Lab3

Low Speed Lift and Drag Calculation for a NACA 0012 Wing Section Using the Coefficient of Pressure on the Wing Surface J. Murray Aerospace Engineering Student, Lab Section 1007, Tempe, AZ, 85287 This experiment was performed to determine the lift and drag coefficients of a NACA 0012 wing section using the coefficient of pressure that is created by the shape of the wing section. The NACA 0012 wing section has been studied in depth and has a precise data base to compare the results of the experiment with. The results from this experiment showed that the maximum coefficient of lift was close to .3, and occurred at an angle of attack of 9 degrees. The lift-to-drag ratio was a maximum at an angle of attack of 5 degrees. These results do not match the documented properties because of the low Reynolds number. This was a known error before the experiment was performed. Even with these errors the experiment was still able to describe what is happening to the wing section as it changes in angle of attack.

Nomenclature AoA α CD CL CP c D Fx Fy L P Pamb P∞ q V∞

= = = = = = = = = = = = = = =

angle of attack angle in degrees coefficient of drag coefficient of lift coefficient of pressure chord drag CD CL lift total pressure ambient pressure static pressure dynamic pressure wind velocity

I.

Introduction

The laboratory procedure was to determine how the pressure distribution around the NACA 0012 wing can be used to find the Clmax and (L/D)max at varying angles of attack. To determine the pressure distribution in this experiment, the NACA 0012 wing section was exposed to a low speed freestream. Because this was a low speed freestream, the Reynolds number was also low. This low Reynolds number caused significant error when determining the maximum values. This error happens because the low Reynolds number does not allow the air to gain much energy. This lack of energy causes a premature stall angle for the wing. Even with the premature stall

angle it is still possible to determine the performance of the wing section. There are three types of stall; these are the trailing edge stall, leading edge stall, and the thin-airfoil stall. In this lab it is possible to see two of these. These are the trailing edge stall and the thin-airfoil stall. These trailing edge stalls are visually illustrated in Fig 1:

Figure 1: Trailing edge laminar flow separation (stall) Using the freestream, it was possible to find the pressure distribution on the top surface of the wing section. Once the pressure distribution was found, it was used to calculate the CP. This was done using the equation (1).

CP 

P  P q

(1)

These calculated values were then used to solve the CL and CD for the location of each pressure tap on the wing surface. To calculate these, the panel method was used. To use the panel method the wing section was divided into segments where the end point of each panel is located in the center of two port points on the x axis. The end point was then placed onto the profile of the wing section. This placed the pressure tap very close to the center of the panel. The only exception was for panel number 2. This was because of the wing section profile which does not allow for a simple division at this point due to its parabolic shape. The panels can be seen super imposed on the airfoil in Fig. (2).

0.4

port location port location panel end point location wing section surface

0.3 0.2

z

0.1 0 -0.1 -0.2 -0.3 0

0.1

0.2

0.3

0.4

0.5 x

0.6

0.7

0.8

0.9

1

Figure 2: Wing panels superimposed on the wing surface Once the panels were determined it was then possible to calculate the lift and drag for each of these panels. For this calculation the equations were (2) and (3):

Fx   Pi ( z (i 1)  zi )

(2)

Fy   Pi ( x(i 1)  xi )

(3)

This calculation provided the lift and drag data for each panel. These values when plotted show how the pressure coefficient changed as the angle of attack changed. Once these values were found it was then possible to calculate the force per unit span of the wing section. This was done by summing the values for each port. When these values were calculated it was then possible to determine the total lift and drag for the wing section at each angle by using the equations (4) and (5):

C L  C y cos   C x sin 

(4)

C D  C x cos   C y sin 

(5)

When doing these calculations it was necessary to leave out any wing surface beyond the wing panels. Because the calculations from these sections did not follow the wing profile, or have any pressure data it created very large errors. The coefficients of lift and drag showed that as the angle of attack increased, so did the lift and drag. The drag increased continuously while the lift dropped once the stall condition appeared. This stall happens at a lower

than expected angle of attack because the low Reynolds number does not allow the freestream to attain the amount of energy it would at normal operating speeds. The polar plot that this data provides is what was expected from the wing section. The results of these calculations make it possible to evaluate the lift and drag on this wing section.

