Aerodynamics Extended Essay

Danylo Malyuta; International School of Geneva: Campus des Nations; Candidate Number: 002169-029

TITLE Identification of the effects of plain trailing-edge flap deflection on wing

properties

RESEARCH

QUESTION

How are the lift, lift coefficient, zero-lift angle of attack, and stalling angle of

attack of a wing with a NACA 2412 airfoil dependent on plain, trailing-edge

flap deflection?

APPROACH An experimental and theoretical approach is taken. The effect of flap deflection

on wing lift is determined first from numerical thin-airfoil theory and then from

empirical data collected in a low-speed wind tunnel. Results are compared and a

conclusion is made in real-world context with uncertainty levels appreciated.

WORD COUNT: 3’999/4’000

AN IB EXTENDED ESSAY IN PHYSICS


Dedicated to my family, the time with which

was foregone in the writing of this essay.


i

The Abstract

This is an experimental study considering the change the lift, lift coefficient, zero-lift angle of attack,

and stalling angle of attack of a wing with a NACA 2412 airfoil and a plain trailing-edge flap when

this flap is deflected. The wing has an aspect ratio of 3.2 and the flap chord to wing chord ratio is 0.36,

hence allowing for both large enough increments in the maximum lift to be measured and a small

enough error to be made in predicting flap deflection effects with thin airfoil theory (this theory is bad

at predicting flap effects at low flap-to-wing-chord ratios). To experimentally test the effect of flap

deflection on the lifting characteristics of my wing in real life, a homemade wind tunnel was made and

lift data in grams (or the equivalent grams-force) was taken by connecting the wing to a 2 D.P. balance

via rods through the tunnel floor. The power plant provided a steady airflow of 4.1 m/s, allowing the

wing to create appreciable lift to be collected as data. This data along with aerodynamic formulae and

graphical analysis were used to convert raw lift into section lift coefficient and the zero-lift angle of

attack values for different flap deflections. This allowed for several data plots to be created which

showed that the maximum lift and lift coefficient were directly and positively proportional to flap

deflection and that the zero-lift angle of attack was negatively, but also directly, proportional to flap

deflection. It was likewise discovered that flap deflection decreases the stall angle of attack, but in so

doing it increases the maximum lift achieved before this angle. Compared against textbook data, this

showed that a homemade wind tunnel can correctly predict theoretical trends but cannot give accurate

experimental values due to design limitations.

Word Count: 296


TABLE OF CONTENTS

The Abstract ............................................................................................................................................. i

Introduction ............................................................................................................................................. 1

A briefing on flight mechanics ................................................................................................................ 1

The airfoil ............................................................................................................................................ 1

Basic forces acting on an airborne body.............................................................................................. 2

The lift equations ................................................................................................................................. 2

Reynolds number ................................................................................................................................. 3

Boundary layer theory ......................................................................................................................... 3

Thin airfoil theory ............................................................................................................................... 5

Thin airfoil theory applied to trailing-edge flaps................................................................................. 5

What are plain trailing-edge flaps? .................................................................................................. 6

Applying flaps to a ‘thin airfoil’ ...................................................................................................... 6

Hypothesis ............................................................................................................................................... 7

Physics-based ...................................................................................................................................... 8

Thin-airfoil based ................................................................................................................................ 8

Experimental design & procedure ......................................................................................................... 11

Design ................................................................................................................................................ 11

Method .............................................................................................................................................. 12

Safety ................................................................................................................................................. 13

Wind tunnel test results ......................................................................................................................... 13

vs. plots ....................................................................................................................................... 14

vs. plots .................................................................................................................................... 19

vs. plots .................................................................................................................................. 21

Conclusion on relationship between and , , and .................................................................. 24

Scrutiny ................................................................................................................................................. 24

Of method and apparatus ................................................................................................................... 25

Of wind tunnel ................................................................................................................................... 27

Bibliography .......................................................................................................................................... 29

Appendices ............................................................................................................................................ 31

A ........................................................................................................................................................ 31

B ........................................................................................................................................................ 33

C ........................................................................................................................................................ 35


D ........................................................................................................................................................ 37

E ........................................................................................................................................................ 41

F ......................................................................................................................................................... 43

Apparatus diagram ........................................................................................................................ 43

Apparatus photographs .................................................................................................................. 44

G ........................................................................................................................................................ 45

H ........................................................................................................................................................ 47

I.......................................................................................................................................................... 49

J ......................................................................................................................................................... 50

K (Nomenclature) .............................................................................................................................. 52


1

INTRODUCTION

Aerodynamics is the study of the motion of air and of objects through air.1 Interaction with air

particles, engineers, and good theories are what airplanes rely on to operate flawlessly.

This paper uses thin-airfoil theory and a wind tunnel to investigate one of the most important aspects

of a wing: flaps. Flaps provide the lift (and drag) necessary for large airplanes to take off and land in

length-constrained spaces. Engineers must know how flaps affect wing lift; with this knowledge, more

efficient flaps can be made which will allow airplanes to take off in shorter distances, saving money,

energy, and space in building smaller airports in a world where space is becoming limited.

I consider the effects of flap deflection on maximum lift, lift coefficient, and stalling and zero-lift

angles of attack of a wing with a NACA 2412 airfoil. These properties are major aircraft performance

factors, and I will measure them using a home-made wind tunnel whose accuracy I will compare to

computational fluid dynamics (CFD) and professional wind tunnels- key tools in aircraft design.

A BRIEFING ON FLIGHT M ECHANICS

Background aerodynamic knowledge was required for this paper. Appendix K (Nomenclature) defines

the symbols used in this paper.

THE A I R F OI L

The airfoil is a cross-section perpendicular to the wing span.

I needed to know about two airfoil properties; the angle of attack, , and the camber line. is the

angle between and the chord line, and the camber line is constructed from points midway between

upper and lower airfoil surfaces measured perpendicular to the camber line itself. Fig.1 illustrates

these:

1 "Aerodynamics Definition." NASA. NASA. Web. 14 July 2011. <http://www.grc.nasa.gov/WWW/K-

12/FoilSim/Manual/fsim001m.htm>.


