thin airfoil theory lab - nd.eduame40431/labnotes/ame30333_lab2_2016.pdf · to start the lab, a...

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Thin Airfoil Theory Lab AME 30333 University of Notre Dame Spring 2016 Written by Chris Kelley and Grady Crahan December 10, 2008 Updated by Brian Neiswander and Ryan Kelly February 6, 2014 Updated by Kyle Heintz February 8, 2016 Abstract The purpose of this report is (1) to measure and analyze pressure distribution around an airfoil, (2) to calculate lift, drag, and moment coefficients for the entire airfoil from discrete, local pressure coefficient measurements, (3) and to compare experimental results with thin airfoil theory. To start the lab, a NACA 0015 symmetric airfoil will be placed in the wind tunnel with pressure taps located at known distances from the leading edge. Then pressure measurements will be taken and recorded to the computer with a pressure transducer connected to a Scanivalve. The local pressure coefficient will be calculated at each pressure tap on both the upper and lower airfoil surface. Next, the normal and axial force coefficients will be tabulated for each pressure tap location which will then be transformed to a Riemann sum to find the total lift, drag, and quarter chord moment coefficients for the entire airfoil. By repeating these measurements at different angles of attack, the lift curve slope can be calculated and compared to the expected value from thin airfoil theory. Similarly, the coefficient of pressure can be used to calculate the moment coefficients at the leading edge and the quarter chord locations. 1

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Page 1: Thin Airfoil Theory Lab - nd.eduame40431/LabNotes/AME30333_Lab2_2016.pdf · To start the lab, a NACA 0015 symmetric airfoil will be placed in the wind tunnel with pressure taps located

Thin Airfoil Theory Lab

AME 30333University of Notre Dame

Spring 2016

Written by Chris Kelley and Grady Crahan December 10, 2008Updated by Brian Neiswander and Ryan Kelly February 6, 2014Updated by Kyle Heintz February 8, 2016

Abstract

The purpose of this report is (1) to measure and analyze pressure distribution around anairfoil, (2) to calculate lift, drag, and moment coefficients for the entire airfoil from discrete,local pressure coefficient measurements, (3) and to compare experimental results with thin airfoiltheory. To start the lab, a NACA 0015 symmetric airfoil will be placed in the wind tunnel withpressure taps located at known distances from the leading edge. Then pressure measurementswill be taken and recorded to the computer with a pressure transducer connected to a Scanivalve.The local pressure coefficient will be calculated at each pressure tap on both the upper and lowerairfoil surface. Next, the normal and axial force coefficients will be tabulated for each pressuretap location which will then be transformed to a Riemann sum to find the total lift, drag, andquarter chord moment coefficients for the entire airfoil. By repeating these measurements atdifferent angles of attack, the lift curve slope can be calculated and compared to the expectedvalue from thin airfoil theory. Similarly, the coefficient of pressure can be used to calculate themoment coefficients at the leading edge and the quarter chord locations.

1

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2

1 Introduction

1.1 Thin Airfoil Theory

Thin Airfoil Theory is the work of Max Munk, a colleague of Ludwig Prandtl. For the interestedreader his original report was first translated into English in 1923 is NACA report 142, GeneralTheory of Thin Wing Sections [1].

Some preliminary kinematics are necessary for thin airfoil theory. The first component isa vortex, defined mathematically as a point that produces purely tangential velocity inverselyproportional to the distance from the vortex:

Vθ = − Γ

2πr. (1)

Vθ is the velocity induced at a point of interest distance r from the vortex with strength Γ, positivewhen producing clockwise velocity. Now assume that this vortex is a line that extends infinitelyinto and out of the page forming a vortex filament. A vortex sheet is formed by having a continuousfunction of vortex filaments. It is this vortex sheet that Prandtl defined which allowed Munk toquantify thin airfoil theory as will be seen.

The circulation, or strength, of the vortex filament per unit length is defined as γ(s). UsingEq. 1, the infinitesimal velocity dV∞ induced due to a vortex filament is described by

dVθ = −γds2πr

, (2)

where ds is the infinitesimal arc length along the vortex sheet. The primary assumption of thinairfoil theory is to assume that a thin airfoil (one where the thickness is on the order of magnitudesmaller than the chord) can be replaced with a vortex sheet. This sheet corresponds to the pathof the camber line of the airfoil which is also a streamline so that the camber line is impermeable.Figure 1 shows this camber line, z(x), where the local slope can be written as arctan

(− dzdx

). In

addition to the vortex sheet there is a free-stream velocity V∞ at some angle of attack α.

