Mae 360 Lab 2

Low Speed Lift and Drag for a NACA 0012 Wing Section J. Murray Aerospace Engineering Student, Lab Section 1007, Tempe, AZ, 85287

The experiment was preformed to examine what effect a change in the Reynolds number would have on the lift and drag characteristics of the NACA 0012 wing section. The wing profile is common and well studied. It has well documented properties that provide precise data to compare with the experimental data. The results of this lab showed the lift and drag characteristics of the wing section. These results were a maximum coefficient of lift was about .8, and happened at 9 to 10 degrees. The drag and lift increase together until around 10 degrees where lift no longer increases but drag does. The lift and angle of attack are linearly related up to point of stall. The results also show that the drag on the wing section increases as the angle of attack increased. The wing section was tested at three data points. These were Reynolds numbers of 50000, 100000, and 150000. The Reynolds numbers were set by the velocity and air density inside the wind tunnel. The data produced a very small error for the results. Because of historical data it was known before the experiment that low Reynolds numbers produce erroneous data. This is caused by premature stall of the wing section. This result was present in our data, even with these results the experiment provided data that is accurate and expected.

Nomenclature A AOA Cd Cl Cy c D L l q Re UCd Ud Uq URe v  

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

area of the wing section angle of attack coefficient of drag coefficient of life force coefficient in the y direction chord Drag Lift chord length dynamic pressure Reynolds number uncertainty of the coefficient of drag uncertainty of drag uncertainty of total pressure uncertainty of Reynolds number velocity atmospheric density viscous forces

I.

Introduction

The laboratory procedure was an exploration of the aerodynamic lift and drag forces experienced by a NACA 0012 wing section at low Reynolds numbers. The model experienced a low speed uniform free stream velocity. This velocity was set by using specific Reynolds numbers. When low Reynolds numbers are used, the data for the velocities will show the characteristics of the laminar-separation bubble on the top surface of the wing section. This

laminar-separation is caused by low velocities and is not representative of actual results found at Reynolds numbers that the wing section would experience in normal operational conditions. The coefficients of lift and drag that are calculated show the increase in these values in response to the increase in AOA, and also the point at which the laminar-separation occurs. Laminar separation can also be referred to as wing stall. The equations to calculate these forces are the Coefficient of Lift (1), and the Coefficient of Drag (2).

Cl 

L qA

(1)

Cd 

D qA

(2)

The coefficient for lift shows the point of separation where the lift curve levels off or falls when it is plotted against the AOA. The separation point is the maximum value of the plot line. The separation occurred when the lift falls suddenly after this point. The polar plot is a plot of Cd versus Cl that shows the best climb or sink rate that the wing section has at the specific Reynolds number. To find the best climb rate of the wing section in this experiment a tangential line is drawn from the origin of the plot to the plot line. This point is the best climb rate the wing section has for the Reynolds number it is being evaluated at. Although we know the Reynolds number that we are testing at, the true number needs to be found for the sake of accuracy. To do this equation (3) is used.

Re 

vl 

(3)

To find  in this equation it is necessary to use Sutherland’s equation (4).

 air  1.458  10 6 

3 2

T T  110.4

(4)

This allowed the calculation of the Reynolds number for each data set. Once all of these values have been found it is important to know what the uncertainty of each one is. To do this, the equation for the uncertainty of drag is (5)

 U    DU a    DU q    2   d    2  qA   qA   q A 2

U Cd

2

  

2

(5)

To find the uncertainty of the coefficient of lift the equation used is (6)

 U    LU a    LU q    2   l    2   qA   qA   q A 2

U Cl

2

  

2

(6)

Then the uncertainty of the Reynolds number was found using the equation (7)

 Re  vl  2 

(7)

These numbers make it possible to evaluate the lift and drag at the different Reynolds numbers and to see how they change with the possible error as the number increase.

II.

Procedure

The Equipment used to perform the lab: NACA 0012 wing section NACA 0012 wing pylon Low speed wind tunnel Computer with LABView program installed Barometer Thermometer Pressure transducer Load Cell The experiment was conducted by first recording the ambient temperature and pressure. This was done using a barometer and thermometer. These values were then entered into the computer which was running the program LABView, with the experiment already programmed into it. The chord length and the span width of the wing section were also recorded and entered into the program. The type of wing section at this time was also noted as being a NACA 0012. The wing section was then placed into the wind tunnel test section. The wing section was fastened onto a pylon that had a cross section that was a NACA 0012. This pylon was attached to a pressure gauge that would measure the force of lift created by the wing section. The pitch of the wing section was then adjusted so that the AOA was zero

degrees. The pitch angle was adjusted by a knob that was located underneath the pylon and outside of the wind tunnel. The next step was to calculate the ambient temperature and viscosity. To calculate this Bernoulli’s equation was used for the pressures, and Sutherland’s equation was used to find the viscousity. The data was acquired with a pitot tube in the test section and a thermometer. This data was used to determine the dynamic pressure in the wind tunnel at each Reynolds number. The data was entered into LABView for automatic calculation of the required values. These calculations used the equations (1), (2), (3), and (4). The wind tunnel was then turned on and adjusted until LABView had a Reynolds number reading of 50000. Once this value was reached and the value that the pressure gauge was recording settled the measurement was written into LABView. This procedure was done for all angles from negative 10 to positive 15 with each step being one degree. The data acquisition was carried out the same way for each Reynolds number that was evaluated. These Reynolds numbers were 50000, 100000, and 150000. All the steps were repeated until the complete data sets were acquired.