2

2

BA S I C F OR CES A CTI NG O N A N A I R BOR NE BODY

By Newton’s second law, an object’s motion depends on the forces/moments acting on it:

( )

Four forces act on an airplane: lift, drag, thrust, and weight. Only lift and drag are aerodynamic forces

as such forces arise from pressure or viscous shear forces only, and lift equals weight while thrust

equals drag in non-accelerated flight.3 Fig.2 shows this:

4

THE LI F T EQUA TI ON S

2 Photograph. Mechanics of Flight. Hoboken, NJ: Wiley, 2010. 27. Print.

3 Phillips, Warren F. "1.1 Introduction and Notation." Mechanics of Flight. Hoboken, NJ: Wiley, 2004. Print.


Figure 2 Forces acting on an aircraft in level flight (angles exaggerated)

Figure 1 Airfoil section terminology


3

Concepts of lift and section lift exist. refers to a wing with no wingtips (an infinite wing or a 2D

airfoil) whilst refers to a finite wing; is hence always greater than because lift decreases near the

wingtips of a finite wing due to induced drag. In my wind tunnel, wingtips were placed very close to

the tunnel walls, so finite affects were minimized, making more appropriate:

So is:

REY NOLDS N UMBER

The Reynolds number is a ratio of the magnitudes of pressure to viscous aerodynamic forces (a higher

value indicates the dominance of pressure forces):

My wing operates at low because of low and short chord length, so viscous forces will be

significant. Thin airfoil theory only analyzes pressure forces, limiting the accuracy of its predictions

for my experiment.

BOUNDA R Y LA Y ER T HE O R Y

This theory states that viscous effects for flow over a wing at low and high are confined to a thin

layer around the wing’s surface. Outside this layer, viscous forces are insignificant. Fig.3 shows this:


4

5

To note:

Pressure forces dominate at high .

Inviscid (pressure-only) flow analysis is accurate for high flow.

Boundary layer separation occurs at high (Fig.4).

6

Fig.4 is referred to as a stall, which is reached after a near-linear increase in as in Fig.5:



Figure 3 Boundary layer flow over an airfoil

Figure 4 Wake formation behind an airfoil beyond the critical angle of attack


5

7

1. increases as a linear function of .

2. Pressure build-up on top of the wing eventually causes partial boundary layer separation,

flattening the slope.

3. At “maximum” , a Fig.4 wake occurs. Airfoil geometry influences the sharpness of this

“cusp”.

4. Lift rapidly decreases. Beyond , lift becomes independent of airfoil shape.8

THI N A I R F OI L TH EOR Y

The thickness of an airfoil with maximum thickness of chord length has little effect on the

pressure forces acting on it. Thin airfoil theory uses this to simplify such an airfoil to a thin filament

(Fig.7, Appendix G).

THI N A I R F OI L TH EOR Y A PPLI ED T O TR A I LI NG-ED G E F LA PS

Deflection of a plain flap changes the wing camber line, and the resulting changes in aerodynamic

characteristics may be calculated from thin airfoil theory assuming zero flow separation.

7 "AeroCFD Operating Instructions." AeroRocket Simulation Software for Rockets and Airplanes. Web. 20 Nov.

2011. <http://aerorocket.com/AeroCFD/manual.html>.

8 Phillips, Warren F. "1.4 Inviscid Aerodynamics." Mechanics of Flight. Hoboken, NJ: Wiley, 2004. Print.

Figure 5 Lift curve of NACA 0012

airfoil

1

2

3

4


6

WH A T A R E P L A I N T R A I L I N G -E D G E F L A P S ?

Some aft portion of an airfoil is hinged to make a plain trailing-edge flap (Fig.6); rotating it about the

hinge axis produces a flap-deflection angle (Fig.7). A downward deflection increases camber, so is

considered positive.

9

A P P L Y I N G F L A P S T O A ‘T H I N A I R F O I L ’

For small and , thin airfoil theory can predict the effects of flap deflection on lift. Development of

below-given equations is in Appendix H.

Fig.7 shows the geometric meaning of terms used in subsequent equations:


Figure 6 An airfoil section with a deflected flap


7

10

Flap chord fraction is:

(

)

Ideal section flap effectiveness is:

Lift coefficient for an airfoil with a plain flap is:

( ) [ ( ) ]

is section flap effectiveness (accuracy evaluated in Appendix B):

and are the section flap hinge efficiency and deflection efficiency (Appendix A).

Using above formulae, one deduces , then from

, and from the lift-curve x-

intercept.

HYPOTHESIS


Figure 7 Thin airfoil flap approximation


8

PHY S I CS-BA S ED

In a positive plain flap deflection, airfoil camber is increased so air over the top surface has “less

space” to travel in than on the bottom surface. From the Law of Continuity:

Hence, decreased cross-sectional area will increase airspeed and decrease static pressure from

Bernoulli’s Law as dynamic pressure

increases:

Lower on top increases lift so a positive correlation with should exist.

Lift is a multiple of lift coefficient; and should also be in positive correlation. should

decrease for positive as extra lift will have to be suppressed with more negative angles of attack;

there should be a negative correlation between and .

THI N-A I R F OI L BA S ED

A spreadsheet (Appendix D) is created using thin airfoil equations to simulate the effect of on ,

, and . Results:


9

-300.0

-200.0

-100.0

0.0

100.0

200.0

300.0

400.0

-20 -15 -10 -5 0 5 10 15 20 25

L (

/gf)

α (/°)

Calculated L versus α for various δ

-10°

-5°

0°

5°

10°

15°

20°

25°


10

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

-20 -15 -10 -5 0 5 10 15 20 25CL (

/dim

ensi

on

less

)

α (/°)

Calculated CL versus α for various δ

-10°

-5°

0°

5°

10°

15°

20°

25°

-16

-14

-12

-10

-8

-6

-4

-2

0

2

4

-15 -10 -5 0 5 10 15 20 25 30

αL

0 (

/°)

δ (/°)

αL0 vs. δ plot

Thin airfoil theory


11

Physics-based reasoning was correct- positive increases lift/lift coefficient and decreases . This is

an accepted textbook fact, so what I am looking for is the accuracy to which my wind tunnel will

match these predictions.

NB: thin airfoil theory ignores stall, so I will be able to determine the effect of on the stall angle of

attack only from wind tunnel tests. Note that lift coefficients greater than 3.0 cannot be achieved on a

NACA 2412 airfoil with a plain flap, and high-end coefficients of 2.0 are unlikely to be recorded in

my limited wind tunnel. The accuracy of thin airfoil values is hence dubious, but the demonstrated

relationship may not be.

