Aerodynamics Extended Essay

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T

ITLE

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

R

ESEARCH

Q

UESTION

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?

A

PPROACH

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

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Dedicated to my family, the time with which was foregone in the writing of this essay.

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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.

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T

ABLE OF

C

ONTENTS

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

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D ... 37 E ... 41 F ... 43 Apparatus diagram ... 43 Apparatus photographs ... 44 G ... 45 H ... 47 I... 49 J ... 50 K (Nomenclature) ... 52

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1

I

NTRODUCTION

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 MECHANICS

Background aerodynamic knowledge was required for this paper. Appendix K (Nomenclature) defines the symbols used in this paper.

T

HE AIRFOIL

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.}

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2

B

ASIC FORCES ACTING ON AN AIRBORNE 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

T

HE LIFT EQUATIONS

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.} 4_{Photograph. Mechanics of Flight. Hoboken, NJ: Wiley, 2010. 261. Print.}

Figure 2 Forces acting on an aircraft in level flight (angles exaggerated) Figure 1 Airfoil section terminology

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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:

̃ ̃

R

EYNOLDS NUMBER

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.

B

OUNDARY LAYER THEORY

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:

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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:

5_{Photograph. Mechanics of Flight. Hoboken, NJ: Wiley, 2010. 14. Print.} 6_{Photograph. Mechanics of Flight. Hoboken, NJ: Wiley, 2010. 15. Print.}

Figure 3 Boundary layer flow over an airfoil

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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

T

HIN AIRFOIL THEORY

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).

T

HIN AIRFOIL THEORY A PPLIED TO TRAILING

-

EDGE FLAPS

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

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6

W

H AT ARE PL AIN TR AIL ING

-

EDGE FLAPS

?

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

PPL YING FL APS TO A

‘

TH I N AIR FOIL

’

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:

9_{Photograph. Mechanics of Flight. Hoboken, NJ: Wiley, 2010. 39. Print.}

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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.

H

YPOTHESIS

10_{Photograph. Mechanics of Flight. Hoboken, NJ: Wiley, 2010. 40. Print.}

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8

P

HYSICS

-

BASED

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 .

T

HIN

-

AIRFOIL BASED

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

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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 ( /g f) α (/°)

Calculated L versus α for various δ

-10° -5° 0° 5° 10° 15° 20° 25°

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10 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 -20_C (L -15 -10 -5 0 5 10 15 20 25 /di m ens io nles s) α (/°)

Calculated C

_L

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 αL0 (/ °) δ (/°)

α

_L0

vs. δ plot

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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.

E

XPERIMENTAL DESIGN

&

PROCEDURE

D

ESIGN

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.

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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.

M

ETHOD

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).}

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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.

S

AFETY

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.

W

IND TUNNEL TEST RESULTS

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.

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14

̃

VS

.

PLOTS

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 ( /g f) α (/°)

L versus α for various δ

-10° -5° 0° 5° 10° 15° 20° 25°

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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 ( /g f) α (/°)

Calculated L versus α for various δ

-10° -5° 0° 5° 10° 15° 20° 25°

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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 (/°)

L

_max

vs. δ

Wind tunnel data

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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

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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 (/°)

α

₀

vs. δ plot

Wind tunnel data

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19

17

̃

VS

.

PLOTS

A ̃ plot was obtained from its equation in the introduction:

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

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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 (/di m ens io nles s) α (/°)

C

_L

versus α for various δ

-10° -5° 0° 5° 10° 15° 20° 25°

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

VS

.

PLOTS

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 Δ CL m ax (/di m ens io nles s) δ (/°)

ΔC

_Lmax

vs. δ

Wind tunnel data

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22 -25 -20 -15 -10 -5 0 5 -15 -10 -5 0 5 10 15 20 25 30 αL0 (/ °) δ (/°)

α

_L0

vs. δ plot

Wind tunnel data Thin airfoil theory

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23 Linear fit lines calculated

with Vernier LoggerPro®

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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 .

C

ONCLUSION ON RELATIONSHIP 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.

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25

O

F METHOD AND APPARATUS

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: “YO U’R E… G O I N G T O F I N D T H A T W I T H A 12X12 I N C H T E S T A R E A… Y O U'R E G O I N G T O S U F F E R I N T E R F E R E N C E F R O M T H E W A L L S, B O T T O M A N D T O P O F T H E T E S T A R E A W I T H A I R F O I L S I N T H E 3 T O 4 I N C H C H O R D R A N G E… I N F U L L S I Z E W I N D T U N N E L S… T H E M O D E L S B E I N G T E S T E D A R E T Y P I C A L L Y L E S S T H A N 1/ 10 T H E S I Z E O F T H E C R O S S S E C T I O N O F T H E T E S T A R E A.”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}

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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>.}

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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.

O

F WIND TUNNEL

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.}

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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.

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29

B

IBLIOGRAPHY

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.

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.

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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>.

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31

A

PPENDICES

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

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32

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

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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

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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

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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.

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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.

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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:

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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

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39 Other factors Airfoil NACA 2412 δ (/°)

η

h

η

d 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=

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40 Method for obtaining the column:

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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

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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

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43

F

A

PPARATUS D IAGR AM

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

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44

A

PPARATUS P HOTOGRAP HS

Perspective Side

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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.

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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 .

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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:

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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

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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

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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

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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.

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52

K

(N

OMENCLATURE

)

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

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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