III.

Results

The results of this experiment were very precise and matched what was expected very closely. The Reynolds numbers were found using equation (3). The true value for each measurement was used to create an average value for the setting. The calculated average value for 50000 was 50356 +/-690. For the setting of 100000 the calculated average value was 100300 +/- 1374. And for the setting of 150000 the calculated average value was 149710 +/2050.5. These values are shown in table (10) in the appendix. The error for these values is close to one percent. This is not a very significant error at the low Reynolds numbers but can become significant if the numbers get close to the realistic values that the wing section would encounter. The lift was plotted against the AOA for each Reynolds number. This provided a plot that shows what the lift on the wing is at each AOA for the specific Reynolds number. This is shown in figure (1).

Coefficient of Lift versus Angle of Attack 1 0.8 0.6 0.4

Cl

0.2 0 -0.2

50000 50000 100000 100000 150000 150000

-0.4 -0.6 -0.8 -15

-10

-5

0 5 Angle of Attack

Reynolds number Renolds number error bar Reynolds number Reynolds number error bar Reynolds number Reynolds number error bar 10

15

20

Figure 1: Coefficient of lift versus Angle of Attack The plot clearly shows that as the angle of attack increases so does the lift of the wing section. The plot also shows the laminar-separation at an AOA of 9 and 10 degrees. The plot also shows that laminar-separation is smaller as the Reynolds number increases. This is because as the Reynolds number increases, the air becomes more viscous. The viscosity in turn keeps the boundary layer attached to the wing surface. With the laminar flow being smoother over the wing surface the lift is greater as the Reynolds number increases. This separation can be seen in figure (3).

Figure 3: Laminar-separation

The coefficient of drag was plotted against the coefficient of lift for figure (2). This plot shows that as the lift increases so does drag.

Cd versus Cl 0.35 50000 Reynolds number 50000 Reynolds number error bar 100000 Reynolds number 100000 Reynolds number error bar 150000 Reynolds number 150000 Reynolds number error bar

0.3

0.25

Cd

0.2

0.15

0.1

0.05

0 -0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Cl

Figure 2: Coefficient of Drag versus Coefficient of Lift Also in figure (2) the plot shows when laminar-separation occurs. This is between the coefficients of lift of .7 to .9. At this value the NACA 0012 wing section no longer provides anymore lift but is creating a large amount of drag. The plot shows that both drag and lift are related. The two coefficients are related by the equation (8).

C d  C do  KCl

(8)

This equation describes that the drag coefficient increases as the parasite drag increases and the coefficient of lift is added to this value.

IV.

Conclusion

The experiment has shown that as the AOA increases so does lift until laminar-separation occurs. The relation between lift and AOA is linear until laminar-separation starts to occur. The experiment data does not match the data from the NACA 0012 data. This is because of the low Reynolds number. If the number was closer to normal conditions the results would match very closely. The lift and drag also increase together as the AOA of the wing section changes. This suddenly changes with laminar-separation. At this point the wing section no longer produces more lift but it does increase the drag as more frontal area of the wing section is exposed to the constant velocity. The errors that happened in the experiment were very obvious in figure (2). The data point that causes a spike does not belong in the set. It could possibly be from recording the data before the velocity stream settled or the vibrations from the wing section settled. It is too far out to have been from a degree adjustment. Where the lift vs.

AOA plot has an increase in lift at -1 AOA could possibly be from a mistaken angle adjustment that was not corrected, or it could have been from the adjustment knob not being used accurately. These errors did not prove to be large enough to affect the required outcome of the experiment.

References Anderson, J.D. A History of Aerodynamics and Its Impact on Flying Machines,Chapter 7 pp. 308.

http://images.google.com/imgres?imgurl=http://content.answers.com/main/content/wp/en/6/67/Flow_separation.jpg &imgrefurl=http://www.answers.com/topic/flowseparation&h=189&w=320&sz=12&hl=en&start=7&um=1&tbnid=doUK_kpUoFyVcM:&tbnh=70&tbnw=118&pr ev=/images%3Fq%3Dlaminar%2Bflow%2Bseparation%2Bon%2Ba%2Bwing%2Bsurface%26svnum%3D10%26u m%3D1%26hl%3Den

Appendix A Table A.1: Data set from 50000 Reynolds number

Lift (N)

Drag (N)

Temp (K)

DyPres (Pa)

0.11 0.18 0.42 0.74 0.89 1 1.14 1.27 1.38 1.46 1.74 1.5 1.42 1.38 1.38 1.38 0.14 0.04 -0.22 -0.47 -0.68 -0.78