EXPERIMENTAL DESIGN & PROCEDURE

DES I G N

A 3m home-made wind tunnel with a 0.30mX0.30mX0.30m test section is used. A ProTronik Pro-

72210 electric motor spinning a 0.254mX0.119m propeller blows air through the tunnel at an average

speed of 4.1 ms-1

.

A NACA 2412 airfoil (Fig.8) is used for the wing; it’s 12% of chord maximum thickness makes it

applicable for thin airfoil approximation. The wing is rectangular with 0.300m span and 0.095m

chord; the aspect ratio is 3.2. It is made from Styrofoam enclosed in plywood and heat-shrink plastic

to reduce friction drag; hinged rods extend through the tunnel floor from its ideal quarter-chord

aerodynamic center to a 2 D.P. balance measuring wing mass. A steel rod is hinged to the wing to

seamlessly control . An acrylic cover encloses the test section to prevent flow leakage. The wing is

mounted at 0.15m above the tunnel floor, almost touching the tunnel sides with the wingtips to reduce

effects of finite wingspan.


12

11

A plain flap is hinged at the trailing edge such that the ratio is 0.36c below which thin-airfoil

theory produces high error (Appendix B) and above which only marginal increase is observed12

.

Flap deflection is controlled manually.

A flow straightener positioned before the test section minimizes the spin the rotating propeller gives to

the airflow. Appendix E shows an apparatus diagram.

METH OD

Independent variable: flap deflection (/°).

Dependent variables: lift (/gf13

).

The method’s aim is to provide a way of collecting lift data accurately (to be converted later to ,

, and stall angle parameters). To collect lift data in /gf:

1. Introduce a 2 D.P. balance underneath the wind tunnel and rest on it an MDF base attached to

the wing via rods through the tunnel floor.

a. Make sure the balance is zeroed prior to step 1.

11 Photograph. UIUC Airfoil Coordinates Database. Web. 20 Nov. 2011. <http://www.ae.illinois.edu/m-

selig/ads/coord_database.html>.

12 Ira H. Abott, Alber E. Von Doenhoff. Theory of Wing Sections. United States: McGraw-Hill Book Company,

Inc., 1959.

13 Grams-force (equivalent to grams).

Figure 8 NACA 2412 airfoil


13

2. The lift force is to be tested in grams for a range of -10°-25° for and -15°-20° for , both in

5° increments, so construct an appropriate table/spreadsheet in Excel.

3. Enter an equation “=massbefore-massafter” in the cell for the appropriate and condition,

where the massbefore value is the wing mass prior to turning on the fan.

4. First test all positive , so begin by adjusting the wing to 0° alpha and -10° delta. Hence, fill

in for massbefore.

a. Adjust by adjusting the wing chord in reference to a protractor mounted in the test

section centered at the wing’s pivot point.

b. Adjust manually with another protractor.

5. Turn on the engine to full throttle and fill in for massafter, completing the equation for the

appropriate spreadsheet cell.

a. Turn off the engine to save battery.

6. Repeat steps 3-5 for all positive alpha 0°-20° and delta -10°-25°.

a. Keep controlled variables constant (Appendix J).

7. Flip the protractor measuring to repeat steps 3-5 for the negative alpha range, also in 5°

increments.

8. Having taken all raw lift measurements, Excel makes it easy to translate them into , ,

and values from graphical analysis and equations in the theoretical section of this paper.

9. Clear up. Process data. Draw a conclusion/evaluation.

SA F ETY

The fan spins at over 6000 RPM so presents a potential hazard. A fine chicken wire mesh installed at

the fan duct prevented accidental contact with the fan’s plane of rotation.

WIND TUNNEL TEST RESU LTS

Appendix E presents raw/processed data tables. Notice the absence of error bars due to only one trial

of data being collected; collecting more trials to enable error bar calculation is a future improvement.


14

VS . PLO TS

Empirical lift curves for -10°-25° :

The lift curves follow standard lift curve form. A linear part for small exists where lift increase is

linear; the curves then level off and the slope becomes negative as the wing stalls at high . This is

-30

-20

-10

0

10

20

30

40

50

60

-20 -15 -10 -5 0 5 10 15 20 25

L (

/gf)

α (/°)

L versus α for various δ

-10°

-5°

0°

5°

10°

15°

20°

25°


15

expected from boundary layer theory as the boundary layer separates from the upper surface beyond

the stall angle. For 0° , stall occurs at 15° 14.

There was evident friction in my data collection mechanism; as a result, my lift data is lower than the

thin airfoil theory predictions, for which the lift slopes are:

Note, however, that thin airfoil theory ignores wing stall, in which case theoretical positive lift

increments beyond 15° would be unachievable even in a 100% accurate wind tunnel experiment.

Experimental and theoretical lift slopes show that positive flap deflections translate the =0° lift curve

vertically upwards, and downwards for negative flap deflections. Therefore, even though the

14 Evaluation check: the stall angle has a ± few degrees uncertainty as was taken at 5° intervals.

-300.0

-200.0

-100.0

0.0

100.0

200.0

300.0

400.0

-20 -15 -10 -5 0 5 10 15 20 25

L (

/gf)

α (/°)

Calculated L versus α for various δ

-10°

-5°

0°

5°

10°

15°

20°

25°


16

magnitude my experimental lift values are below theoretical results, the upward shift of the lift slope is

definitely correct as it is backed by both theory and experiment.

Judging from data for small , this horizontal translation is almost constant each time15

. As proof, one

graphs against (in degrees):

The average linear best-fit shows what the theoretical variation of could be which, from

inspection of Fig. 9, is most likely the case.

15 Discrepancies at larger alpha may be caused by increasing friction between wing rods and tunnel floor due to

higher lift and drag forces

0

10

20

30

40

50

60

70

-20 -10 0 10 20 30

Lm

ax (/g

f)

δ (/°)

Lmax vs. δ

Wind tunnel data

Linear (Wind tunnel data)


17

16

My data also shows that the angle of maximum lift coefficient generally decreases with positive flap

deflection- this is a trend likewise demonstrated by Fig. 9. Plotting change in angle, , versus

:

16 Photograph. Theory of Wing Sections. United States: McGraw-Hill Book, 1959. 195. Print.

Figure 9 Typical trend for a NACA 66(215)-216 airfoil with 0.20c plain flap

Notice the very

consistent shift

upwards of the lift

curves with 𝛿.