II.

Procedure for Experiment

The Equipment used to perform the lab: NACA 0012 wing section with pressure taps Multiple manometer Pitot-static tube Pressure Transducer Low speed wind tunnel Computer with LABView program The experiment was conducted by first measuring the chord length of the wing section. At this time the location of the pressure taps was also measured. The location measurement was from the leading edge of the wing section to the center point of the pressure tap. The next step was to setup and calibrate the Pitot-static tube, pressure transducer, and the multiple manometer in the test section of the wind tunnel. After this was done the wing section was then placed into the test section of the wind tunnel on a gimble that allowed it to have its angle of attack adjusted. The pressure taps were then connected to the multiple manometer with rubber hoses. One of the multiple manometer ports was then connected to a pressure tap located on the floor of the wind tunnel test section. This pressure tap provided the data for the static gauge pressure in the test section. All of the gauges were then connected to the computer for LABView to record the readings that were acquired. Next the ambient room temperature and pressure were recorded using a barometer and thermometer. The values from these measurements were then entered into the LABView computer program. This completed the setup of the experiment. Once the setup was complete the wind tunnel was started and set to a speed of 40 Hz. The wing section was then set to an angle of attack at 0 degrees. Once the wind speed had settled, static pressure and velocity of the wind tunnel test section were recorded. Data from the pressure taps on the wing section surface were also recorded at this

time. This provided the data for the pressure distribution on the wing sections surface. It was assumed that because the wing section is symmetrical that the pressure on the lower surface was identical to the upper surface. The wing section was then rotated through the angles of attack of 5, 9, and 13 degrees. At each of these angles the wind velocity was allowed to stabilize. After it stabilized the data for the gauges was measured and the wing section was rotated to the next angle of attack. The data collected from these measurements was then used to calculate the lift and drag the wing section created.

III.

Results

The results of this experiment very closely matched what was expected of this wing section at a low Reynolds number. The only place where error was substantial is in the first and second panels. This is possibly because of the panel shape. The first panel was very small in size and the pressure coefficient was very large at the leading edge. This would make the pressure appear higher than it should be. The second panel was very large and the port was not centered on this panel. The large size probably made the calculation appear abnormally low. The values for the upper surface were calculated using equation (1). These values are shown in Table (1): 0 degree angle of attack

5 degree angle of attack

9 degree angle of attack

13 degree angle of attack

-0.2468 -1.2361 -1.8866 -0.3497 -0.2094 -0.6829 -0.8294 -0.2681 -0.3398 -0.5663 -0.7119 -0.3168 -0.1867 -0.321 -0.4816 -0.3597 -0.1505 -0.2587 -0.2688 -0.362 -0.1191 -0.2012 -0.146 -0.361 -0.1031 -0.1291 -0.0793 -0.3656 -0.0556 0.0282 0.1271 -0.3506 0.0392 0.0919 0.1616 -0.332 Table 1: CP at the pressure tap for each angle evaluated on the upper surface

The values for the lower surface were calculated with equation (1) and are shown in table (2): 0 degree angle of attack

5 degree angle of attack

9 degree angle of attack

13 degree angle of attack

0.6789 0.5502 0.8566 -0.247 0.4658 -0.2094 0.2507 0.5002 0.137 0.0631 0.1595 -0.3398 0.0459 -0.1867 0.023 0.0902 0.0223 -0.0124 0.0301 -0.1505 -0.0716 -0.0128 0.1046 -0.1191 -0.1437 -0.1031 -0.0116 -0.024 -0.1338 -0.0193 -0.0304 -0.0556 -0.1392 0.0392 0.0724 0.0488 Table 2: CP at the pressure tap point along the x axis for each angle evaluated on the lower surface The results from these calculations are what were to be expected with the exception of the first 2 panels. These show a very high coefficient of pressure at the leading edge and almost zero press on the trailing edge. This data is displayed in Fig. (3): -0.35 -0.3 -0.25