0.12 0.14 0.17 0.18 0.19 0.21 0.23 0.27 0.28 0.31 0.13 0.43 0.48 0.53 0.59 0.65 0.11 0.1 0.09 0.08 0.08 0.07

310.18 309.92 310.13 310.95 311.07 310.99 311.09 311.39 311.29 311.13 310.81 310.2 310.48 310.26 310.32 310.47 310.3 310.39 310.92 310.55 311.03 310.93

67.71 67.52 66.98 66.29 66.38 67.28 67.24 67.1 67.07 66.51 65.78 65.07 64.99 65.63 65.71 66.79 67.34 67.61 67.58 67.49 67.06 66.4

Ambient Pres (Pa) 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500

Angle of attack

Renumber

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -1 -2 -3 -4 -5 -6

50720.11 50761.38 50440.12 50197.04 50048.35 50469.94 50574.89 50210.14 50349.39 50184.36 49988.88 49751.92 49709.91 49926.27 50002.92 50341.16 50657.43 50664.95 50699.85 50675.4 50389.84 50033.63

-0.89 -1 -1.13 -1.21

Lift (N) 0.69 1.86 2.35 2.9 3.57 4.81 5.28 5.71 6.19 6.56 6.09 5.88 5.86 5.89 5.79 5.86 0.09 -0.68 -1.42 -2.13 -2.8 -3.35 -3.85 -4.38 -4.72 -5.35

Lift (N) 2.32 7.06 8.14 9.31 10.48 11.55 12.47 13.55 14.49

0.07 0.07 0.07 0.07

311.13 310.8 311.06 311.11

66.95 67.16 67.26 67.63

96500 96500 96500 96500

-7 -8 -9 -10

Table A.2: Data set from 50000 Reynolds number Ambient Angle Drag Temp Dy Pres Pres of (N) (K) (Pa) (Pa) attack 0.48 310.69 268.69 96500 0 0.54 310.86 268.42 96500 1 0.58 310.82 267.8 96500 2 0.59 310.17 265.58 96500 3 0.7 310.47 264.56 96500 4 0.86 310.77 261.79 96500 5 0.94 310.86 264 96500 6 1.02 311.25 263.01 96500 7 1.15 310.8 262.25 96500 8 1.26 311.34 260.97 96500 9 1.79 311.29 260.36 96500 10 2.04 311.48 264.84 96500 11 2.25 311.54 268.37 96500 12 2.47 311.57 268.67 96500 13 2.6 311.95 267.3 96500 14 2.83 312.21 266.97 96500 15 0.42 311.68 268.23 96500 -1 0.39 311.75 268.16 96500 -2 0.35 311.64 267.87 96500 -3 0.31 312.28 268.28 96500 -4 0.31 312 268.18 96500 -5 0.29 312.15 267.19 96500 -6 0.3 312.26 266.36 96500 -7 0.31 311.88 267.34 96500 -8 0.32 312.14 265.63 96500 -9 0.37 312.49 266.56 96500 -10 Table A.3: Data Set for 150000 Reynolds number Ambient Angle Drag Temp Dy Pres Pres of (N) (K) (Pa) (Pa) attack 0.98 311.47 593.67 96500 0 1.34 311.53 588.95 96500 1 1.46 311.14 588.8 96500 2 1.59 311.33 585.16 96500 3 1.74 310.97 585.22 96500 4 1.9 310.78 584.64 96500 5 2.04 310.95 582.95 96500 6 2.23 310.85 582.78 96500 7 2.4 311.17 579.47 96500 8

50264.53 50456.63 50459.38 50528.61

Re number 100954.5 100850.1 100922.6 100856.4 100438.5 99976.99 99871.72 99605.3 99714.39 99457 99110.12 99884.32 100494.5 100831 100122.7 99965.75 100400.3 100388.1 100413 100083.5 100123.8 100205.8 99970.83 100173 100100.1 100010.5

Re number 149290.8 148800 148837.3 148777.1 148862.5 148882.2 148659.8 148565.9 148200.8

15.49 13.73 13.37 13.39 13.24 13.31 13.16 1.08 -2.26 -3.72 -5.71 -6.85 -7.97 -9.22 -10.54 -11.59 -12.81

2.67 4.18 4.71 5.1 5.65 6.04 6.3 0.9 0.78 0.71 0.66 0.65 0.61 0.61 0.62 0.66 0.73

311.43 310.87 311.23 311.48 311.48 311.62 311.14 311.43 311.26 310.61 310.85 311.23 311.2 311.42 312.09 311.78 311.62

581.51 582.51 586.74 591.16 589.57 590.99 589.22 593.2 596.51 594.11 593.07 595.78 594.4 595.07 594.41 596.59 596.65

96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500 96500

9 10 11 12 13 14 15 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

148109.5 148362.6 148676.1 149075.9 149525.9 149317 149111.9 149443.9 150182.1 150060.6 149999.4 149820.5 149894.3 149831.3 149337.7 149384.6 150047.7

