11 43.4 12 49.4 Task 1: Exercises
Divide exercises 1.1 – 1.7 among the different group members. Next to calculations you are also asked to make a sketch of the situation (like Figure 5) with all relevant numbers and parameters in it.
Each subgroup should prepare a short presentation about the exercise they did for the rest of the project group. The purpose of the presentation is to tell the rest of the group how you have
Aerofoil type NACA0020 Aerofoil chord 63 mm Aerofoil wingspan 49 mm Effective surface area 0.0031 m2
Figure 4: The NACA0020 aerofoil [Markland (2009)]
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solved the exercise. Use the whiteboard to explain the steps you have taken. In the preparation you should think about how you are going to explain your work to the rest in a consistent way. 1.1 Explain why the width, A, of the tube in a manometer is not relevant in the calculations. 1.2 In Figure 5 you see a simple representation of the wind tunnel you will be using next
week. It consists of a settling chamber and a convergent duct. A fan generates a flow in the settling chamber and due to the contraction, the air is accelerated through the duct. In the wind tunnel there are 3 pressure holes, one in the settling chamber (P ), one in 1 the contraction (P ) and one in the duct (2 P ). The pressure at point 1 and 3 is 3 measured using a manometer. The pressures are measured with respect to the atmosphere and given in mm H2O. At point 1 there is a difference of 25mm and at point 3 there is no difference. It is further given that 1mm H2O is equal to 9.81N/m2 and the wind tunnel is located at sea level in a standard atmosphere on Earth. Assume that the cross-sectional area of the settling chamber is very large compared to the cross section of the duct.
Calculate the speed of the air in the duct. What assumption did you use? What is the Mach number of the airflow?
Why is it not possible to use the pressure at point 2 instead of point 1 to calculate the speed of the air in the duct?
Assume the wind tunnel is placed on the planet Mars. Calculate how high the pressure should be in the settling chamber to get the same speed as calculated earlier. Use the data for the standard atmosphere for Mars given in Excel sheet MarsAtmosphere.xls that can be found on BrightSpace.
How much higher is pressure in the settling chamber with respect to the atmosphere (in %), on Mars? And on Earth?
What is the speed of sound on Mars at sea level in the standard atmosphere described above?
1.3 Consider the incompressible flow of water through a divergent duct. The inlet velocity and area are 1.524 m/s and 0.929 m2, respectively. If the exit area is 4 times the inlet area, calculate the water flow velocity at the exit and the pressure difference between the exit and the inlet. (Based on question 4.1 and 4.2 from Anderson (2008))
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1.4 Consider an airplane flying with a velocity of 30 m/s at a standard altitude of 3 km on Mars. At a certain point on the wing, the airflow velocity is 35 m/s. Calculate the pressure and the Mach number at this point (use the speed of sound calculated in 1.2). (Based on question 4.3 from Anderson (2008))
1.5 Consider a low-speed wind tunnel with a nozzle contraction ratio of 1:20. One side of a mercury manometer is connected to the settling chamber, and the other side to the test section. The pressure and temperature in the test section are 1 atm and 300 K, respectively. What is the height difference between the two columns of mercury when the test section velocity is 80 m/s? (Question 4.18 from Anderson (2008))
1.6 A Pitot tube is mounted in the test section of a low-speed subsonic wind tunnel. The flow in the test section has a velocity, static pressure and temperature of 241.4 km/h, 1 atm and 21.1 °C, respectively. Calculate the pressure measured by the Pitot tube. (Based on question 4.20 in Anderson (2008))
1.7 A wind meter that is to be used on Mars is tested in a wind tunnel. The wind meter can measure dynamical pressure with an accuracy of 0.03 Pa. What is the minimum wind speed at which an accuracy in wind velocity of 10% is reached? Assume that the density is known without error.
Task 2: The wind tunnel experiment
During this wind tunnel experiment you are going to perform measurements on a NACA0020 profile. You will be doing pressure measurements on the aerofoil at different angles of attack. These measurements will be further processed in the following chapter.
During the experiment don’t forget to measure the air temperature [K], using the thermometer attached to the wind tunnel and the barometric pressure [N/m2]. The barometric pressure can
be taken from the KNMI website observations for the location of Rotterdam. You need these data later to determine the air density.