𝛼 decreases linearly with 𝛿


18

Since I was measuring lift data for 5° increments, I could not capture the gradual decrease in ;

hence a linear trend line was plotted which shows the likely change in as hypothesized by Fig. 9.

Note from Fig. 10 that the accurate stall angle for the NACA 2412 airfoil is in fact around 14°-16°,

which shows that my collected data is roundabout accurate in terms of stall angle at 0° .

Hence, increases the maximum lift of a wing as shown by the positive correlation between

and . Additionally, there is negative correlation between and .

0

2

4

6

8

10

12

14

16

18

-20 -10 0 10 20 30

α0 (

/°)

δ (/°)

α0 vs. δ plot

Wind tunnel data



19

17

VS . PLO TS

A plot was obtained from its equation in the introduction:

17 Photograph. Theory of Wing Sections. United States: McGraw-Hill Book, 1959. 478. Print.

Figure 10 Lift slope for the NACA 2412 airfoil (𝛿=0°)


20

The slope of the linear section of graphs above is used to compare lift curves; a steeper linear part

suggests that a wing creates more lift per alpha. The discrepancy in lift slopes for =0° between my

results and those from NACA (Fig. 10) (their slope: 0.105 deg-1

, mine: 0.024 deg-1

) may be attributed

to friction in my data collection mechanism which would impeded lift increase by friction factor .

Notice that is a multiple of from the section lift equation, so all analysis and evaluation from the

previous section applies and will not be repeated here.

My experiment has also shown the following relationship between and the flap deflection (in

degrees):

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-20 -15 -10 -5 0 5 10 15 20 25

CL

(/d

imen

sio

nle

ss)

α (/°)

CL versus α for various δ

-10°

-5°

0°

5°

10°

15°

20°

25°


21

increases the maximum lift coefficient as shown by the positive correlation between and

.

VS . PLO TS

My data confirms the hypothesized negative correlation between and .

Comparing wind tunnel data to thin airfoil predictions for and graphing:

Flap deflection (/°)

αL0 (/°) Empirical

αL0 (/°) Theory

-10 2 3

-5 0 1

0 -2 -2

5 -4 -5-

10 -6 -7

15 -7 -10

20 -19 -12

25 N/A -15

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-20 -10 0 10 20 30

ΔC

Lm

ax (/

dim

ensi

on

less

)

δ (/°)

ΔCLmax vs. δ

Wind tunnel data



22

-25

-20

-15

-10

-5

0

5

-15 -10 -5 0 5 10 15 20 25 30

αL

0 (

/°)

δ (/°)

αL0 vs. δ plot

Wind tunnel data

Thin airfoil theory


23

Linear fit lines calculated

with Vernier LoggerPro®

An outlier (ignored for linear fit)


24

The theoretical vs. slope (=-0.517) appears more negative than the experimental slope (=-0.371).

Notice that thin airfoil theory overestimates for negative and underestimates for positive , and

equals experimental for . This may be due to the theory’s neglect for viscous effects which

perhaps increase with flap deflection due to higher drag.

There is a large jump in between the 15°-20° deflections, and no data for =25°. This is because

my lift curves flatten for negative alpha of large . As higher increases lift/drag, the rods connecting

the wing to the 2D.P. balance would rub more against their tunnel floor holes, preventing seamless lift

force transmission- producing the “leveling off” effect. Thus at large the extrapolated value

appears unnaturally low, creating the jump seen above. My results should not be a benchmark for

the theory or the theory for my results- my tunnel lacks accuracy due to friction in the data

collection mechanism whereas thin airfoil theory wrongfully ignores viscous effects. The values from

both are therefore dubious in magnitude, but it is certain that there is a negative correlation between

and .

CONCLUSION ON RELATIO NSHIP BETWEEN AND , , AND

True to my hypothesis, there exists a positive correlation between and , a positive

correlation between and , a negative correlation between and , and a negative

correlation between and . It is of great interest for the engineer and the pilot to know that the

stall angle of attack decreases linearly with positive flap deflection; as a result, it is crucial that the

airplane takes off and lands at lower than usual so as to avoid a stall and a subsequent crash. For a

further investigation in flap deflection effects on aerodynamic properties, I should investigate drag. I

predict there would be a proportional rise in the section drag and drag coefficients with increased flap

deflection as flow separation and frontal area facing the free stream airflow would increase.

SCRUTINY


25

OF MET HOD A ND A P PA R A TUS

The Reynolds number in my experiment was . This is smaller by 2 orders of magnitude from

the values used in for Fig. 1018

. Notice that the lowest comparable19

value from graph (ο) in Fig.

10 was achieved under the lowest of . The small magnitudes of my may therefore be

explained by my lower Reynolds number. I may increase my values by raising through

increasing airspeed, the chord of the wing, or the fluid viscosity by using something like water

(although this is unrealistic for my tunnel). These improvements are questionable in their usefulness as

I am merely proving a trend; however, engineers designing planes like the Airbus A380 will need

accurate values.

The test chamber I used also prevented me from achieving entirely accurate data. From a forum about

my tunnel:

“YOU’RE… GOING TO FIND THAT W ITH A 12X12 INCH TEST AREA… YOU 'RE GOING

TO SUFFER INTERFERENCE FROM THE WAL LS , BOTTOM AND TOP OF THE TEST

AREA WITH AIRFOILS I N THE 3 TO 4 INCH CHORD RANGE… IN FULL SIZE WIND

TUNNELS… THE MODELS BEING TES TED ARE TYPICALLY LESS THAN 1/10 THE SIZE

OF THE CROSS SECTION OF THE TEST AREA .”20

Compression of airflow and wingtip vortices has thus likely offset the accuracy of my results. To

improve, I would need to reduce wing size to satisfy above conditions (in which case wingtip vortices

would be present), and/or increase the size of the test section (this is not feasible as my tunnel is

stationed in school).

18 Fig. 10 is not my own data.

19 The shallowest curve (Δ) is for a rougher, incomparable wing section.

20 BMatthews. RC Universe. s.d. 19 July 2011

<http://www.rcuniverse.com/forum/m_10501474/anchors_10506889/mpage_1/key_/anchor/tm.htm#10506889>.


26

One of the largest shortcomings was friction in the data collection mechanism of wing rods against

tunnel floor and the wing against the acrylic container in trying to reduce wingtip effects21

. The wing

rods also had turbulent circular cross-sections; they could be reshaped into symmetric airfoils such as

the NACA 0012. The steel rod for changing was also circular. To avoid this, one could buy

university-grade apparatus, but a more feasible solution may be to implement simple harmonic motion

(Appendix I).