Appendix B Equations used

L qA D Cd  qA vl Re   Cl 

 air  1.458  10 6 

3 2

T T  110.4

 U    DU a    DU q    2   d    2  qA   qA   q A  Re  vl   2  Re  vl   2 2

U Cd

2

2

  

        P   err   U P    U T  RT        

2

2

Appendix C Table C.1: Density and Density error Density for Re=50000

Density for Re=100000

Density for Re=150000

Density error for Re=50000

Density error for Re=100000

Density error for Re=150000

1.0848 1.0857 1.0849 1.0821 1.0816 1.0819 1.0816 1.0805 1.0809 1.0814 1.0825 1.0847 1.0837 1.0845 1.0843 1.0837 1.0843 1.084 1.0822 1.0835 1.0818 1.0821 1.0814 1.0826 1.0817

1.0852 1.0846 1.0848 1.087 1.086 1.0849 1.0846 1.0832 1.0848 1.0829 1.0831 1.0824 1.0823 1.0822 1.0808 1.0799 1.0818 1.0815 1.0819 1.0797 1.0807 1.0801 1.0798 1.0811 1.0802

1.0862 1.0859 1.0873 1.0866 1.0878 1.0885 1.0879 1.0882 1.087 1.0862 1.0881 1.0869 1.0861 1.0861 1.0856 1.0873 1.0863 1.0869 1.0892 1.0883 1.087 1.0871 1.0863 1.084 1.0851

0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037

0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037

0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037

Table C.2: Velocity and velocity error Velocity for Re=50000

Velocity for Re=100000

Velocity for Re=150000

Velocity error for Re=50000

Velocity error for Re=100000

Velocity error for Re=150000

11.1731 11.1527 11.1118 11.0691 11.0787 11.1521 11.1506 11.1444 11.1401 11.0907 11.024 10.9536

22.2525 22.2474 22.2203 22.1051 22.0734 21.9685 22.064 22.0365 21.9888 21.9542 21.9269 22.121

33.0629 32.9352 32.9104 32.8191 32.8018 32.7757 32.7375 32.7275 32.6518 32.7225 32.721 32.8579

0.131 0.1312 0.1317 0.1324 0.1324 0.1315 0.1316 0.1318 0.1318 0.1323 0.1329 0.1335

0.0832 0.0831 0.0832 0.0834 0.0834 0.0834 0.0833 0.0832 0.0834 0.0833 0.0833 0.0831

0.0895 0.0892 0.0895 0.0892 0.0894 0.0895 0.0893 0.0894 0.0891 0.089 0.0894 0.0893

10.9518 11.0017 11.0094 11.1022 11.1447 11.1686 11.1757 11.1616 11.1346 11.0779 11.1273 11.1388 11.1517 11.1832

22.2696 22.2831 22.2399 22.2355 22.2688 22.2684 22.2525 22.2923 22.2782 22.2425 22.2119 22.2391 22.1773 22.2284

32.9939 32.9498 32.9966 32.9221 33.0478 33.1302 33.0294 33.0134 33.1085 33.0688 33.099 33.1163 33.1601 33.1533

0.1336 0.1329 0.1329 0.1319 0.1314 0.1312 0.1313 0.1313 0.1318 0.1323 0.1319 0.1316 0.1316 0.1313

0.083 0.083 0.0829 0.0829 0.083 0.0829 0.083 0.0828 0.0829 0.0829 0.0829 0.0829 0.0829 0.0828

0.0894 0.0893 0.0893 0.0895 0.0895 0.0897 0.09 0.0898 0.0897 0.0897 0.0896 0.0891 0.0894 0.0895

Table C.3: Calculated Reynolds numbers and calculated errors reynolds reynolds Reynolds Reynolds Reynolds reynolds number number number number number number error for error for for for for error 150000 100000 150000 100000 50000 50816 101120 150090 696.388 1385.5 2055.6 50798 101000 149450 696.1899 1383.8 2046.8 50551 100900 149670 692.7709 1382.4 2050 2042 50123 100750 149090 686.7629 1380.5 2045.2 50132 100430 149310 686.8751 1376.1 2045.9 50488 99780 149350 691.7563 1367.1 50452 100170 149030 691.2535 1372.3 2041.4 2042 50338 99820 149070 689.6446 1367.4 2033.3 50348 99860 148450 689.7862 1368.1 2034.7 50169 99390 148560 687.3706 1361.6 49958 99300 149030 684.5262 1360.3 2041.3 49811 100070 149350 682.61 1370.8 2045.6 2051.1 49723 100710 149760 681.3677 1379.6 2048.4 50013 100760 149560 685.3655 1380.2 50031 100350 149660 685.6061 1374.4 2049.6 1372 2050.7 50410 100180 149720 690.7743 2055.1 50652 100630 150050 694.1224 1378.4 2062.4 50735 100590 150580 695.2438 1377.8 50615 100580 150670 693.5063 1377.7 2064 1375 2060 50657 10040 150390 694.1478 50397 100490 150500 690.505 1376.4 2061.4 1373 2059.2 50169 100240 150350 687.3919 50335 100040 150300 689.6419 1370.2 2058.5 2051.4 50482 100380 149810 691.7034 1374.9 1369 2057.9 50466 99960 150270 691.4455 50594 99990 150380 693.197 1369.3 2059.4