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Before the measurements can be done, the aerofoil pressure holes should be connected to the manometer. Check if connections are done correct. To understand the results better, you should connect tappings 1, 3, 5, 7, 9 and 11 (lower surface) to the first set of manometer tubes. Tappings 2, 4, 6, 8, 10 and 12 (upper surface) can then be connect to the next set of tubes. The 13th tube should be left open so all pressures are shown with respect to the atmosphere. The final tube is used for measuring the static pressure in the airbox and at the inlet. This set-up can be seen in Figure 6.
If the manometer is set-up correctly, you can start the wind tunnel, as described earlier. The valve controlling the flow is already open and should not be changed. Now the wind tunnel is running and ready for the measurements to start. You have about one hour to complete the measurements.
The angle of attack of the aerofoil can be adjusted after loosening a screw at the back of the test section. For the first measurement set the angle of attack to 0° (zero). Do not forget to tighten the screw again.
For the different angles of attack the following should be done (NOTE: all pressure measurements should be done with respect to the manometer tube with the atmospheric pressure) and in the following order:
Measure the ambient pressure,patm, and the air temperature,Tair. Since the wind tunnel draws its air from the room, the air temperature is assumed to be the same as the temperature in the test section.
Connect the static pressure tapping at the top of the wind tunnel (air box) to the last manometer tube and measure the pressure,pt.
Change from the static pressure tapping at the top of the wind tunnel to the static pressure tapping in the inlet and measure this pressure,p.
Measure all the pressures from the tappings at the aerofoil.
Repeat all these measurements for the following angles of attack -5°, 0°, 5°, 12.5°, 20° and 25°. This brings the amount of measurements, including the first measurement at 0°, to 6 in total. You can write your measurements down in a table like Table 3, use for every measurement a new table. The grey cell in the table should be filled out during the measurement. A sheet with tables where you can enter your measurements should be available at the wind tunnel.
Also at high angles of attack, you will see that the pressure changes abruptly. Write down the angle of attack at which this happens. Also write down the pressure distribution (from the manometer) for this angle of attack.
Remark: please note the behaviour of the surface mounted tuft during the measurements as this may indicate flow separation (further discussed later on)
19
Table 3: Results wind tunnel measurements Angle of attack,
[]Air temperature, Tair [K] Ambient pressure,patm [N/m2]
Tapping No. h[mm Hp 2O] - p patm h h [mm H2O] atm p p [N/m2] p [N/m2] t p p 1 3 5 7 9 11 patm 2 4 6 8 10 12
Task 3: Results of the wind tunnel experiment
The following questions must be worked out after the wind tunnel experiment in the project room.
3.1 For the measurements you did at the wind tunnel, you must calculate the pressure p in N/m2 for all tappings on the aerofoil, in the airbox (p
0) and the inlet (p∞). Use the intermediate steps indicated in the table. This is a similar exercise as you have done in task 2.
3.2 To check the behaviour of the pressure distribution over the model you should draw the data points in a figure like Figure 7 (available on BrightSpace) or in a likewise plot. Each group should draw the pressure distribution at a different angle of attack.
20
Figure 7: Sample figure for the pressure distribution over the NACA0020 profile 3.3 Calculate the Mach number in the inlet. Considering this, is it allowed to assume
incompressible flow?
3.4 Where would you expect the highest Mach number at aerofoil? How high is it? At which angle of attack? How much higher is it compared to the flow at the inlet?
3.5 Calculate the Reynolds number. Can you achieve the same Reynolds number on Mars (use the excel sheet MarsAtmosphere.xls)?
3.6 What diameter should a wind tunnel for Martian conditions have, to achieve laminar flow at 10 m/s? (Note: assume that turbulent flow occurs for a Reynolds number larger than 104). Which wind tunnels (see section ‘About the AE1111-I project’) that simulate Martian conditions will likely exhibit turbulent flow?
3.7 What happens at high angles of attack? How can you see that in the pressure distribution?
3.8 Why is water and not Mercury used in the measurements of pressure differences? What are the advantages of water in comparison to Mercury? When would you use Mercury? Now the pressure distribution has been determined along the aerofoil, it is an easy step to convert this into forces, such as lift and drag. These topics will be treated in the subsequent chapters. A well-constructed Excel sheet might come in useful, as future exercises may need to be added and that the Excel sheet may need to be updated.