Figure 1: Vortex Sheet [2]

The velocity induced by the vortex sheet, w(s), at any point along the camber line is equal andopposite to the free-stream velocity normal to the camber line, V∞,n. When these two velocities areequal and opposite the camber line lies along a streamline because the flow is everywhere parallel

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1.1 Thin Airfoil Theory 3

to streamlines by definition. Next the assumption is made that the vortex sheet instead of beingaligned along the camber line, is along the chord line. Therefore w′(s) ≈ w(x) because the distancebetween the camber line is very small, and for a symmetric airfoil the difference is zero. So for thenormal free-stream to be cancelled out by the induced velocity of the vortex sheet,

V∞,n + w(x) = 0. (3)

Figure 2: Vortex Sheet Approximated along Chord Line [2]

From trigonometry and Figure 2, the local normal free-stream velocity V∞,n can be written as

V∞,n = V∞ sin

(α+ arctan

(−dzdx

)). (4)

Assuming small angles of attack and a thin airfoil, a small angle approximation is used (α ≈ sinα ≈arctanα), and so (4) becomes

V∞,n = V∞ sin

(α− dz

dx

). (5)

Using (5) and integrating from the leading edge (LE) to the trailing edge (TE) with the dummyvariable along the chord line ξ, the total induced velocity due to the vortex sheet w(x) can then befound as follows:

w(x) =

∫ c

0

γ(ξ)

2π(x− ξ)dξ. (6)

Plugging (6) and (4) into (3) with small rearrangement yields the fundamental equation of thinairfoil theory for an uncambered airfoil:

1

∫ c

0

γ(ξ)

(x− ξ)dξ = V∞α. (7)

In this lab the NACA 0015 symmetric airfoil is being used, so the uncambered assumptionthat dz

dx = 0 is accurate, and remember that in thin airfoil theory α is always measured in radians.Next it is necessary to make a change of variables from ξ and x to θ such that ξ = c

2(1 − cos θ),dξ = c

2 sin θdθ, and x = c2(1 − cos θ0). Substituting into (7), and noting that the bounds of

integration must change such that θ = 0 at the leading edge and that θ = π at the trailing edge.

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1.1 Thin Airfoil Theory 4

The solution of the integral of (7) is then

γ(θ) = 2αV∞1 + cos θ

sin θ. (8)

Now it is necessary to find the total circulation due to the vortex sheet.

Γ =

∫ c

0γ(ξ)dξ =

c

2

∫ π

0γ(θ) sin θdθ = παcV∞ (9)

Equation (9) is quite useful when combined with the Kutta-Joukowski theorem: lift per unit span(2D lift) L′ is equal to ρ∞V∞Γ. Hence the lift coefficient from thin airfoil theory is proportional toangle of attack:

Cl =L′

12ρ∞V

2∞

=ρ∞V∞παcV∞

12ρ∞V

2∞c

= 2πα (10)

Equation (10) implies that the constant of proportionality between lift and angle of attack is thelift slope Clα :

Clα =∂Cl∂α

=∂

∂α(2πα) = 2π. (11)

Another coefficient that can be obtained from thin airfoil theory is the moment coefficientabout both the leading edge and the quarter chord. Consider one vortex filament of strengthγ(ξ)dξ. Again from the Kutta-Joukowski theorem the differential lift from this vortex filament isdL = ρ∞V∞dΓ. See Figure 3 for a diagram of this increment of lift.

Figure 3: Induced Lift Increment [2]

To find the total moment per unit span about the leading edge integrate across the entirechord:

M ′LE = −

∫ c

0ξ(dL) = −ρ∞V∞

∫ c

0ξγ(ξ)dξ = −1

2ρ∞V

2∞c

2πα

2. (12)

By definition of the moment coefficient per unit span (2D) and using the previous equation forthe leading edge moment, the following is found:

CmLE =M ′LE

12ρ∞V

2∞c

2= −πα

2= −Cl

4. (13)

More importantly the moment coefficient at the quarter chord can be found from the leading

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1.2 Aerodynamic Forces and Moments 5

edge since the moment about the aerodynamic center is where all the lift acts:

Cmc/4 = CmLE +Cl4

= 0 (14)

Thin airfoil theory predicts that the moment coefficient about the quarter chord is 0, whichmeans that the aerodynamic center coincides with the center of pressure. No moment is requiredto hold a symmetric airfoil at constant angle of attack about the quarter chord.