The accuracy of and measurement could also be improved. As a protractor of 1° uncertainty was

used, and and was measured in 5° increments, this presented over 20% uncertainty in the

readings! More accurate methods of measuring these variables are implemented in university-grade

apparatus in attaching the wing to a device which sets it at an accurate angle while measuring the

forces acting on the wing, such as in Fig. 11.

22

Thin airfoil theory, as previously mentioned, is also worth evaluating. This theory is a tremendous

simplification which allows quick performance predictions for aircraft operating at high . My wing

operated at low so the considerable flow separation and viscous forces acting on it were ignored by

this theory! Hence whereas both empirical and theoretical results agreed on the trend between and

, they rarely agreed on the magnitudes of these values. Thin airfoil theory also ignores stall- an

21 This means reducing wingtip vortices to make the section lift coefficient applicable

22 TecQuipment. s.d. 19 July 2011 <http://www.tecquipment.com/prod/AFA3.aspx>.

Figure 11 A three-component balance designed for use with the AF100 Subsonic Wind Tunnel.


27

important performance factor- so could not predict factors like factors such as . The turbulent flow

separation over my wing at low also explains the relative irregularity of my data compared to the

linearity of thin airfoil theory that assumes no flow separation.

OF WI ND T UNNEL

Found through this paper are textbook facts. Presented alone, these give limited value to the extended

essay as flap deflection effects on lift are common engineering knowledge. My essay goes beyond

information in such books as it evaluates a homemade wind tunnel’s abilities to predict a known

aerodynamic relationship.

The good. Lift data collected in a homemade wind tunnel enable an engineer to extrapolate results to a

wide range of performance features- lift coefficient, maximum lift alpha, effect of zero-lift alpha, etc.

These relationships have been compared against recognized sources in this essay and all were correct.

An engineer seeking experimental evidence from a wind tunnel will also find these trends; he/she will

also have more time than I did and, by collecting data for more alpha and perhaps buying a more

powerful engine to increase , he/she will make patterns in this essay more refined (such as for )

and thus get better results than I did.

The bad. A wind tunnel’s job is to show both trends and values of correct/accurate magnitude. In

designing aircraft, it is of little use to an engineer to merely know the trend- he/she must also know the

value, such as the maximum lift coefficient. He/she may design a failed aircraft thinking is 0.7

from an inaccurate homemade wind tunnel test when it is in fact 1.4. As a result, CFD23

methods will

be much more useful as they will reveal both patterns and correct values.

What makes wind tunnels still useful is their ability to predict what CFD cannot- a wind tunnel must

hence be more accurate than a computer code. In the case of my homemade wind tunnel, even though

it reveals the trends, its construction is painstaking and will not pay off as its accuracy will be inferior

23 CFD: Computational Fluid Dynamics.


28

to that of computer simulation. In my case, CFD will ultimately be both faster and more accurate than

a wind tunnel- and this is a topic for a whole another extended essay.

Nevertheless, the data achieved in my tunnel enabled me to draw conclusions consistent with those

achieved in much more expensive, professional-grade aerodynamic testing facilities. To what extent

can a homemade wind tunnel predict the effect of on wing performance? To the extent of trends, not

of correct values; as a result, what I found is that my wind tunnel enabled me to find the trends I was

looking for, and they were correct trends. Never mind that I did not get accurate actual magnitudes of

or - my wind tunnel enabled me to construct an experimentally-supported argument, and that was

its aim. As for an engineer, he/she should seek more elaborate testing facilities and CFD methods to

fulfill his/her needs.

Word Count: 3’999/4’000


29

BIBLIOGRAPHY

Internet:

"Aerodynamics Definition." NASA. NASA. Web. 14 July 2011. <http://www.grc.nasa.gov/WWW/K-

12/FoilSim/Manual/fsim001m.htm>.

BMatthews. RC Universe. s.d. 19 July 2011

<http://www.rcuniverse.com/forum/m_10501474/anchors_10506889/mpage_1/key_/anchor/tm.htm#1

0506889>.

Books:

Phillips, Warren F. "1.1 Introduction and Notation." Mechanics of Flight. Hoboken, NJ: Wiley, 2004.

Print.

Phillips, Warren F. "1.4 Inviscid Aerodynamics." Mechanics of Flight. Hoboken, NJ: Wiley, 2004.

Print.

Phillips, Warren F. "1.6 Incompressible Flow over Airfoils." Mechanics of Flight. Hoboken, NJ:

Wiley, 2004. Print.

Ira H. Abott, Alber E. Von Doenhoff. Theory of Wing Sections. United States: McGraw-Hill Book

Company, Inc., 1959.

Video/audio/photo:

TecQuipment. s.d. 19 July 2011 <http://www.tecquipment.com/prod/AFA3.aspx>.

Photograph. Mechanics of Flight. Hoboken, NJ: Wiley, 2010. 27. Print.





30

"AeroCFD Operating Instructions." AeroRocket Simulation Software for Rockets and Airplanes. Web.

20 Nov. 2011. <http://aerorocket.com/AeroCFD/manual.html>.



Photograph. Theory of Wing Sections. United States: McGraw-Hill Book, 1959. 195. Print.

Photograph. Theory of Wing Sections. United States: McGraw-Hill Book, 1959. 478. Print.

Database photos :

Photograph. UIUC Airfoil Coordinates Database. Web. 20 Nov. 2011. <http://www.ae.illinois.edu/m-

selig/ads/coord_database.html>.


31

APPENDICES

A

Section flap hinge efficiency, , and deflection efficiency, , are used in the equation for calculating

section flap effectiveness, :

The values for these parameters are deduced from the following graphs:

Figure A.1 Section flap hinge efficiency for sealed trailing-edge flaps. For unsealed flaps (as in my

case) these values should be reduced by 20%.


32

Figure A.2 Section flap deflection efficiency for sealed trailing-edge flaps. For deflections less than

10°, a deflection efficiency of 1.0 should be used.


33

B

As seen in Fig. B.1, the section flap effectiveness predicted by thin airfoil theory correlates well to the

more complex vortex panel method which takes thickness into account (this method is beyond the

scope of this text), but overestimates the actually flap effectiveness calculated from experiment (Fig.