21
LI
Lift and weight
A flying wing needs lift to stay in the air. This lift is generated by a distribution of pressure over the wing. You have learned about the pressure distribution along an aerofoil in the previous chapter.
Part 1 - Lift distribution
In the previous chapter, you have measured pressure distributions of an aerofoil at different angles of attack. The next step is to use these pressure distributions to determine the lift coefficient of the aerofoils. In this chapter, you will learn how this can be done. After this we will look at the relation between the lift coefficient and the angle of attack of a certain aerofoil. You will probably guess that the lift will increase when the angle of attack increases. But will this also follow from the measurements and calculations? And how will a plot of the lift coefficient as a function of the angle of attack look like? When you finished this chapter, you will have the answers to these questions.
Part 2 – Lift and weight
In a few weeks you will have to design your flying wing. To be able to make choices about the design parameters of the aircraft (wingspan, chord length, thickness, sweep, etc.), you should understand what influence these parameters have on your flight performance. In this chapter, we will focus on the lift and weight of the aircraft. In this respect the main question to be answered is: will the wing generate enough lift to counteract the weight of the aircraft? If this is not the case the aircraft simply will not fly. At the end of this chapter you will be able to estimate the lift force of your flying wing.
Learning
objectives
The student should be able to:
Understand the relation between pressure distribution and lift.
Convert the pressure distributions to lift coefficient.
Get insight in the typical lift curve of 2-dimensional aerofoils.
Understand the balance between wing lift and weight. Understand the difference between infinite and finite wings. Build a (well structured) excel sheet for lift analysis.
Schedule
(30 min.) Preparation and familiarizing with the theory, including reading the related chapters in Anderson (2008). (15 min.) Divide tasks amongst group members. (165 min.) Work out tasks.
(30 min.) Share results amongst group members
Deliverables
Make sure that you write clearly on everything you hand in who has worked on it. Worked out exercises and produced graphs [Task 1, 2, 3and 5].
Lift curve and important values (Table 2) of NACA 0020 with the entire group [Task 3.1 and Task 4].
Worked out calculation of [Task 5].
Worked out calculation of E and the total excel sheet (lift and weight) [Task 6 and 7].
Table with the results for lift and weight [Task 8 and 9]. Clear table containing work and hour distributions of all
22
Theoretical background
Pressure coefficientStart with reading Chapter 5.6 of Anderson (2008) to get familiar with the definition of pressure coefficient. You should be able to find and understand the following formula for the pressure coefficient:
(
) (
)
2
1
2
atm atmp p
p
p
p p
p p
C
p
q
q
V
(1) Where: pC
= Pressure coefficient (dimensionless)p
= Pressure at a given point (tapping) (N/m2)p
= Undisturbed static pressure (N/m2)q
= Undisturbed dynamic pressure (N/m2)
= Density (kg/m3)V
= Undisturbed free stream velocity (m/s)The aerofoil is in a duct in which the boundary layer thickness over the wall increases. This gives a smaller effective area around the aerofoil, leading to a free stream velocity which is, in stream wise direction, higher at the aerofoil than it is at the inlet. This can be explained using the continuity equation as shown in Chapter 2 of Anderson (2008). To correct for this, you must find the ‘effective static pressure’ (
p
eff ), around the aerofoil, and then use this to find the correct free stream velocity around the aerofoil.To find the effective static pressure, you must interpolate between the duct inlet pressure (some distance in front of the aerofoil) and atmospheric pressure (behind the aerofoil). This means that you assume the pressure as a function of distance from the inlet to the outlet to be linear. Now you will be able to estimate the effective pressure at each point between the inlet and the outlet. The duct inlet tapping is 135 mm upstream of the exit of the duct. The centre of the aerofoil is 85 mm downstream from this tapping. So:
85
(
)
135
eff atm
p
p
p
p
(2)where
p
is the pressure at the duct inlet andp
atm is the atmospheric pressure.For each angle of attack, you can use
p
eff to find the correct free stream velocity (V
) with Bernoulli’s equation: 21
2
t effp
p
V
(3)For each tapping point on the aerofoil, find the pressure coefficient with the equation for pressure coefficient as mentioned above using:
2
1
2
eff pp p
C
V
(4)23
Lift coefficient
The next step is to use pressure distributions to calculate the lift coefficients of the different aerofoils under different angles of attack. The definition of lift coefficient should have been treated in the lectures, but if the meaning of lift coefficient is still a bit vague for you, start with reading Chapter 5.3 of Anderson (2008).