Table C.4: Coefficient of lift and coefficient of lift error

Coefficient of lift 50000

Coefficient of lift 100000

Coefficient of lift 150000

Coefficient of lift error for 50000

-0.5632 -0.5289 -0.4687 -0.4185 -0.3698 -0.3192 -0.2192 -0.1025 0.0186 0.0654 0.0511 0.0839 0.1974 0.3514 0.422 0.4679 0.5337 0.5958 0.6477 0.691 0.8327 0.7256 0.6878 0.6619 0.6611 0.6504

-0.6318 -0.5593 -0.5157 -0.455 -0.3947 -0.3287 -0.2499 -0.1669 -0.0798 0.0106 0.0808 0.2181 0.2762 0.3437 0.4248 0.5784 0.6296 0.6834 0.743 0.7913 0.7363 0.6989 0.6873 0.6901 0.6819 0.6909

-0.6758 -0.6115 -0.5582 -0.4877 -0.4221 -0.3619 -0.3031 -0.1971 -0.1193 0.0573 0.123 0.3773 0.4352 0.5008 0.5637 0.6219 0.6734 0.7319 0.7871 0.8385 0.742 0.7173 0.713 0.7069 0.7089 0.7031

0.0023 0.0023 0.0024 0.0025 0.0026 0.0026 0.0027 0.0028 0.0029 0.0029 0.0032 0.003 0.003 0.0029 0.0029 0.0029 0.0023 0.0023 0.0023 0.0024 0.0025 0.0025 0.0026 0.0026 0.0027 0.0027

Coefficient of lift error for 100000 0.0006 0.0008 0.0009 0.001 0.0012 0.0016 0.0017 0.0018 0.002 0.0021 0.0019 0.0018 0.0018 0.0018 0.0018 0.0018 0.0006 0.0006 0.0007 0.0009 0.001 0.0011 0.0013 0.0014 0.0015 0.0017

Coefficient of lift error for 150000 0.0004 0.001 0.0011 0.0013 0.0014 0.0016 0.0017 0.0018 0.002 0.0021 0.0019 0.0018 0.0018 0.0018 0.0018 0.0018 0.0003 0.0004 0.0006 0.0008 0.0009 0.0011 0.0012 0.0014 0.0016 0.0017

Table C.5: Coefficient of drag and coefficient of drag error Coefficient Coefficient Coefficient Coefficient Coefficient Coefficient of drag of drag of drag of drag of drag of lift error error error error 150000 100000 50000 0.0326 0.0434 0.0387 0.0023 0.0006 0.2952 0.0326 0.0376 0.035 0.0023 0.0006 0.3217 0.0326 0.0364 0.0327 0.0024 0.0006 0.3309 0.0327 0.0352 0.0324 0.0024 0.0006 0.3436 0.033 0.0342 0.0326 0.0024 0.0006 0.3564 0.0383 0.0365 0.0346 0.0024 0.0007 0.3713 0.0384 0.0363 0.0352 0.0024 0.0007 0.3856 0.0436 0.0411 0.0378 0.0024 0.0007 0.4044 0.0484 0.0464 0.0418 0.0024 0.0007 0.4242 0.0526 0.0508 0.0486 0.0024 0.0007 0.4514 0.0558 0.0562 0.052 0.0024 0.0008 0.6259 0.0653 0.0633 0.0716 0.0025 0.0008 0.6862 0.0799 0.0682 0.0781 0.0025 0.0009 0.7291 0.0855 0.0699 0.0855 0.0025 0.0009 0.7998 0.0901 0.0833 0.0936 0.0025 0.001 0.847

0.0983 0.1077 0.1267 0.1314 0.1467 0.0622 0.208 0.2325 0.2542 0.2826 0.3063

0.1034 0.1121 0.1221 0.138 0.152 0.2164 0.2425 0.2639 0.2894 0.3062 0.3337

0.1023 0.1102 0.1205 0.1304 0.1445 0.2259 0.2527 0.2716 0.3017 0.3217 0.3366

0.0025 0.0023 0.0023 0.0023 0.0023 0.0023 0.0024 0.0024 0.0023 0.0023 0.0023

0.001 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006 0.0006

Table C.6: Calculated Reynolds numbers Reynolds Reynolds Reynolds number number number at at at 50000 100000 150000 50816 101120 150090 50798 101000 149450 50551 100900 149670 50123 100750 149090 50132 100430 149310 50488 99780 149350 50452 100170 149030 50338 99820 149070 50348 99860 148450 50169 99390 148560 49958 99300 149030 49811 100070 149350 49723 100710 149760 50013 100760 149560 50031 100350 149660 50410 100180 149720 50652 100630 150050 50735 100590 150580 50615 100580 150670 50657 100400 150390 50397 100490 150500 50169 100240 150350 50335 100040 150300 50482 100380 149810 50466 99960 150270 50594 999900 150380 Average 50356 100300 1497100