B.1 is not my own data).

As we can see, thin airfoil theory always overestimates actual section flap effectiveness. This is

because, in reality, the hinge mechanism about the flap causes local boundary layer separation and

flow leakage from the high-pressure bottom surface to the lower-pressure top surface, and this causes

the section flap effectiveness to reduce as it is a factor of the section flap hinge efficiency as in

. In addition, at deflections greater than 10°, error associated with the ideal section flap

Figure B.1 Section flap effectiveness compared amongst thin airfoil theory, vortex panel method, and

experimental data.


34

effectiveness becomes significant, resulting in the implementation of the factor in the equation

for whose value is calculated from Appendix A.

From Fig. B.1 we see that for a flap chord fraction like mine of about 0.36, the discrepancy in

theoretical and experimental results is only about 7%, whereas for smaller flap chord fractions of

about 0.1 the error rises to almost 25%. As a result, a flap chord fraction like mine is a good choice for

the application of thin airfoil theory as it reaches a good compromise which allows for a small enough

error yet provides large enough maximum section lift coefficient increments to reduce percentage

error in data as per Abbott and Doenhoff24

and yet does not dominate too much of the airfoil geometry

by the flap.

24 Ira H. Abott, Alber E. Von Doenhoff. Theory of Wing Sections. United States: McGraw-Hill Book Company,

Inc., 1959.


35

C

As was mentioned in the main body of the text, thin airfoil theory is a method belonging to a field of

inviscid aerodynamics that ignores viscous effects on the forces acting on airborne bodies.

Thin airfoil theory is in good agreement with experimental data for low speeds and small for airfoils

of thickness <12%. Figs. C.1 and C.2 compare the results given by thin airfoil theory to those given by

the vortex panel method (ignored here) and experimental data:

At small , thin airfoil theory agrees well with experimental data. However, as it predicts a linear

relationship between lift coefficient and the angle of attack, thin airfoil theory overestimates and

becomes flat out incorrect for lift coefficients beyond the stall angle. Also, viscous effects become

important for airfoils thicker than 12%, so thin airfoil theory predictions deviate from experimental

data as seen in Fig. C.2. However, notice than thin airfoil theory and experimental data agree very

closely around the zero lift angle of attack; we hence see that thickness distribution has little effect on

Figure C.1 Section lift coefficient for NACA

2412 predicted by thin airfoil theory vs. vortex

panel method vs. experimental data.

Figure C.2 Section lift coefficient for NACA

2421 predicted by thin airfoil theory vs. vortex

panel method vs. experimental data.


36

lift produced by any airfoil around the zero-lift angle of attack. It is hence interesting to notice that, in

my own experiment, thin airfoil theory agreed very well with my data for the vs. plot- in fact, it

agreed with it more than for any other plot that I generated for this extended essay.

The reason for why thin airfoil theory predicts lift so well over such a wide range of airfoil thicknesses

is because thickness increases the lift slope, whereas viscous effects decrease it. As the theory neglects

both, these errors tend to cancel, giving the theory a wide usability range. However, thin airfoil theory

cannot predict section drag- which makes it impossible to use if I were to extend the scope of this

essay to collect drag data as well for the NACA 2412 airfoil.

An alternative to thin airfoil theory is a computation fluid dynamics (CFD) solution; however, CFD

experiments require high computational power and will increase computation time by several orders of

magnitude.


37

D

Excel was used to create spreadsheets which were used in plotting the vs. , the vs. , and the

vs. curves as predicted by thin airfoil theory. In doing so, spreadsheets were created that are

demonstrated on the following pages as part of Appendix D.

To create these spreadsheets, the following functions were create within excel and used for each

appropriate cell. These are shown below.

Equation for calculating the ČL(α,δ) column:

=2*PI()*(RADIANS(A18)-RADIANS($B$30)+G30*RADIANS($A$16))= [ ( ) ]

Where the flap deflection angle , the zero-lift angle of attack with no flap deflection ( ), and the

angle of attack are all in radians.

Equation for calculating the column:

=B18*0.5*$B$34*$B$35^2*(95/1000)*101.971621298=

Where the factor of 1000 is because “95” is the airfoil chord in millimeters and the factor of

101.971621298 are to convert Newtons, the force unit of lift which the lift equation outputs by default,

to grams-force, which is the gram multiplied by the acceleration of gravity and is equal to the gram

measured by the 2 D.P. digital balance used during the experiment.

Equation for calculating the cell is:

=ACOS(2*B33-1)= (

)

Equation for calculating the cell is:

=1-((B32-SIN(B32))/PI())=


38

Raw theoretical lift: Flap Deflection (/°)

-10 -5 0 5 10 15 20 25

α (/°) Lift (/g) α (/°) Lift (/g) α (/°) Lift (/g) α (/°)

Lift (/g) α (/°)

Lift (/g) α (/°)

Lift (/g) α (/°)

Lift (/g) α (/°)

Lift (/g)

-15 -193.2 -14 -156.9 -13 -130.7 -12 -104.5 -11 -78.3 -10 -52.1 -9 -25.9 -8 0.3

-10 -142.9 -9 -106.6 -8 -80.4 -7 -54.2 -6 -28.0 -5 -1.8 -4 24.4 -3 50.6

-5 -92.6 -4 -56.4 -3 -30.2 -2 -4.0 -1 22.2 0 48.4 1 74.7 2 100.9

0 -42.4 1 -6.1 2 20.1 3 46.3 4 72.5 5 98.7 6 124.9 7 151.1

5 7.9 6 44.2 7 70.4 8 96.6 9 122.8 10 149.0 11 175.2 12 201.4

10 59.5 11 95.1 12 120.7 13 146.2 14 171.8 15 197.3 16 222.9 17 248.4

15 111.9 16 146.4 17 170.9 18 195.4 19 219.9 20 244.4 21 268.9 22 293.4

20 164.4 21 197.8 22 221.2 23 244.6 24 268.0 25 291.3 26 314.7 27 338.1

Calculated theoretical lift coeff.:

Flap Deflection (/°)

-10 -5 0 5 10 15 20 25

α (/°) ČL(α,δ) α (/°) ČL(α,δ) α (/°) ČL(α,δ) α (/°) ČL(α,δ) α (/°) ČL(α,δ) α (/°) ČL(α,δ) α (/°) ČL(α,δ) α (/°) ČL(α,δ)