Read Chapter 5.7 of Anderson (2008) as well. Here you can find that the area between the two curves (upper and lower surface) in the pressure distribution graph is equal to the normal force coefficient. In other words, the integral of the upper surface Cp-curve minus the lower surface Cp-curve gives the normal force coefficient. This is represented by the following formula:
1
, , , , 0 01
c n p l p u p l p ux
c
C
C
dx
C
C
d
c
c
(5)where
c
n is the normal force coefficient,c
the chord length,C
p l, the lower surface pressure coefficient curve andC
p u, the upper surface pressure coefficient curve. The next step is to convert this normal force coefficient into a lift coefficient. In Anderson (2008) you can find that these terms relate as follows:cos
sin
l n a
c
c
c
(6)Where
c
l is the lift coefficient,
is the angle of attack andc
a is the axial force coefficient (also often symbolized asc
t). In Anderson (2008) it is stated that for small angles of attack( 0
5
) you can assume thatc
l
c
n, since thencos
1
andsin
0
. Although some of your measurements were performed at angles of attack higher than 5 degrees, for simplification you can assumec
l
c
n in all your calculations. Know this assumption is not allowable in more advanced aerodynamic experiments, since it can differ greatly from the real case!In the previous chapter, you have learned to determine the lift coefficient of an aerofoil from its pressure distribution. The next step is to convert this lift coefficient to a lift force. In Chapter 5.3 and 5.4 of Anderson (2008) you can find the following relation:
2
1
2
LL C
V S
(7) Where:L
= Lift force [N]
= Air density [kg/m3] (varies with altitude)V
= Flight speed [m/s]L
C
= Lift coefficient [dimensionless]S
= Wing area [m2]24
Tasks
Task 1: Compute the pressure distributions of the wind
tunnel experiment
Perform the following two subtasks for all 6 different angles of attack that where measured in the wind tunnel. Use Excel for your calculations and plots. A typical form of a pressure distribution on an aerofoil can be found in Chapter 5.7 of Anderson (2008). Make sure that eventually all angles of attack are covered, since they are needed in Task 3.
Calculate the pressure coefficients of all measured points along the surface
Make one graph of the pressure distributions over both the upper and lower surface. Plot negative
C
p in the positive y-direction. Connect the points with straight lines.Task 2: Exercises
Now you should have 6 graphs of pressure distributions of the aerofoil measured in the wind tunnel, each for a different angle of attack. Answer the following questions:
2.1 Why do we plot
C
p (negative) instead ofC
p (positive) upward?2.2 At which Mach number did you perform the wind tunnel experiment? What would happen with the pressure distribution graphs if your measurements where performed at (a) half this speed and (b) Mach 0.8?
2.3 Is there a relation between the location of the thickest part of the aerofoil and the lowest pressure (in your data)? Explain.
2.4 What is the highest value of
C
p that is to be found in the test and why? How do we call this point on the aerofoil?Task 3: Compute the lift coefficients and the lift curve
3.1 Wind tunnel experiment
In Task 1 you have produced graphs of the pressure distributions over the upper and lower surface of the aerofoil, at six different angles of attack. Use the theory above to determine the lift coefficient at each angle of attack.
Combine the results to plot these 6 lift coefficients as a function of the angle of attack, using Excel.
3.2 MH44 with use of the Trapezoid Rule
Download the file ‘PressureDistributionMH44.xls’ from BrightSpace (AE1-100 -> Course Documents). In this file you will see the following column titles:
Table 1: Part of ‘PressureDistributionMH44.xls’
Upper Surface
x/c Cp Step size Average height Area covered 0 0.99093
0.00505 0.58711 0.01449 0.20079 0.0282 -0.06389 ……….. ………..
25
As you can see the first two columns are already filled in. The first column represents the tapping position from the leading edge, divided by the chord length. The second column is the pressure coefficient at that tapping position. The pressure distribution can be plotted with this information. Further down the excel sheet (not visible in the table above), also the pressure coefficients of tapping points on the lower surface are given.
This pressure distribution will be integrated using the Trapezoid Rule, which is explained in Appendix A of this reader. Be aware that the step size between the tapping points (referred to as h in Appendix A) is not constant! Therefore Equations A.5 and A.6 are not valid.