0.8826 0.2909 0.2832 0.2811 0.2795 0.2778 0.2768 0.2765 0.2772 0.2778 0.2808

The MATLab Code: function lab2=lift() clc; clear; load lift1.txt; load lift2.txt; load lift3.txt; Chord=.0794;%m Ucord=.001;%m Span=.4001;%m Uspan=.001;%m Ulift=.001;%N Utemp=.1;%C Upress=.01;%1 percent A=Chord*Span; % Calculate Re Temp=lift1(:,3); Temptwo=lift2(:,3); Tempthree=lift3(:,3); L=lift1(:,1); z=length(L); Ltwo=lift2(:,1); Lthree=lift3(:,1); D=lift1(:,2); Dtwo=lift2(:,2); Dthree=lift3(:,2); R=287; t=length(Temp); P_atm=lift1(:,5); P_atmtwo=lift2(:,5); P_atmthree=lift3(:,5); q=lift1(:,4); qtwo=lift2(:,4); qthree=lift3(:,4); P=P_atm + q; Ptwo=P_atmtwo + qtwo; Pthree=P_atmthree + qthree; rho=zeros(size(P)); l=Chord; Vf=zeros(size(Temp)); mu=1.458e-6; for i = 1:t Vf(i)=(mu*(Temp(i)^1.5))/(Temp(i)+110.4); Vftwo(i)=(mu*(Temptwo(i)^1.5))/(Temptwo(i)+110.4); Vfthree(i)=(mu*(Tempthree(i)^1.5))/(Tempthree(i)+110.4); rho(i)=P(i)/(R*Temp(i)); rhotwo(i)=Ptwo(i)/(R*Temptwo(i)); rhothree(i)=Pthree(i)/(R*Tempthree(i)); V(i)=sqrt((2*q(i))/rho(i)); Vtwo(i)=sqrt((2*qtwo(i))/rhotwo(i)); Vthree(i)=sqrt((2*qthree(i))/rhothree(i)); Re(i)=(rho(i)*V(i)*l)/Vf(i); Retwo(i)=(rhotwo(i)*Vtwo(i)*l)/Vftwo(i); Rethree(i)=(rhothree(i)*Vthree(i)*l)/Vfthree(i); i=i+1; end Reynolds_Number =sum(Re)/z; Reynolds_Number_two = sum(Retwo)/z; Reynolds_Number_three = sum(Rethree)/z;

%Calculate Re with error positive Temp=lift1(:,3); Temptwo=lift2(:,3); Tempthree=lift3(:,3); R=287;

