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### Transcript of Aero Final Report

Pedagogical 2D Panel Methods

For

Embry Riddle Aeronautical University

Daytona Beach Florida

By Megha Bafna December 2, 2011

1 Flow past an ellipse1.1 Different numbers of panels were used to plot the pressure distribution shown below:

Figure 1: The CP for the 512 panels was found to be -2.989 while the CP for 8 panels was found to be -1.697. When is 90. The CP is found to be -3. CP= 1 4sin2 Hence, CP= -3 The error in minimum CP should be less that 4%. Hence, the appropriate number of panels that should be used would be 58.

1.2The percent error is calculated by taking the theoretical value that is -3 and comparing it to the experimental value obtained. For example:

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The table below consists the different number of panels used the and the percent error for each:

Number of Panels 8 16 32 40 58 80 128 256 512 2. NACA 4-digit library2.1

The coefficient of pressure -1.697 -2.476 -2.78 -2.831 -2.901 -2.923 -2.954 -2.978 -2.989

Percent Error 43.43333333 17.46666667 7.333333333 5.633333333 3.3 2.566666667 1.533333333 0.733333333 0.366666667

The NACA 4-digit airfoils mean the following: The first digit expresses the camber in percent chord, the second digit gives the location of the maximum camber point in tenths of chord, and the last two digits give the thickness in percent chord. Thus 4412 has a maximum camber of 4% of chord located at 40% chord back from the leading edge and is 12% thick, while 0006 is a symmetrical section of 6% thickness. (Abbott, 1949)

Figure 2: The airfoil Geometry (Source: Applied Aerodynamics, A digital textbook) 2.2 The effect of the three shape parameters thickness, camber and camber location taken separately on Cl and Cm can be seen in the following cases. The chosen airfoil is NACA 4312.

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Figure 2: The NACA 4312 airfoil with 58 panels can be seen above. In the first case the value of camber is changed, while the camber position and the thickness are kept the same. NACA

Coefficient of lift 2312 3312 4312 5312 6312 7312 8312 9312 0.2352 0.3528 0.4699 0.5867 0.7031 0.819 0.9342 1.0488

Coefficient of moment -0.0455 -0.0682 -0.0908 -0.1133 -0.1357 -0.158 -0.1801 -0.202

In the second case the value of camber position is varied while the camber and the thickness are kept constant NACA

Coefficient of lift 4212 4312 4412 4512 4612 4712 4812 4912 0.4436 0.4699 0.5051 0.5524 0.6182 0.7154 0.8751 1.1911

Coefficient of moment -0.0727 -0.0908 -0.1093 -0.1297 -0.1539 -0.1854 -0.232 -0.3167

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Lastly the thickness is changed and the camber and camber position remain the same. NACA

Coefficient of lift 4304 4305 4306 4307 4308 4309 4310 4311 4312 4313 4314 4315 4316 4317 4318 4319 4320 0.433 0.4375 0.4421 0.4467 0.4513 0.456 0.4606 0.4653 0.4699 0.4746 0.4793 0.4839 0.4886 0.4933 0.498 0.5027 0.5073

Coefficient of moment -0.0894 -0.0895 -0.0897 -0.0899 -0.0901 -0.0903 -0.0905 -0.0906 -0.0908 -0.0909 -0.0911 -0.0912 -0.0913 -0.0914 -0.0915 -0.0916 -0.0917

All the lift coefficients and the moment coefficients are plotted in different graphs:

Variable Maximum Camber1.2 1 0.8 Coefficient 0.6 0.4 0.2 0 2 4 6 8 10 Cl Cm

-0.2 0 -0.4

Value of camber

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Variable Maximum Camber Position1.5 1 Coefficient 0.5 0 0 -0.5 2 4 6 8 10 Cl Cm

Value of max camber position

Variable Thickness0.6 0.5 0.4 Coefficient 0.3 0.2 0.1 0 -0.1 0 -0.2 5 10 15 20 25 Cl Cm

Value of thickness

From the graphs above it can be seen that as we vary the camber the lift and moment coefficients change. As we increase the camber the lift coefficient increases but the moment coefficient increases. The same effect is seen on both the coefficients when the camber position is varied. When the thickness of the airfoil is changed it is seen that the lift coefficient increases while the moment coefficient decreases. The moment coefficient changes very slightly each time the thickness is changed. 2.3 As seen in the previous part, the coefficient of lift increases with increase in the values of thickness, camber and camber location, it can be said that the maximum Cl occurs when all the three values are the highest.

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Figure 3: The graph above shows the NACA 9920 airfoil and the linear vortex. It has Cl of 2.336 As seen in the above graph NACA 9920 produces highest Cl of 2.336. This allows for the largest maximum camber located at the rear of the airfoil with a thickness of 20%. The above given solution is not realistic because the Kutta condition must be applied at the trailing edge. At this point, the local vortices must be equal and if the maximum camber is at this location, there will be a adverse pressure gradient as the air flow tries to adhere to this boundary layer. The constraint needed is to restrict the maximum camber location to the first half of the airfoil. This will let V1=V2 at the trailing edge of the airfoil and the Kutta condition will be satisfied by this airfoil. The solution given in the graph above is not physically relevant as we are analyzing inviscid flow. Viscosity should be added as a constraint for accurate optimization. Optimiser is the parameter that affects the coefficient of lift the most, and in this case it seems like its the camber position. Optimum is the maximum value. In this case the maximum coefficient of lift is 1.1911.