Plot the pressure distribution from the provided data in excel. Upper and lower surfaces should be plotted in the same graph.
Use the Trapezoid Rule to find the lift coefficient of the MH44 aerofoil. The three column titles are given to point you in the right direction.
3.3 Pressure distribution given by a function
In some cases, it is useful to evaluate theoretical pressure distributions which are given by a function. By integration of that function it is then possible to get an exact result for the lift coefficient, instead of an estimation by using the Trapezoid Rule or another method.
Consider an aerofoil under zero angle of attack with chord length
c
and the running distancex
measured along the chord. The leading edge is located atx c
/
0
and the trailing edge at/
1
x c
. The pressure coefficient distributions over the upper and lower surfaces are given, respectively, as: 2 ,1 200
p ux
C
c
x
for 0
0.13
c
,2.47
2.63
p ux
C
c
x
for 0.13
1.0
c
,1 0.84
p lx
C
c
x
for 0
1.0
c
Calculate the lift coefficient. Type out your calculation steps clearly. Task 4: Determine important values from the lift curve
From a lift curve, some important characteristics of an aerofoil can be determined. One is the angle of attack at which no lift is generated (
L0). Another one is the lift slope (dc d
l/
)When these two values and the angle of attack are known, the lift coefficient can easily be calculated:
(
0)
l l Ldc
c
d
(8)Of course, this function only holds for the linear part of the lift curve! Another important property of an aerofoil is the maximum lift coefficient (
c
l,max). This is the top of the lift curve. Evaluate the lift curve of the aerofoil tested in the wind tunnel to determine the three characteristics mentioned above.26
Table 2: Lift curve characteristics NACA 0020 (Wind tunnel)
0 l
c
/
ldc d
max lc
Task 5: ExercisesAnswer the following questions:
Does the NACA 0020 profile have a lift curve which is going through the origin? Explain why lift curves of certain aerofoils go through the origin and others do not.
Which phenomenon occurs when the angle of attack gets too high? Make a sketch of the flow of this for a sample aerofoil.
What is the ratio of the lift coefficients on Mars and on Earth for an aircraft with the same mass, wing surface area and cruise speed?
Search your book and the internet for the theoretical value of the lift slope (
dc d
l/
) forthin aerofoils. Does this value agree with the value you have found for the NACA 0020 aerofoil?
Task 6: Determine the lift coefficient of a solar airplane on Mars in cruise flight
Consider the Sky-Sailor autonomous solar powered Martian airplane with characteristics given in the paper Noth et al. (ASTRA proceedings, 2004):
Draw the forces acting on an aircraft in flight. Determine the relations between those forces in steady horizontal flight.
Calculate the lift coefficient of the Sky-Sailor in cruise flight.
Theoretical background
Infinite and finite wings
In the Wind Tunnel Experiment 2D you tested a so-called infinite wing, since the wing was clamped between the walls of the wind tunnel. However, the flying wing will be a finite wing, since it has wing tips. Chapter 5.13 of Anderson (2008) explains the difference between the lift coefficient of an infinite and a finite wing. Read this chapter before you continue.
With the aerofoil lift coefficients found in the previous chapter we can calculate the lift per unit span:
2
1
(
)
2
lL per unit span
V c c
(9)where c is the chord length. You should understand now that we cannot just multiply the lift per unit span with the wingspan to find the total lift:
(
)
27
2
1
2
lL
V c S
(11)The only difference between equations 7 and 11 is the lift coefficient. The aerofoil lift coefficient (Cl) is not equal to the wing lift coefficient (CL). As we will see later the lift distribution of a straight finite wing can sometimes be approximated by an elliptical distribution (see chapter 5.14 of Anderson (2008)). This elliptical lift distribution can now be used to model the actual lift distribution over your wing. We can use this model to estimate the relation between the aerofoil lift coefficient and the wing lift coefficient.
Shown on the left side of Figure 2 is the case when a section of an infinite lift distribution is assumed. On the right side an elliptical lift distribution is shown. This is a far more realistic representation of the actual lift distribution over a finite wing. Note that the maximum lift per unit span lift (in the middle of the wing) of the elliptical lift distribution is equal to the lift per unit span of the infinite lift distribution.