Finally take note that drag has been implicitly defined as zero in the derivation of thin airfoiltheory. The Kutta-Joukowski theorem relates circulation to lift for a potential flow. Using thinairfoil theory and the Kutta-Joukowski theorem, a real airfoil can be replaced by a streamlinesegment with a vorticity distribution in the incompressible flow.

1.2 Aerodynamic Forces and Moments

1.2.1 Integral Coefficients

The main idea of this lab is to calculate the lift, drag, and moment coefficient from the pressuretaps and compare them to those predicted by the above thin airfoil theory results. All forces andmoments acting on a body are solely due to the pressure and shear distribution. For this lab itwill be useful to decompose the drag and lift coefficients into axial and normal force coefficients.The axial direction, A, is parallel to the chord line, and the normal direction, N , is parallel to thechord line (Figure 4). Therefore, the axial and normal coefficients only differ from drag and liftcoefficients by the angle of attack, α.

Cl = Cn cosα− Ca sinα (15)

Cd = Cn sinα+ Ca cosα (16)

By definition the pressure coefficient, force coefficients, lift coefficient, drag coefficient, and momentcoefficient are defined as follows:

Cp =plocal − p∞

q∞, Cn =

N ′

q∞c, Ca =

A′

q∞c, Cl =

L′

q∞c, Cd =

D′

q∞c, CmLE =

M ′LE

q∞c2(17)

where q∞ is the dynamic pressure and the indicates per unit span since airfoil analysis is, bydefinition, 2-dimensional. By breaking the airfoil into a lower section (denoted by subscript l) andupper section (denoted by subscript u) the total normal force coefficient per unit span is found.

N ′ = −∫ TE

LE(pu cos θ + τu sin θ)dsu +

∫ TE

LE(pl cos θ − τl sin θ)dsl (18)

Here θ is the local angle between a pressure normal vector and the line normal to the chord line. θ ispositive when measured clockwise from the normal, and negative when measured counterclockwise.See Figure 4 for a diagram of this coordinate system.

And similarly for the axial force coefficient per unit span:

A′ =

∫ TE

LE(−pu sin θ + τu cos θ)dsu +

∫ TE

LE(pl sin θ + τl cos θ)dsl (19)

The assumption is made that the pressure terms are of greater significance than the shear stress

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1.2 Aerodynamic Forces and Moments 6

Figure 4: Normal and Axial Coordinates

terms. This reduces equations (18) and (19) to simply

N ′ = −∫ TE

LE(pu cos θ)dsu +

∫ TE

LE(pl cos θ)dsl (20)

A′ =

∫ TE

LE(−pu sin θ)dsu +

∫ TE

LE(pl sin θ)dsl (21)

The moment about the leading edge per unit span (M ′LE) is found by integrating the axial and

normal differential force components multiplied by each locations moment arm.

M ′LE =

∫ TE

LEdMn +

∫ TE

LEdMa =

∫ TE

LE(pu cos θ x− pu sin θ y)dsu +

∫ TE

LE(−pl cos θ x− pl sin θ y)dsl

(22)Positive pitching moment is defined as leading edge up or clockwise direction about the leading edge.As will be seen in the next section when these equations are discretized, it will be advantageous tohave the axial and tangential force coefficients and the leading edge moment coefficient in terms ofthe pressure coefficient. First substitute equation (20) into equation (17) to get the following forthe normal force coefficient:

Cn = −∫ c

0

puq∞

cos θ d(suc

)+

∫ c

0

plq∞

cos θ d(slc

)(23)

Seeing that ds cos θ = dx then the normal force coefficient in terms of the local pressure is

Cn = −∫ c

0

puq∞

d(xc

)+

∫ c

0

plq∞

d(xc

)=

∫ c

0

(plq∞

− puq∞

)d(xc

)=

∫ c

0(Cpl − Cpu)d

(xc

). (24)

By doing the same operations the axial force and leading edge moment coefficients in terms of thelocal pressure coefficients follow:

Ca =

∫ c

0(Cpl − Cpu)d

(yc

), (25)

CmLE =

∫ c

0(Cpu − Cpl)

x

cd(xc

)+

∫ c

0(Cpl − Cpu)

y

cd(yc

). (26)