Cp

-0.2 -0.15 -0.1 -0.05 0 0.05

0

0.1

0.2

0.3

0.4 x/c

0.5

0.6

0.7

0.8

Figure 3: Coefficient of Pressure vs. x for 0 degrees angle of attack The next results for the angle of attack of 5 degrees were also very close to expected values. The coefficient of pressure went down for all locations on the upper surface. At the same time the coefficient of pressure on the lower surface increased. Because the lower surface did not increase as much as the upper surface decreased, the net result was the wing section had more lift than it had at zero degrees angle of attack. With this data it is possible to see the trailing edge stall start to form on the last two panels. On the data plot this is the area where the pressure lines cross. The pressure coefficients for this angle of attack can be seen in Fig 4:

-1.4 Bottom Surface Top Surface

-1.2 -1 -0.8

Cp

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

0

0.1

0.2

0.3

0.4 x/c

0.5

0.6

0.7

0.8

Figure 4: Coefficient of Pressure vs. x for 5 degrees angle of attack As the angle of attack increased to 9 degrees the coefficient of pressure reached its lowest value on the leading edge of the upper surface. The lower surface pressure also increases but not as much as the upper surface. At this angle of attack the wing section provided its highest lift. Also at this angle of attack the trailing edge stall has increased from in size with the change in angle of attack. The pressure coefficients for this angle of attack can be seen in fig (5): -2 Lower Surface Upper Surface -1.5

Cp

-1

-0.5

0

0.5

1

0

0.1

0.2

0.3

0.4 x/c

0.5

0.6

0.7

0.8

Figure 5: Coefficient of Pressure vs. x for 9 degrees angle of attack The final angle of attack was 13 degrees. At this angle the wing section was in stall and was not producing as much lift. This is the stage when the freestream has separated from the top surface of the wing section. The upper surface was producing far less negative pressure than the lower surface was producing positive pressure. The pressure coefficients can be seen in Fig. 6:

-0.4

-0.2

0

Cp

0.2

0.4

0.6

0.8

1

Lower surface Upper Surface 0

0.1

0.2

0.3

0.4 x/c

0.5

0.6

0.7

0.8

Figure 6: Coefficient of Pressure vs. x for 13 degrees angle of attack When the total CD and CL were calculated for each angle of attack, the numbers were within the expected range. The calculated lift and drag coefficients when plotted made a polar plot in the general form that it should have been. The values for the integrations are provided in table (2). angle of attack

Lift

Drag

0

0

-0.0106

5

0.2751

0.0099

9

0.3639

0.0396

13 0.2971 0.0921 Table 2: CD and CL as functions of angle of attack This plot has shows the premature stall due to the low Reynolds number. The maximum coefficient of lift is at a 9 degree angle of attack instead of the expected 12 degrees. The drag acted as expected and increased as more surface area was exposed to the freestream. The data can be seen in Fig. (7). The data for the CL/ CD vs. angle of attack shows that the highest lift was at 9 degrees angle of attack. This data is plotted in Fig (8).

0.4 Coefficient of Lift Coefficient of Drag

0.35 0.3 0.25

Cp

0.2 0.15 0.1 0.05 0 -0.05

0

2

4

6 8 Angle of attack

10

12

14

Figure 7: Polar plot of CD and CL vs. α 30

25

Cp

20

15

10

5

0

0

2

4

6 8 Angle of attack

10

12

14

Figure 8: CL/ CD vs. angle of attack.

IV.