-16 -2.1 -15 -1.7 -15 -1.4 -15 -1.1 -15 -0.9 -15 -0.6 -15 -0.3 -15 0.0

-11 -1.6 -10 -1.2 -10 -0.9 -10 -0.6 -10 -0.3 -10 0.0 -10 0.3 -10 0.6

-6 -1.0 -5 -0.6 -5 -0.3 -5 0.0 -5 0.2 -5 0.5 -5 0.8 -5 1.1

-1 -0.5 0 -0.1 0 0.2 0 0.5 0 0.8 0 1.1 0 1.4 0 1.6

4 0.1 5 0.5 5 0.8 5 1.1 5 1.3 5 1.6 5 1.9 5 2.2

9 0.6 10 1.0 10 1.3 10 1.6 10 1.9 10 2.2 10 2.4 10 2.7

14 1.2 15 1.6 15 1.9 15 2.1 15 2.4 15 2.7 15 2.9 15 3.2

19 1.8 20 2.2 20 2.4 20 2.7 20 2.9 20 3.2 20 3.4 20 3.7


39

Other factors

Airfoil NACA 2412

δ (/°) ηh ηd εf

δ (/°) αL0(δ)

αL0(0) -2.0

-10 0.74 1.00 0.52

-10 4

εfi 0.70666159

-5 0.74 1.00 0.52

-5 2

θf 1.87548898

0 0.74 1.00 0.52

0 0

cf/c 0.35

5 0.74 1.00 0.52

5 -2

ρ∞ (kg/m^3) 1.126098

10 0.74 1.00 0.52

10 -3

V∞ (m/s) 4.1

15 0.74 0.98 0.51

15 -5

20 0.74 0.94 0.49

20 -7

25 0.74 0.89 0.46

25 -8

30 0.74 0.85 0.44

35 0.74 0.81 0.42

40 0.74 0.76 0.40

45 0.74 0.71 0.37

50 0.74 0.66 0.34

The “other factors” table provides values of variables that are needed for the calculation of other parameters which were used in this paper.

Equation for calculating the column:

=E30*F30*$B$31=

Where and are taken from Appendix A.


40

Method for obtaining the column:

The column was obtained through analyzing the plot for the x-intercept of the curves for each flap deflection .


41

E

To those whom it may concern, below are raw and collated data tables of all values which were used

to present the different plots shown throughout the main body of this text. Note that lift was collected

in grams, or its equivalent “grams-force”, and therefore a factor of 0.00980665 was used to convert the

lift in grams into lift in Newtons when calculating the section lift coefficient values .

Aux. Data

Chord of wing (/mm) 95

Flap deflection (/°) αL0 (/°)

Chord of flap (/mm) 34

-10 2

Flap fraction 0.36

-5 0

Wing airfoil NACA 2412

0 -2

ρ∞ (kg/m^3) 1.126098

5 -4

V∞ (m/s) 4.1

10 -6

Airspeed right pole 3.7

15 -7

Airspeed left pole 3.6

20 -19

Airspeed middle 5

25 N/A

μ air (kg m/s) 0.00001983

Re (dimensionless) 22119

2.2*104


42

Raw theoretical lift: Flap Deflection (/°)

-10 -5 0 5 10 15 20 25

α (/°) Lift (/g) α (/°) Lift (/g) α (/°)

Lift (/g)

α (/°)

Lift (/g)

α (/°)

Lift (/g)

α (/°)

Lift (/g)

α (/°)

Lift (/g)

α (/°)

Lift (/g)

-15 -20 -15 -14 -15 -10 -15 -9 -15 -8 -15 -4 -15 3 -15 7

-10 -15 -10 -9 -10 -7 -10 -6 -10 -4 -10 -2 -10 7 -10 10

-5 -10 -5 -5 -5 -3 -5 -1 -5 1 -5 3 -5 11 -5 14

0 -3 0 0 0 3 0 7 0 12 0 14 0 19 0 22

5 6 5 10 5 15 5 21 5 27 5 30 5 33 5 36

10 13 10 19 10 29 10 35 10 50 10 53 10 54 10 56

15 16 15 22 15 39 15 43 15 45 15 46 15 48 15 50

20 13 20 19 20 25 20 31 20 34 20 36 20 41 20 44

Calculated theoretical lift coeff.:

Flap Deflection (/°)

-10 -5 0 5 10 15 20 25

α (/°) ČL(α,δ) α (/°) ČL(α,δ) α (/°) ČL(α,δ) α (/°) ČL(α,δ)

α (/°) ČL(α,δ)

α (/°) ČL(α,δ)

α (/°) ČL(α,δ)

α (/°) ČL(α,δ)

-15 -0.2 -15 -0.2 -15 -0.1 -15 -0.1 -15 -0.1 -15 0.0 -15 0.0 -15 0.1

-10 -0.2 -10 -0.1 -10 -0.1 -10 -0.1 -10 0.0 -10 0.0 -10 0.1 -10 0.1

-5 -0.1 -5 -0.1 -5 0.0 -5 0.0 -5 0.0 -5 0.0 -5 0.1 -5 0.2

0 0.0 0 0.0 0 0.0 0 0.1 0 0.1 0 0.2 0 0.2 0 0.2

5 0.1 5 0.1 5 0.2 5 0.2 5 0.3 5 0.3 5 0.4 5 0.4

10 0.1 10 0.2 10 0.3 10 0.4 10 0.5 10 0.6 10 0.6 10 0.6

15 0.2 15 0.2 15 0.4 15 0.5 15 0.5 15 0.5 15 0.5 15 0.5

20 0.1 20 0.2 20 0.3 20 0.3 20 0.4 20 0.4 20 0.4 20 0.5


43

F

A P P A R A T U S D I A G R A M

Air Intake

Wind tunnel base Sample airflow path

Excel data collection

NACA 2412 wing

Rod to vary angle

of attack

Wing-balance rods

Wing-balance base

2 D.P. balance

Test section

Alpha-measuring protractor

Airflow straightener


44

A P P A R A T U S P H O T O G R A P H S

Perspective Side

Inside Me & The Wind Tunnel!


45

G

The thickness distribution of an airfoil of maximum thickness only slightly affects the

pressure forces and moments acting on it. The net aerodynamic force on such airfoils is largely

dependent on and the camber line geometry, which is why, as shown in Fig. 7, thin airfoil theory

assumes an airfoil of zero thickness about its camber line.