Let us define E as a factor that relates the area of the elliptical lift distribution to the infinite lift distribution: infinite ellipse
A
E
A
(12)Where Aellipseis the surface area of the elliptical lift distribution and Ainfinitethe surface area of the
infinite lift distribution (dashed area in Figure 2). Since the surface of the lift distribution represents the total lift of the wing, the following relation can be found (using Equation 9):
2 2
1
2
1
2
l l L Lc
V bc
c
E
C
C
V S
(13)So, if we can determine a value for E and the aerofoil lift coefficient (cl) is known, the wing lift coefficient (CL) can easily be calculated. For reference, further information about the relations between infinite and finite wing can be found in Chapter 5.13 and 5.15 of Anderson (2008). Weight
After you finished the design your flying wing will be cut out from a block of foam. This is not the usual foam used to isolate houses, but a special engineered plastic foam material which is a lot stronger and less brittle than the usual foam. This very light weight material is called EPP (Expanded Polypropylene) and has a density of 20 kg/m3.
Figure 2: Representation of a lift distribution over an infinite wing (left) and a finite wing (right)
28
Next to the weight of the wing itself the following items should be added to calculate the total weight of the aircraft:
- Battery - Propellers - Electronics
The total mass of these three items is 17 grams.
Keep in mind that it could be possible that more weight must be added to the aircraft in a later stadium, i.e. stabilizing fins or duct tape.
Task 7: Build an Excel sheet which calculates the total lift force
In this task, you will have to build an Excel sheet which calculates the total lift of the flying wing when all parameters are known. Consider your aircraft as a rectangular wing with a certain airspeed under a certain angle of attack, as shown in Figure 3 and Figure 4.
Your Excel sheet should contain only the following input variables:
- Factor between elliptical and infinite lift distribution (E) [dimensionless] Dimensional characteristics
- Chord length (c) in [mm] - Wingspan (b) in [mm] Aerofoil characteristics
- Zero-angle-of-attack lift coefficient (cl(α0)) [dimensionless] - Lift slope (dcl/dα) in [rad-1]
Flight characteristics
- Airspeed (V∞) in [m/s] - Angle of attack (α) in [rad] - Air density (ρ) in [kg/m3]
- Gravitational acceleration (g) in [m/s2]
With these variables, the following parameters should be calculated: - Wing area (S) in [m2]
- Wing lift coefficient (CL) [dimensionless] Perform the following subtasks:
V∞
b
c
S
Figure 3: Top view of the flying wing Chord line
α
V
∞c
29
Build the excel sheet capable of calculating the total lift force in Newton. Make sure your sheet is well structured; it should be clear which values are input variables and which values are calculated.
Calculate E using Equation 12.
Task 8: Extend your Excel sheet with weight calculation Add the following input variables to your excel sheet:
- Area of aerofoil (Aaerofoil) in [mm2] - Density of EPP (ρEPP) in [kg/m3]
- Weight of battery, propellers and electronics (Wbpe) in [g] - Extra weight (Wextra) in [g]
To estimate the area of the aerofoil (
A
airfoil) you can use the ratio between the maximum thickness (t
max) and chord of the aerofoil. The ratio is given by the aerofoil name: NACA 0020 means that the percentage of thickness to chord is 20%. Extend your Excel sheet so it calculates the weight of the aircraft in Newton. Task 9: Calculate lift and weight with use of excel sheet
Consider two rectangular wings flying in a standard atmosphere on sea level with the characteristics as stated in Table 3.
Table 3: Characteristics of two flying wings Flying Wing 1 Flying Wing 2
Aerofoil NACA 0020 NACA 0020
Wingspan 600 mm 400 mm
Chord 120 mm 75 mm
Airspeed 3 m/s 6 m/s
Angle of attack 3° 3°
Calculate the lift and weight (in Newton) of both flying wings, using your Excel sheet. Check with your fellow group members if everyone has found the same results.
Lift and weight have now been discussed thoroughly, but how about the drag an aircraft has to overcome? Furthermore, lift and drag may also contribute to the longitudinal stability of the aircraft. This will be treated in the next chapter.
References
Noth, A, Bouabdallah, S, Michaud, S, Siegwart, R, Engel W, 2004, The design of an autonomous solar powered Martian airplane, In: Proceedings of the 8th ESA Workshop on Advanced Space Technologies for Robotics, (ASTRA 2004), Noordwijk, Netherland,2004
30
DR Drag and Longitudinal stability
As discussed in the previous chapter, there are four forces acting on an aircraft. Lift and weight have been treated already and in this chapter, drag and thrust will be discussed.