Conclusion

The lab showed that the maximum Cp was at a 9 degree angle of attack. This is not the same as the expected value. The reason for this is that the Reynolds number is too low to simulate actual conditions and model scaling also affects the performance of the wing section. This lab could be made to be more accurate by increase the wind velocity. With an increased Reynolds number the freestream would have more energy. This higher energy would keep the boundary layer on the wing section at higher angles of attack. This would have the effect of increasing the Reynolds number. This increase would delay the stall until the expected angle of attack. Although the results are not

quite the same as the expected, they are close enough to describe what is happening to the wing section as it changes angles of attack. When this data is compared to lab 2 at a similar Reynolds number, the maximum lift is at the same angle of attack, but the Cp for the maximum lift is about half or slightly less. The slope of the maximum lift is what would be expected from the data. The data only differs beyond the maximum lift region. This is probably due to the low Reynolds number not accurately modeling the wing section in a stall condition. The lift-to-drag ratio reaches a maximum ratio at a 5 degree angle of attack. This is not the same as the angle of attack for maximum lift. This is because beyond 5 degrees the drag increases faster than the lift decreases. It would be possible to change this to a higher expected angle of attack again by increasing the Reynolds number. A problem with the lab was accurately modeling the Cp over the surface of the airfoil. This could be solved with more time invested in tinkering with the panels and extrapolating the data over the ends of the profile.

Appendix A Table A.1: Port locations on the wing section Distance of each port from leading edge of airfoil x

xnondim

Port0

0.4

0.04

Port1

1

0.1

Port2

2

0.2

Port3

3

0.3

Port4

4

0.4

Port5

5

0.5

Port6

6

0.6

Port7

7

0.7

Port8

8

0.8

Chord 10.0 cm

10

Table A.2: Calculated lift and drag coefficients angle of attack Lift Drag 0

0

-0.0106

0

5

0.0392

0.0099

3.959596

9

0.013

0.0396

0.328283

13

P_trans Pa

α °

P_scan inH2O

697.94276

0

-3.494

700.07592

0

-3.399

704.07559

0

-3.787

705.67546

0

-3.362

706.87536

0

-3.265

703.5423

0

-3.161

704.07559

0

-3.118

708.60856

0

-3.003

706.74204

0

-2.726

757.13791

5

-6.797

763.13742

5

-5.156

759.67104

5

-4.777

756.87126

5

-4.014

759.93768

5

-3.84

762.07084

5

-3.675

760.60429

5

-3.448

762.33748

5

-2.974

760.60429

5

-2.773

805.2673

9

-9.332

805.93391

9

-5.919

808.73368

9

-5.558

807.80042

9

-4.805

803.5341

9

-4.093

801.80091

9

-3.689

0.0046

0.0921

0.049946

Table A.3: Calculated Cp for the pressure taps Scan P_scan port V P_atm P_atm P Pa # m/s kPa Pa Pa 870.31662 0 35.377 96.2 96200 95329.683 846.65317 1 35.547 96.2 96200 95353.347 943.29966 2 35.489 96.2 96200 95256.7 837.43688 3 35.693 96.2 96200 95362.563 813.27526 4 35.666 96.2 96200 95386.725 787.37001 5 35.576 96.2 96200 95412.63 776.65919 6 35.533 96.2 96200 95423.341 748.01397 7 35.725 96.2 96200 95451.986 679.01634 8 35.667 96.2 96200 95520.984 1693.0573 0 37.011 96.2 96200 94506.943 1284.3024 1 37.053 96.2 96200 94915.698 1189.8977 2 36.91 96.2 96200 95010.102 999.84284 3 37.011 96.2 96200 95200.157 956.50138 4 37.055 96.2 96200 95243.499 915.40171 5 37.009 96.2 96200 95284.598 858.85853 6 36.919 96.2 96200 95341.141 740.79039 7 36.993 96.2 96200 95459.21 690.72352 8 36.985 96.2 96200 95509.276 2324.4976 0 38.04 96.2 96200 93875.502 1474.3572 1 38.099 96.2 96200 94725.643 1384.4361 2 38.153 96.2 96200 94815.564 1196.8722 3 38.136 96.2 96200 95003.128 1019.5209 4 38.067 96.2 96200 95180.479 918.88895 5 38.025 96.2 96200 95281.111

P_inf Pa

CP []