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7

1.2.2 Discretized Coefficients

In this lab the pressure is not known as a function of x and y, but the pressure is known at each taplocation. Using the integral form of the equations for the normal, axial, and leading edge momentcoefficients, we can approximate them with a Riemann sum since the pressure taps are quite closetogether. Using the trapezoidal rule in defining the summations, equations (24), (25), and (26),become the following:

Cn =

#taps∑i=LE

1

2

(Cpi + Cpi+1

) (xi+1

c− xi

c

), (27)

Ca = −#taps∑i=LE

1

2

(Cpi + Cpi+1

) (yi+1

c− yic

), (28)

CmLE = −#taps∑i=LE

1

2

(Cpi

xic

+ Cpi+1

xi+1

c

)(xic− xi+1

c

)−

#taps∑i=LE

1

2

(Cpi

yic

+ Cpi+1

yi+1

c

)(yic− yi+1

c

).

(29)Splitting the upper and lower surfaces, the discretized equations become

Cn =∑lower

1

2

(Cpi + Cpi+1

) (xi+1

c− xi

c

)−∑upper

1

2

(Cpi + Cpi+1

) (xi+1

c− xi

c

)(30)

Ca =

[−∑lower

1

2

(Cpi + Cpi+1

) (yi+1

c− yic

)]−

[−∑upper

1

2

(Cpi + Cpi+1

) (yi+1

c− yic

)](31)

CmLE =

[−∑upper

1

2

(Cpi

xic

+ Cpi+1

xi+1

c

)(xic− xi+1

c

)−∑upper

1

2

(Cpi

yic

+ Cpi+1

yi+1

c

)(yic− yi+1

c

)]−[

−∑lower

1

2

(Cpi

xic

+ Cpi+1

xi+1

c

)(xic− xi+1

c

)−∑lower

1

2

(Cpi

yic

+ Cpi+1

yi+1

c

)(yic− yi+1

c

)](32)

2 Description of Experiment

2.1 Pressure Transducer

A pressure transducer is a device that converts a pressure into a quantity that can be measured.One quantity is a voltage that can be measured using an analog-to-digital converter or a digitalvoltmeter, and another quantity would be a height difference in a U-Tube manometer.

Two common types of transducers are strain gage and capacitance types. The strain gagepressure transducer, which is the one used in this lab, consists of a thin circular diaphragm on thebottom of which are bonded tiny strain gages wired as a Wheatstone bridge. When the diaphragmexperiences a pressure on its exposed upper surface that is different from the pressure in a smallcavity under the diaphragm it deflects, and the resulting bridge imbalance is a measure of the

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2.2 Scanivalve 8

deflection. This deflection is usually very small and will need amplification after it is converted toa voltage.

The capacitance-based pressure transducer has a stretched membrane clamped between twoinsulating discs, which also support capacitive electrodes. A difference in pressure across thediaphragm causes it to deflect, increasing one capacitor and decreasing the other. These capacitorsare connected to an electrical, alternating-current (AC) bridge circuit, producing a high level ofvoltage output.

Strain gage transducers can be made small, hence they can be internally mounted in a windtunnel model. Also, they have reasonably good frequency response because of the small mass of thediaphragm and the short distance between the pressure tap and the diaphragm face. Capacitancetransducers usually are not well suited for internal mounting and such systems do not have a fastresponse.

In this experiment, the pressure transducer measures the static pressure at each tap on theairfoil (controlled by Scanivalve) referenced to the freestream static pressure. The freestream stag-nation (total) pressure is measured after the Scanivalve has stepped through all ports on the airfoil.This final measurement is the very last column of data in the saved .mat files saved.

2.2 Scanivalve

A Scanivalve will take multiple pressure taps and allow them to be measured using only one pressuretransducer. The tubes are all connected to one stationary disk that is on top of a moving disk. Themoving disk has a hole in it and will rotate the stainless steel tubes (coinciding with each pressuretap) that will be measured by the pressure transducer. Pictured in Figure 5 is the 4SDS-1124Scanivalve as installed with a stepper motor and the pressure tap tubing associated with this lab.

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2.3 Airfoil 9

Figure 5: Scanivalve

2.3 Airfoil

A NACA 0015 airfoil is being used for this experiment. It has a thickness of 22.86 mm, and a chordof 152.4 mm. This is a symmetric airfoil that is relatively thin, so thin airfoil theory will applyreasonably well. Pressure taps have been placed on this airfoil at the x and y locations in Table 1,with the origin at the leading edge. These values will be useful in calculations of Cn and Ca.