On aircraft, such as flying wings, drag is considered unwanted as it needs to be overcome by thrust. Thrust is generally created by either a piston engine or a jet/turbofan engine. Both come in many different variants, but that is not the scope of this chapter.
Learning
objectives
The student should be able to:
become familiar with the concept of drag and be able to explain which contributions make up the drag on a wing Calculate the total drag of a wing
Get insight in the drag behaviour of a 3-dimensional wing Be familiar with the concept of (longitudinal) stability. Describe how longitudinal stability can be achieved.
Schedule
(30 min) Preparation and familiarizing with the theory, including reading the related chapters in Anderson (2008) (15 min) Divide the tasks over the group
(45 min) Extend the excel sheet with drag and thrust calculations
(120 min) Working out the exercises. (30 min) Present results.
Deliverables
Make sure that you write clearly on everything you hand in who has worked on it. Excel sheet with the drag/thrust calculations (hand indigitally) [Task 1 & 2]
Worked out exercises [Task 3]
Worked out exercises and produced graphs per sub group [Task 4].
Clear table containing work and hour distributions of all team-members.
Theoretical background of drag
Let us first consider drag. The more drag an aircraft experiences, the more thrust is needed to propel the aircraft. So, for an aircraft designer it is important to try to minimize the drag of an aircraft. But what is drag, which components does drag have? In the next section only a short summary on drag will be given. The related chapters in Anderson (2008) where more information can be found, are indicated in the text.
Infinite wing
In the previous chapter the step has been made from an infinite to a finite wing, here we first take a step back and consider again an infinite wing, the drag for such a wing consists of three parts. f p w D D D D (1) where: D = total drag f
D = skin friction drag
p
D = pressure drag due to flow separation
w
D = wave drag
31
The wave drag can be neglected at subsonic speeds, it only plays a role at transonic and supersonic speeds. The remaining drag components are called the profile drag, because both heavily depend on the shape of the object.
The skin friction drag (Df) is present because the airflow over a solid object creates a boundary layer. For a detailed explanation, please read the related Chapters (3.15 – 4.17) in Anderson (2008) on viscous flows and laminar- and turbulent boundary layers.
The final contribution to the drag comes from the separation of the flow over a surface. If you look at a wing with a high angle of attack, it is possible the flow is not able to follow the shape of the aerofoil and separates at a certain location (Figure 1). This separation causes a large decrease in lift and an increase in drag and is something that should be avoided. More on pressure drag can be found in Chapter 4.20 in Anderson (2008).
Measurements on drag of a wing can easily be done in a wind tunnel, but exact calculations are much harder to perform. For the skin friction in a laminar boundary layer, there are formulas from laminar boundary layer theory to calculate the drag. But for turbulent boundary layers there is no such theory. Turbulent boundary layers and their drag can only be estimated by approximate formulae, derived from experiments. Also, pressure drag due to separation cannot be calculated exactly. Although it must be said that nowadays numerical calculations can approximate the reality very well.
Finite wing
If we now go back to the finite wing, there is another contribution to the total drag. This is caused by the fact that a wing is finite and that there is a pressure difference between the upper and lower surface. Due to this pressure difference some air will flow around the tips from the high pressure to the low-pressure side. This flow will create a circular motion that trails down from the wing tips. This motion is called a vortex. Due to these vortices, the oncoming flow of air near the wing will be tilted slightly down, this is called the downwash. This has two effects, the angle of attack of the aerofoil is reduced, with respect to the undisturbed flow, due to the downward component of the oncoming flow. The second effect is an increase in drag, called the induced drag. This induced drag has at least three physical interpretations:
The vortices change the pressure distribution over the wing, which leads to an increase in drag
Due to the downwash and the change in stream direction the lift vector is tilted back slightly. Therefore, a contributing component in the drag direction occurs.
The circular motion of the vortices themselves contain a certain amount of energy, which has to come from somewhere and thus is a source of drag
More on induced drag can be found in Chapters 5.13 and 5.14 in Anderson (2008), please study this carefully. The derivation of the formula for induced drag coefficient is given in Chapter 5.14 and will not be repeated here, only its final form will be given.
2 , L D i C C eAR