95491.391

0.2469742 0.2093734 0.3397704 0.1867167 0.1505214 0.1191509 0.1030906 0.0556096

95493.258

0.0392303

95442.862

-1.236128 0.6829241 0.5663328

95502.057 95499.924 95495.924 95494.325 95493.125 95496.458 95495.924

95436.863 95440.329 95443.129

95439.396

-0.321021 0.2586577 0.2012029 0.1291792

95437.663

0.0282645

95439.396

0.0918753 1.8866162 0.8293773 0.7118566 0.4816434

95440.062 95437.929

95394.733 95394.066 95391.266 95392.2 95396.466 95398.199

-0.268796 0.1460313

798.7345

9

-3.461

803.26746

9

-2.815

801.00098

9

-2.696

773.66989

13

-4.192

774.3365

13

-3.942

775.00311

13

-4.097

862.09668 701.18525 671.54367 1044.1807 981.90844 1020.5172

6

38.076

96.2

96200

95337.903

95401.266

0.0793282

7

37.948

96.2

96200

95498.815

95396.733

0.1270837

8

37.968

96.2

96200

95528.456

95398.999

0

37.166

96.2

96200

95155.819

95426.33

1

37.26

96.2

96200

95218.092

95425.664

2

37.282

96.2

96200

95179.483

95424.997

-1053.397

3

37.389

96.2

96200

95146.603

95425.264

-1047.917 1057.3824 1057.6315 1048.1661 1041.1916 292.43037 484.47791 606.28238 630.94218 654.10745 652.36383 652.11474 656.10016 595.82065 87.679293 306.13026 -516.1122 557.95914 598.06245 552.23009 629.19856 -

4

37.317

96.2

96200

95152.083

95430.596

5

37.437

96.2

96200

95142.618

95423.13

0.1616194 0.3496463 0.2680643 0.3167911 0.3596842 0.3619861 0.3610809

774.73647

13

-4.229

769.40357

13

-4.207

776.86962

13

-4.245

774.46982

13

-4.246

6

37.288

96.2

96200

95142.369

95425.53

7

37.304

96.2

96200

95151.834

95423.93

8

37.47

96.2

96200

95158.808

95418.331

-0.36562 0.3506082 0.3320105

776.06969

13

-4.208

781.66923

13

-4.18

650.21334

-5

-1.174

0

34.123

96.2

96200

95907.57

95549.787

0.5502547

646.61364

-5

-1.945

1

34.065

96.2

96200

95715.522

95553.386

0.2507459

647.14693

-5

-2.434

2

34.075

96.2

96200

95593.718

95552.853

0.0631457

645.81371

-5

-2.533

3

33.991

96.2

96200

95569.058

95554.186

4

34.041

96.2

96200

95545.893

95553.92

5

34.015

96.2

96200

95547.636

95555.919

6

33.985

96.2

96200

95547.885

95555.386

7

34.001

96.2

96200

95543.9

95556.319

0.0230276 0.0124243 0.0128607 0.0116363 0.0192947

646.08035

-5

-2.626

644.08051

-5

-2.619

644.6138

-5

-2.618

643.68055

-5

-2.634

642.34732

-5

-2.392

8

34.02

96.2

96200

95604.179

95557.653

0.0724323

611.54985

-9

-0.352

0

33.208

96.2

96200

96112.321

95588.45

0.8566277

612.48311 614.08297

-9 -9

-1.229 -2.072

1 2

33.136 33.222

96.2 96.2

96200 96200

95893.87 95683.888

95587.517 95585.917

0.5001817 0.15954

613.28304

-9

-2.24

3

33.265

96.2

96200

95642.041

95586.717

0.0902094

616.6161

-9

-2.401

4

33.273

96.2

96200

95601.938

95583.384

0.0300895

616.74942

-9

-2.217

5

33.238

96.2

96200

95647.77

95583.251

0.1046119

614.48294 617.14939

-9 -9

-2.526 -2.553

6 7

33.243 33.317

96.2 96.2

96200 96200

95570.801 95564.076

95585.517 95582.851

-0.023948 -

615.14955 638.48097 637.41439 639.28091 638.34765 638.48097 638.08101 637.68104 639.94752 637.81436