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2.4 Pressure Tap Design 10

Upper LowerTap Number x (mm) y (mm) Tap Number x (mm) y (mm)

1 0.00 0.00 16 0.84 -2.442 0.84 2.44 17 3.33 -4.683 3.33 4.68 18 7.46 -6.714 7.46 6.71 19 13.17 -8.455 13.17 8.45 20 20.42 -9.856 20.42 9.85 21 29.11 -10.837 29.11 10.83 22 39.14 -11.358 39.14 11.35 23 50.42 -11.399 50.42 11.39 24 62.82 -10.9610 62.82 10.96 25 76.20 -10.0911 76.20 10.09 26 90.41 -8.8012 90.41 8.80 27 105.31 -7.1513 105.31 7.15 28 120.71 -5.1614 120.71 5.16 29 136.47 -2.8715 136.47 2.87

Table 1: Tables of NACA 0015 coordinates for the UPPER surface on the left and the LOWERsurface on the right.

2.4 Pressure Tap Design

The motivation for the second portion of the lab is to study the effects of varying pressure tapgeometry. Pressure taps are holes drilled into a surface with which the local static pressure canbe measured. Tygon tubing is usually used to connect these taps to a pressure transducer. Thebasic assumption being made is that the static pressure within the tubing is the same as the staticpressure at the wall of the tap location.

Careful design of pressure taps will minimize error between measured static pressure and theactual static pressure. The diameter of the pressure tap orifice should be twice as great as thetubing diameter. Also the ratio of the depth of the orifice to orifice diameter should be between 0.1and 1.75 to reduce variation of pressure error. [3] However, as is the case with many measurementdevices, measuring the flow usually affects the flow. For example, if a burr was created in thedrilling of the pressure tap, it may cause the pressure in the tubing to be higher or lower than thetrue local static pressure because the streamlines bend around the burr. As seen in Figure 6, anobstruction upstream of a pressure tap causes the streamlines to have curvature concave down overthe pressure tap. In the second portion of this lab, the presence of burrs near taps is emulated byeither placing tape upstream or downstream of a tap.

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11

Figure 6: Upstream Burr

Conversely, a downstream obstruction will cause concave up streamlines over the pressure tap.

Figure 7: Downstream Burr

By using the Euler-N equation one could decide how to change the geometry near the tap, andhence the streamlines, to cause a pressure reading not indicative of the local static pressure. Alsoif a pressure gradient exists, a larger pressure tap will give an average pressure over a larger area,thus reducing its preciseness.

3 Procedure

3.1 Part 1 - Thin Airfoil Theory

1. Check that the pitot tube is oriented parallel to the flow inside the tunnel.

2. Check the pressure system. The pitot tube static pressure port (parallel to flow) is hooked upto the TOP (reference) port on the transducer. The Scanivalve is hooked up to the BOTTOM(measurement) port on the transducer.

3. Check the Analog In system. The BNC cable from the Scanivalve is connected to the AI0channel.

4. Check the Digital Out system. The Scanivalve’s “STEP” BNC is connected to DO1 and the“HOME” BNC is connected to DO3.

5. Turn on the Scanivalve controller and send it to the home setting by pressing the “Home”button.

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3.2 Part 2 - Effect of Modified Pressure Tap 12

6. Start the Aerodynamics DAQ Utility program in MATLAB.

7. Make sure Analog In is enabled. Set “Channels” to 0.Set “Sampling [Hz]” toSet “No. Samples” to

8. Make sure Digital Out is enabled. Set “No. Steps” to 1. Leave “Direction” as “Pos.”.

9. Set “Repitions” to 30.Set “Timeout [s]” to

10. Turn tunnel fan on to 25 Hz.

11. Set Airfoil to -5 degrees angle of attack.

12. Click the “Run” button. The program will automatically acquire the mean pressure trans-ducer voltages for each of the 29 pressure taps and the upstream stagnation pressure. Thedata collected is displayed in “Plot 2”.

13. When the acquisition is complete, click “Save .MAT” in the “Plot 2 - All Data” panel. Namethe file with the appropriate angle of attack (“aoa n05”, “aoa 00”, “aoa 05”, “aoa 07p5”,“aoa 10”, “aoa 12p5”, “aoa 15”, “aoa 17p5”).