t=length(Temp); t=length(Temptwo); t=length(Tempthree); P_atm=lift1(:,5); P_atmtwo=lift2(:,5); P_atmthree=lift3(:,5); q=lift1(:,4); qtwo=lift2(:,4); qthree=lift3(:,4); P=P_atm + q; Ptwo=P_atmtwo + qtwo; Pthree=P_atmthree + qthree; rho=zeros(size(P)); rhotwo=zeros(size(Ptwo)); rhothree=zeros(size(Pthree)); l=Chord; Vf=zeros(size(Temp)); Vftwo=zeros(size(Temptwo)); Vfthree=zeros(size(Tempthree)); mu=1.458e-6; for i = 1:t Ua=sqrt((.0794*.001)^2+(.4001*.001^2)^2); Vf(i)=(mu*(Temp(i)^1.5))/(Temp(i)+110.4); Vftwo(i)=(mu*(Temptwo(i)^1.5))/(Temptwo(i)+110.4); Vfthree(i)=(mu*(Tempthree(i)^1.5))/(Tempthree(i)+110.4); rho(i)=P(i)/(R*Temp(i)); rhotwo(i)=Ptwo(i)/(R*Temptwo(i)); rhothree(i)=Pthree(i)/(R*Tempthree(i)); V(i)=sqrt((2*q(i))/rho(i)); Vtwo(i)=sqrt((2*qtwo(i))/rhotwo(i)); Vthree(i)=sqrt((2*qthree(i))/rhothree(i)); Upatm(i)=P_atm(i)*.01+P_atm(i); Upatmtwo(i)=P_atmtwo(i)*.01+P_atmtwo(i); Upatmthree(i)=P_atmthree(i)*.01+P_atmthree(i); Utemp(i)=Temp(i)+.1; Utemptwo(i)=Temptwo(i)+.1; Utempthree(i)=Tempthree(i)+.1; Uchord=Chord*.001+Chord; Uspan=Span*.001+Span; Ulift=.005; Udrag=.005; Uq(i)=.001*.001*q(i); Uqtwo(i)=.001*.001*qtwo(i); Uqthree(i)=.001*.001*qthree(i); Umu(i)=((.000002187*(Utemp(i))^.5)/(Utemp(i)+110.4)-(.000001458*Utemp(i)^1.5/(Utemp(i)+110.4)^2)*.1); Umutwo(i)=((.000002187*(Utemptwo(i))^.5)/(Utemptwo(i)+110.4)-(.000001458*Utemptwo(i)^1.5/(Utemptwo(i)+110.4)^2)*.1); Umuthree(i)=((.000002187*(Utempthree(i))^.5)/(Utempthree(i)+110.4)(.000001458*Utempthree(i)^1.5/(Utempthree(i)+110.4)^2)*.1); Urho(i)=sqrt((-Upatm(i)/(R*(Temp(i))^2)*.1)^2+(1/(R*Utemp(i)))*1.1^2); Urhotwo(i)=sqrt((-Upatmtwo(i)/(R*(Temptwo(i))^2)*.1)^2+(1/(R*Utemptwo(i)))*1.1^2); Urhothree(i)=sqrt((-Upatmthree(i)/(R*(Tempthree(i))^2)*.1)^2+(1/(R*Utempthree(i)))*1.1^2); Uv(i)=sqrt(.5*(sqrt(2)/(sqrt(q(i)/rho(i))*rho(i))*1.1)^2+(-.5*(sqrt(2)*q(i)/(sqrt(q(i)/rho(i)))*(rho(i))^2)*Urho(i))^2); Uvtwo(i)=sqrt(.5*(sqrt(2)/(sqrt(qtwo(i)/rhotwo(i))*rhotwo(i))*1.1)^2+(.5*(sqrt(2)*qtwo(i)/(sqrt(qtwo(i)/rhotwo(i)))*(rhotwo(i))^2)*Urhotwo(i))^2); Uvthree(i)=sqrt(.5*(sqrt(2)/(sqrt(qthree(i)/rhothree(i))*rhothree(i))*1.1)^2+(.5*(sqrt(2)*qthree(i)/(sqrt(qthree(i)/rhothree(i)))*(rhothree(i))^2)*Urhothree(i))^2); UCl(i)=sqrt((Ulift/(q(i) * A))^2 + ((-L(i) * Ua)/(q(i) * A^2))^2 + ((-L(i) * Uq(i))/(A * q(i)^2))^2); UCltwo(i)=sqrt((Ulift/(qtwo(i) * A))^2 + ((-Ltwo(i) * Ua)/(qtwo(i) * A^2))^2 + ((-Ltwo(i) * Uqtwo(i))/(A * qtwo(i)^2))^2); UClthree(i)=sqrt((Ulift/(qthree(i) * A))^2 + ((-Lthree(i) * Ua)/(qthree(i) * A^2))^2 + ((-Lthree(i) * Uqthree(i))/(A * qthree(i)^2))^2); UCd(i)=sqrt((Udrag/(q(i) * A))^2 + ((-D(i) * Ua)/(q(i) * A^2))^2 + ((-D(i) * Uq(i))/(A * q(i)^2))^2); UCdtwo(i)=sqrt((Udrag/(qtwo(i) * A))^2 + ((-Dtwo(i) * Ua)/(qtwo(i) * A^2))^2 + ((-Dtwo(i) * Uqtwo(i))/(A * qtwo(i)^2))^2); UCdthree(i)=sqrt((Udrag/(qthree(i) * A))^2 + ((-Dthree(i) * Ua)/(qthree(i) * A^2))^2 + ((-Dthree(i) * Uqthree(i))/(A * qthree(i)^2))^2); Urerho(i)=(((V(i)*.0749)/Vf(i))*Urho(i))^2; Urerhotwo(i)=(((Vtwo(i)*.0749)/Vftwo(i))*Urhotwo(i))^2; Urerhothree(i)=(((Vthree(i)*.0749)/Vfthree(i))*Urhothree(i))^2; UreV(i)=(((Urho(i)*.0749)/Vf(i))*Uv(i))^2; UreVtwo(i)=(((Urhotwo(i)*.0749)/Vftwo(i))*Uvtwo(i))^2; UreVthree(i)=(((Urhothree(i)*.0749)/Vfthree(i))*Uvthree(i))^2; UreUl(i)=(((rho(i)*V(i))/Vf(i))*.001)^2; UreUltwo(i)=(((rhotwo(i)*Vtwo(i))/Vftwo(i))*.001)^2;