-9 13 13 13 13 13 13 13 13 13

-2.349 -0.823 -1.367 -2.215 -2.445 -2.506 -2.745 -2.928 -2.913 -2.917

635.92396 585.10983 205.00016 340.50453 551.73191 609.02236 624.21678 683.74903 -729.3323 725.59597 726.59232

0.0304214 8

33.259

96.2

96200

95614.89

95584.85

0.0488332

0

33.936

96.2

96200

95995

95561.519

0.6789252

1

33.865

96.2

96200

95859.495

95562.586

0.4658035

2

33.864

96.2

96200

95648.268

95560.719

0.1369492

3

33.875

96.2

96200

95590.978

95561.652

0.0459394

4

33.861

96.2

96200

95575.783

95561.519

5

33.858

96.2

96200

95516.251

95561.919

6

33.925

96.2

96200

95470.668

95562.319

7

33.865

96.2

96200

95474.404

95560.052

8

33.867

96.2

96200

95473.408

95562.186

0.0223408 0.0715709 0.1437259 0.1338367 0.1391909

Appendix B

CP 

P  P q

Equations used in the lab (1)

Fx   Pi ( z (i 1)  zi )

(2)

Fy   Pi ( x(i 1)  xi )

(3)

C L  C y cos   C x sin 

(4)

C D  C x cos   C y sin 

(5)

P  Pamb  Manometer

(6)

P  Pamb  q

(7)

Pi  C p

(8)

Appendix C

Code used to find CL and CD: function [L D] = lab3_cp(x,z_u, z_l, Pc, alpha) Pc_u = zeros(1,length(Pc)/2); Pc_l = Pc_u; for k = 1:length(Pc_u) Pc_u(k) = Pc(k); Pc_l(k) = Pc(length(Pc_u)+k); end % Calculate constant term in pressure coefficient equation % (Pc * const + P_atm = P_i, where const = .5 * rho * v_inf^2) rho = 1.008; v_inf = 35.6; const = (1/2) * rho * (v_inf^2); const = 1; fx = zeros(length(Pc),1); fz = zeros(length(Pc),1); % Calculate force per unit length for k=1:length(Pc_u) Pi_u(k) = Pc_u(k) * const; Pi_l(k) = Pc_l(k) * const; fx_u(k) = Pi_u(k) * (z_u(k+1) fx_l(k) = Pi_l(k) * (z_l(k+1) fz_u(k) = - Pi_u(k) * (x(k+1) fz_l(k) = - Pi_l(k) * (x(k+1) end

for each segment

-

z_u(k)); z_l(k)); x(k)); x(k));

%Calculate force per unit span (multiply by length) Fx = sum(fx_u)-sum(fx_l); Fz = sum(fz_u)-sum(fz_l); % Change coordinate axis from wing to wind (angle of attack = 3°) alpha = alpha * pi / 180; L = Fz * cos(alpha) - Fx * sin(alpha) D = Fx * cos(alpha) + Fz * sin(alpha) % Debugging for k=1:length(x)-1 x_av(k) = (x(k) + x(k+1))/2.0; end x_av = [0.04 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8]; x_av = zeros(length(x)-1,1);

for k=1:length(x)-1 x_av(k) = (x(k) + x(k+1))/2.0; end plot(x_av,Pc_l,'r-', x_av,Pc_u, 'b-'); set(gca,'YDir','reverse');