14. Once the data has been saved, press “Clear Data” in the “Plot 2 - All Data” panel. Click“Yes” and “Yes” to the popup dialog boxes.

15. Repeat steps 11-14 for angles of attack of 0, 5, 7.5, 10, 12.5, 15, and 17.5 degrees usingappropriate the filenames for each.

16. Turn off the wind tunnel.

3.2 Part 2 - Effect of Modified Pressure Tap

1. Once the tunnel is off, place three thin strips of electrical tape between pressure taps 5 and6. Be sure not to cover up any of the taps.

2. Make sure that the data is cleared in “Plot 2 - All Data”. If it is not, press “Clear Data” andclick “Yes” and “Yes” to the popup dialog boxes.

3. Set the airfoil to 5 degrees angle of attack.

4. Turn on the wind tunnel to 25 Hz.

5. When the acquisition is complete, click “Save .MAT” in the “Plot 2 - All Data” panel. Namethe file “aoa 05 tape”.

6. Turn off the wind tunnel.

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13

4 Data Analysis and Discussion

The data processing and plotting instructions are given below.

1. Pressure Coefficient Plots:

(a) Calculate the pressure coefficient CP at each pressure tap location. The position datafor each pressure tap is given in Table 1.

(b) Plot the negative of the pressure coefficients −Cp of both the suction and pressure sidesversus the chord-wise location in mm: −Cp versus x. Make a separate plot for eachangle of attack. (7 plots)

2. Lift, Drag, and Quarter-Chord Moment Plots:

(a) For each angle of attack, calculate the axial force coefficients Ca, normal force coefficientsCn, and leading edge moment coefficients Cm,LE using the split discretized equations.

(b) Calculate the lift coefficients Cl, drag coefficients Cd, and the quarter chord momentcoefficients Cm,c/4.

(c) Plot the lift coefficient versus angle of attack in degrees: Cl versus α. In the same graph,plot the 2π/rad theoretical slope on your plot (be careful with your units!). Providebrief comments on the data and how it compares to theory. (1 plot)

(d) Plot the drag coefficient versus angle of attack in degrees: Cd versus α. Provide briefcomments on the data. (1 plot)

(e) Plot quarter-chord moment versus angle of attack in degrees: Cm,c/4 vs. α. Providebrief comments on the results. (1 plot)

3. Pressure Port Biasing Plot:

(a) For the data set with tape on the airfoil (“aoa 05 tape.mat”) calculate the pressurecoefficient CP at each pressure tap location.

(b) Using your Cp data from step 1(a) and 3(a), plot the negative pressure coefficients veruschord-wise position in mm for α = 5 degrees with and without tape: −Cp versus x. Howdoes the tape affect the pressure upstream and downstream from it? How do you expectthe tape to affect the pressure? If you do not see any effect, try to explain why thismight be. (1 plot)

4. Include your processing code, eg. MATLAB .m file(s).

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REFERENCES 14

Summary of Report Requirements

1. −Cp versus chordwise location showing both the upper and lower surface. There should beone for each angle of attack (8 total).

2. Cl versus α in degrees. Overlay the 2π slope line that is predicted by thin airfoil theory.Comment on what your data shows and how well it compares with thin airfoil theory.

3. Cd versus α in degrees. Comment on what your data shows.

4. Cm,c/4 versus α in degrees. Comment on what your data shows.

5. −Cp versus chordwise location showing just the upper surface for α =5o with and withouttape on the same plot. Comment on what your data shows including how the tape affectsthe pressure at the nearby ports. If you do not see any effect due to the tape, try to explainwhy this may be? How do you exect the tape to affect the pressure.

6. Include your MATLAB code.

IMPORTANT:

• Make sure plots are printed large enough to see everything clearly.

• Make sure all plots have appropriate titles and axes labels.

• If there are multiple lines on a single plot, make sure they are labeled and dis-tinguishable by line style, markers, and/or colors.

References

[1] Munk M.M., ”General Theory of Wing Sections,” Tech. Rep. 142, NACA, 1923.

[2] Anderson, J. D., Fundamentals of Aerodynamics, McGraw-Hill, 4th ed., 2007.

[3] McKeon, B. and Engler, R., ”Pressure Measurement Systems,” Springer Handbook of Experi-mental Fluid Mechanics, 2007, pp. 179-214.

Updated: 2016-02-08 11:50