Thus, the aerodynamic forces about such airfoils can be approximated by a vortex sheet placed along

the camber line whose strength is allowed to vary with distance along the camber line, as shown in

Fig. 6.25

The making of such a sheet is beyond the scope of this paper and is not required background

knowledge; as of raw interest, the equation governing the construction of this vortex sheet is:

∫ ( )

( )

After several derivations, thin airfoil theory suggests that the section lift coefficient of an airfoil of

thickness is equal to:

(

∫ ( )

) ( )

25 Phillips, Warren F. "1.6 Incompressible Flow over Airfoils." Mechanics of Flight. Hoboken, NJ: Wiley, 2004.

Print.

Figure 7 An airfoil simplified by the thin airfoil theory


46

Note that thin airfoil theory performs an inviscid analysis- one which does not take into account

viscous effects of an airfoil- and is thus suited for approximation at high .


47

H

For small and , thin airfoil theory may be used to predict the effects of flap deflection on lift. The

development of equations to do so is described below.

Let be the y-position of the camber line of an airfoil with a deflected flap as shown in Fig. 9:

So, from the last given equation for without flap deflection, we may write as:

(

∫

( )

) ( )

And from small-angle approximation:

{

Where is the undeflected camber line and is the flap chord length and is the flap deflected in

radians. Another important quantity is the flap chord fraction:

(

)

And the ideal section flap effectiveness:

Figure 9 Thin airfoil flap approximation (Fig. 1.7.2)


48

Notice that depends only on the flap to total chord ratio and is independent of camber line

geometry or flap deflection; using this, one finds that the zero lift angle of attack for an airfoil with an

ideal flap varies linearly with flap deflection:

( ) ( )

Eventually, one finds that the lift coefficient for an airfoil with a trailing edge flap varies linearly with

and :

( ) [ ( ) ]

Where is the section flap effectiveness:

and are the section flap hinge efficiency and deflection efficiency, and are deduced from graphs

given in Appendix A. An evaluation of the accuracy with which thin airfoil theory predicts is given

by Appendix B.

As a result, using , , and values in the equation for ( ), one may deduce the section lift

coefficient and hence the lift from the aforementioned lift equation, and from graphical analysis of the

lift slopes the value may be found. A more careful critique of thin airfoil theory in terms of its

discarding of viscous effects is given by Appendix C.


49

I

To avoid buying expensive university-grade apparatus, a more feasible solution of using simple

harmonic motion could be used. The wing would be suspended by 4 springs at the wing planform

edges from bottom and top, and as it generates lift it would oscillate up and then down as the force

from the springs pulls it back to equilibrium. Using a 32 fps camera, this could be recorded, digitized,

and by means of a scale the displacement of the wing from equilibrium could be measured. With this

and the spring constant known, the lift force could be determined. If the wing also exhibits horizontal

motion backwards, even drag force could be determined. Diagram:

However, springs would introduce a lot of turbulence and such a setup would make it difficult to vary

, limiting the amount of data that could be collected.

Drag Lift 32 fps camera Springs Test section


50

J

The wind tunnel experiment carried out for this paper had the following controlled variables, all of

which were successfully controlled by methods outlined below:

1. The test section must be closed off from surroundings by an acrylic cover for each trial

a. This was achieved by ensuring that the acrylic cover was mounted over the test

section with 4 screws for each trial.

2. The wing used has a NACA 2412 cross-section

a. This was controlled by using the same model wing for each trial (only one such model

was made, so everything related to the wing was controlled). All results were also

collected in the space of two days, so it was ensured that the wing model did not

degrade significantly in quality across trials.

3. The flap to wing chord ratio is 0.36c

a. This is likewise controlled by using the same wing model.

4. The same lift measurement mechanism must be used for each trial

a. This is controlled by, essentially, using the same mechanism- the same rod length

connecting the wing to the 2 D.P. digital balance underneath the wind tunnel, the

same 2 D.P. balance itself, and the same wire to control the wing angle of attack.

5. The same engine speed setting must be used for each trial to ensure the same airflow speed

inside the wind tunnel

a. As the engine is radio-controlled, this variable is controlled by running the engine at

full-throttle on the radio-control transmitter controlling the engine. This ensures that

the engine is running air through the wind tunnel at the maximum possible speed, thus

improving the accuracy of collected data and the “fair test” factor across all trials in

the experiment.

6. The wing surface texture must be the same

a. Controlled by using the same wing model.


51

7. The wing must have the same span of 0.30m and aspect ratio of 3.2

a. Controlled by using the same wing model.

8. The same 2 D.P. digital balance must be used for measuring wing mass

a. This is controlled by using the same 2 D.P. digital balance available in the school

physics laboratory.

9. The wing must be positioned at the same height in the test section

a. This is controlled by using the same 2 D.P. balance, positioning the wind tunnel at the

same height above ground for each period of data collection, and using the same rods

connecting the wing to the base on the 2 D.P. digital balance, thus ensuring that the

wing is propped up at the same height in the wind tunnel test section.

This hence outlines all the main controlled variables and how they were controlled in the experiment

carried out for this extended essay. The idea behind controlling controlled variables is to ensure

constant/identical conditions across all trials of the experiment to make collected data comparable

and thus conclusions drawn more accurate.


52

K (NOM ENCLA TUR E)

Angle of attack

Lift coefficient

Section lift coefficient

Lift

Section lift

Air density

Free stream density

Air pressure

Free stream pressure

Velocity

Free stream Velocity

Reynolds number

Aspect ratio

Wingspan

Mean geometric chord

Area

Angle of flap deflection

Thrust vector

Weight vector

Drag

Reynolds number

Dynamic (absolute) viscosity of fluid

y-coordinate of airfoil camber

Change in y-coordinate of camber with the x-coordinate (camber line shape)

Zero-lift angle of attack

y-coordinate of airfoil camber with flap deflected


53

Flap-deflected camber line shape

Flap chord

Flap chord to total chord ratio

Ideal section flap effectiveness

( ) Zero-lift angle of attack when flap is deflected

( )

Zero-lift angle of attack when flap deflection is 0

( )

Section lift coefficient at an angle of attack and flap deflection angle

Section flap effectiveness

Section flap hinge efficiency

Section flap deflection efficiency

Area

Dynamic pressure

Static pressure

Aspect ratio

Documents

Aerodynamics Extended Essay