The code for the airfoil profile and the panels: function [x z] = lab3_airfoil() % Defines the airfoil, as defined by the NACA equations, and puts it into % machine-readable form. % Static port data on airfoil, as given by PortLocations.txt x_p = [0.4; 1; 2; 3; 4; 5; 6; 7; 8]; c = 10; % We know that the ports are in the middle of the panels; as a result, we % need to find where the panel endpoints are. First, add x = 0 and x = 1 to % the airfoil endpoints: x_i(1)=0; %(x_p(1)-x_p(2)/2) for i = 1:length(x_p) x_i(i+1) = x_p(i); end x_i(i+2) = 0.9*c; % Now, non-dimensionalize by dividing by the chord length c x_i = x_i / c; x_p = x_p / c; % Calculate the midpoints between the existing port points and edges; these % will be the endpoints of the panels. for i = 1:length(x_i)-1 x(i) = 0.5*(x_i(i) + x_i(i+1)); end x(2) = x_p(2) - (x(3)- x_p(2)); x(1) = x_p(1) - (x(2) - x_p(1)); % Solving for Z's, using the NACA 0012 equation on the HW 2 handout z_u=naca0012(x_i); z_p=naca0012(x_p); z = naca0012(x); % Plot it, for debugging purposes plot(x_i, z_u, 'ko', x_p, z_p, 'kx', x ,z, 'bo',0:.01:1,naca0012(0:.01:1),'k-');axis equal % % Exit the function, to return x and z end function z = naca0012(x) % The function uses the NACA 0012 equation, as shown in HW #2's handout, % and uses it to calculate the z-coordinate of a given x-coordinate. Works

% with array arithmetic too. z = 0.6.*(0.2969.*sqrt(x)-0.126.*x-0.3516.*x.^2+0.2843.*x.^3-0.1015.*x.^4); end Code used for panles and port locations: function [x z] = lab3_airfoil() % Defines the airfoil, as defined by the NACA equations, and puts it into % machine-readable form. % Static port data on airfoil, as given by PortLocations.txt x_p = [0.4; 1; 2; 3; 4; 5; 6; 7; 8]; c = 10; % We know that the ports are in the middle of the panels; as a result, we % need to find where the panel endpoints are. First, add x = 0 and x = 1 to % the airfoil endpoints: x_i(1)=0; %(x_p(1)-x_p(2)/2) for i = 1:length(x_p) x_i(i+1) = x_p(i); end x_i(i+2) = 0.9*c; % Now, non-dimensionalize by dividing by the chord length c x_i = x_i / c; x_p = x_p / c; % Calculate the midpoints between the existing port points and edges; these % will be the endpoints of the panels. for i = 1:length(x_i)-1 x(i) = 0.5*(x_i(i) + x_i(i+1)); end x(2) = x_p(2) - (x(3)- x_p(2)); x(1) = x_p(1) - (x(2) - x_p(1)); % Solving for Z's, using the NACA 0012 equation on the HW 2 handout z_u=naca0012(x_i); z_p=naca0012(x_p); z = naca0012(x); % Plot it, for debugging purposes plot(x_i, z_u, 'ko', x_p, z_p, 'kx', x ,z, 'bo',0:.01:1,naca0012(0:.01:1),'k-');axis equal % % Exit the function, to return x and z end function z = naca0012(x) % The function uses the NACA 0012 equation, as shown in HW #2's handout, % and uses it to calculate the z-coordinate of a given x-coordinate. Works % with array arithmetic too. z = 0.6.*(0.2969.*sqrt(x)-0.126.*x-0.3516.*x.^2+0.2843.*x.^3-0.1015.*x.^4); end

Acknowledgments Carlos Ballesteros for working in the coding of MATLab and the finding the final values. The James for providing extra information to complete the lab write up

References http://images.google.com/imgres?imgurl=http://people.eku.edu/ritchisong/554images/Eddy_flap_1.gif&imgrefur l=http://people.eku.edu/ritchisong/554notes2.html&h=407&w=368&sz=53&hl=en&start=13&um=1&tbnid=SQUa mVVB0EXVmM:&tbnh=125&tbnw=113&prev=/images%3Fq%3Dtrailing%2Bedge%2Bdrag%2Bon%2Ba%2Bwi ng%26svnum%3D10%26um%3D1%26hl%3Den Your manuscript cannot be published by AIAA if Extra information provided by J. Villarreal AKA The James