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3 General Airfoil Library3.1

Figure 4: The figure above shows the Airfoil N001035 with 58 panels. It has a lift coefficient of 0 when alpha is 0.

Figure 5: The figure above shows the Airfoil RAE2822with 58 panels. It has a lift coefficient of 0.253 when alpha is 0.

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Figure 6: The figure above shows the Airfoil FOIL31 with 58 panels. It has a lift coefficient of 0.6503 when alpha is 0. Cl Airfoil N001035 RAE2822 FOIL31 0.0000 0.2530 0.6503 Cm 0.0000 -0.0736 -0.1528 Xcp 0.00 0.54 0.48

The Cp curve in each case is very different. The Airfoil N001035 looks and behaves very symmetric. The RAE2822 has a higher lift coefficient while the FOIL31 has the highest lift coefficient. The N001035 airfoil has a curve that goes above the airfoil, it looks like a semicircle almost. The N001035 has zero lift coefficients and zero coefficients of moment and Xcp.

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3.2

Figure 7: The figure above shows the Airfoil VEZBL32 with 58 panels. It has a lift coefficient of 0.4649 and a coefficient of pressure of -0.7578 when alpha is 0.

Figure 8: The figure above shows the Airfoil VEZCAN with 58 panels. It has a lift coefficient of 0.7763 and a pressure coefficient of -1.525 when alpha is 0.

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Figure 9: The figure above shows the Airfoil VEZWLTR with 58 panels. It has a lift coefficient of 0.7681 and a coefficient of pressure of -1.105 when alpha is 0. In the airfoil VEZBL32 the lower surface is more curved while in the VEZCAN the upper surface is more curved. In the VEZWLTR the upper surface is more curved but also it appears to be thinner in comparison with the other two airfoils.

Figure 10: The figure above shows the Airfoil KORN with 100 panels. It has a lift coefficient of 0.1842 and a coefficient of pressure of -0.5169 when alpha is 0.

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Figure 11: The figure above shows the Airfoil KORN with 200 panels. It has a lift coefficient of 0 and a coefficient of pressure of -2.54e+012 when alpha is 0. The airfoil trailing edge is very sharp in the KORN airfoil, the lift coefficient changed when the number of panels are changed and also the coefficient of pressure curve behaves very differently in the two cases mentioned above. 3.3

Figure 12: The figure above shows the Airfoil NACA23024 with 58 panels. It has a lift coefficient of 0.1606 and a coefficient of pressure of -2.54e+012 when alpha is 0.

12

Figure 13: The figure above shows the experimental data for Airfoil NACA23024. When the angle of attack (alpha) is zero the coefficient of lift is found to be approximately 0.11. Hence, the experimental value of the lift coefficient is close to 0.11 and the one obtained from Pablo is 0.1606. The percent error is found to be 31.5%.

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4 Effect of the angle of attack VEZCAN Angle of attack (degrees) -10 -8 -6 -4 -2 0 2 4 6 8 10 Coefficient of lift -0.4612 -0.2151 0.0321 0.28 0.5283 0.7763 1.0239 1.2704 1.5154 1.7586 1.9995 NACA 4312 Coefficient of lift -0.7125 -0.4778 -0.2418 -0.005 0.2324 0.4699 0.7071 0.9435 1.1787 1.4122 1.6437 ONERAM6 Coefficient of lift -1.1599 -0.9305 -0.6994 -0.467 -0.2338 0 0.2337 0.4669 0.6993 0.9304 1.1598

Thin, Medium and Thick airfoil2.5 2 Coefficient of lift 1.5 1 0.5 0 -10 -5 -0.5 -1 -1.5 Alpha 0 5 10 15 y = 0.1233x + 0.7735 y = 0.1181x + 0.4682 y = 0.1162x - 5E-05 VEZCAN NACA4312 ONERAM6 Linear (VEZCAN) Linear (NACA4312) Linear (ONERAM6)

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The slopes for the above plots was found and the angles in radians can be put in the following table: Airfoil VEZCAN NACA4312 ONERAM6 Angles in radians 0.0022 0.0021 0.0020

For very thin airfoils the slope of the curves is 2, which is 6.283. As the airfoil becomes thinner the slope approaches to 6.283 radians.

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5 Effect of the type of singularity5.1 Symmetric Airfoil: NACA 0012

Figure 14: The figure above shows the Airfoil NACA0012 with 58 panels. It has a lift coefficient of 0 when alpha is 0. The constant source method is used.

Figure 15: The figure above shows the Airfoil NACA0012 with 58 panels. It has a lift coefficient of 0 when alpha is 0. The constant doublet method is used.

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Figure 16: The figure above shows the Airfoil NACA0012 with 58 panels. It has a lift coefficient of 0 when alpha is 0. The linear vortex method is used. All the three methods have similar results in terms of the c