UreUlthree(i)=(((rhothree(i)*Vthree(i))/Vfthree(i))*.001)^2; UreUmu(i)=(((-rho(i)*V(i)*.0749)/(Vf(i)^2))*Umu(i))^2; UreUmutwo(i)=(((-rhotwo(i)*Vtwo(i)*.0749)/(Vftwo(i)^2))*Umutwo(i))^2; UreUmuthree(i)=(((-rhothree(i)*Vthree(i)*.0749)/(Vfthree(i)^2))*Umuthree(i))^2; Ure(i)=sqrt(Urerho(i)+UreV(i)+UreUl(i)+UreUmu(i)); Uretwo(i)=sqrt(Urerhotwo(i)+UreVtwo(i)+UreUltwo(i)+UreUmutwo(i)); Urethree(i)=sqrt(Urerhothree(i)+UreVthree(i)+UreUlthree(i)+UreUmuthree(i)); i=i+1; end uncertainty_Reynolds_Number = sum(Ure)/z uncertainty_Reynolds_Number_two = sum(Uretwo)/z uncertainty_Reynolds_Number_three = sum(Urethree)/z % Cl vs. Angle of Attack AOA=lift1(:,6); AOAtwo=lift2(:,6); AOAthree=lift3(:,6); AOApositive=zeros(size(16)); Lpositive=zeros(size(16)); AOAnegative=zeros(size(10)); Lnegative=zeros(size(10)); for i = 1:16 AOApositive(i)=AOA(i); AOApositivetwo(i)=AOAtwo(i); AOApositivethree(i)=AOAthree(i); Lpositive(i)=L(i); Lpositivetwo(i)=Ltwo(i); Lpositivethree(i)=Lthree(i); i=1+i; end for i = 1:10 AOAnegative(i)=AOA(i+16); AOAnegativetwo(i)=AOAtwo(i+16); AOAnegativethree(i)=AOAthree(i+16); Lnegative(i)=L(i+16); Lnegativetwo(i)=Ltwo(i+16); Lnegativethree(i)=Lthree(i+16); i=1+1; end %Coefficient of lift Clpositive=zeros(size(16)); for i = 1:16 Clpositive(i)=Lpositive(i)/(q(i)*A); Clpositivetwo(i)=Lpositivetwo(i)/(qtwo(i)*A); Clpositivethree(i)=Lpositivethree(i)/(qthree(i)*A); i=1+i; end Clnegative=zeros(size(10)); for i = 1:10 Clnegative(i)=Lnegative(i)/(q(i+16)*A); Clnegativetwo(i)=Lnegativetwo(i)/(qtwo(i+16)*A); Clnegativethree(i)=Lnegativethree(i)/(qthree(i+16)*A); i=i+1; end FCln=fliplr(Clnegative); FClntwo=fliplr(Clnegativetwo); FClnthree=fliplr(Clnegativethree); FAOAn=fliplr(AOAnegative); FAOAntwo=fliplr(AOAnegativetwo); FAOAnthree=fliplr(AOAnegativethree); Cltotal= [FCln Clpositive]; Cltotaltwo= [FClntwo Clpositivetwo]; Cltotalthree= [FClnthree Clpositivethree]; AOAtotal=[FAOAn AOApositive]; AOAtotaltwo=[FAOAntwo AOApositivetwo]; AOAtotalthree=[FAOAnthree AOApositivethree]; figure(1) hold on

plot(AOAtotal,Cltotal,'r'); errorbar(AOAtotal,Cltotal,Ulift,'r'); plot(AOAtotaltwo,Cltotaltwo,'g'); errorbar(AOAtotaltwo,Cltotaltwo,Ulift,'g'); plot(AOAtotalthree,Cltotalthree); errorbar(AOAtotalthree,Cltotalthree,Ulift);

% Coefficient of drag Vs. Coefficient of lift Dpositive=zeros(size(16)); Dnegative=zeros(size(10)); for i = 1:16 Dpositive(i)=D(i); Dpositivetwo(i)=Dtwo(i); Dpositivethree(i)=Dthree(i); i=1+i; end for i = 1:10 Dnegative(i)=D(i+16); Dnegativetwo(i)=Dtwo(i+16); Dnegativethree(i)=Dthree(i+16); i=1+1; end %Coefficient of Drag Cdpositive=zeros(size(16)); for i = 1:16 Cdpositive(i)=Dpositive(i)/(q(i)*A); Cdpositivetwo(i)=Dpositivetwo(i)/(qtwo(i)*A); Cdpositivethree(i)=Dpositivethree(i)/(qthree(i)*A); i=1+i; end Cdnegative=zeros(size(10)); for i = 1:10 Cdnegative(i)=Dnegative(i)/(q(i+10)*A); Cdnegativetwo(i)=Dnegativetwo(i)/(qtwo(i+10)*A); Cdnegativethree(i)=Dnegativethree(i)/(qthree(i+10)*A); i=i+1; end FCdn=fliplr(Cdnegative); FCdntwo=fliplr(Cdnegativetwo); FCdnthree=fliplr(Cdnegativethree); Cdtotal= [FCdn Cdpositive]; Cdtotaltwo= [FCdntwo Cdpositivetwo]; Cdtotalthree= [FCdnthree Cdpositivethree]; figure(3) Hold on plot(Cltotal,Cdtotal,'r'); errorbar(Cltotal, Cdtotal, UCd,'r'); plot(Cltotaltwo,Cdtotaltwo,'g'); errorbar(Cltotaltwo, Cdtotaltwo, UCdtwo,'g'); plot(Cltotalthree,Cdtotalthree); errorbar(Cltotalthree, Cdtotalthree, UCdthree); end

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