Pedagogical 2D Panel Methods

For

Dr. Vladimir Golubev

Embry Riddle Aeronautical University

Daytona Beach Florida

By

Megha Bafna

December 2, 2011

1 Flow past an ellipse

1.1

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 θ is 90. The CP is found to be -3.

CP= 1 – 4sin2θ

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.

1.2

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

Percent error=3−2.9893

∗100=0.367

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

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Number of PanelsThe coefficient of pressure Percent Error

8 -1.697 43.4333333316 -2.476 17.4666666732 -2.78 7.33333333340 -2.831 5.63333333358 -2.901 3.380 -2.923 2.566666667

128 -2.954 1.533333333256 -2.978 0.733333333512 -2.989 0.366666667

2. NACA 4-digit library

2.1

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 Coefficient of moment2312 0.2352 -0.04553312 0.3528 -0.06824312 0.4699 -0.09085312 0.5867 -0.11336312 0.7031 -0.13577312 0.819 -0.1588312 0.9342 -0.18019312 1.0488 -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 Coefficient of moment4212 0.4436 -0.07274312 0.4699 -0.09084412 0.5051 -0.10934512 0.5524 -0.12974612 0.6182 -0.15394712 0.7154 -0.18544812 0.8751 -0.2324912 1.1911 -0.3167

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

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NACA Coefficient of lift Coefficient of moment4304 0.433 -0.08944305 0.4375 -0.08954306 0.4421 -0.08974307 0.4467 -0.08994308 0.4513 -0.09014309 0.456 -0.09034310 0.4606 -0.09054311 0.4653 -0.09064312 0.4699 -0.09084313 0.4746 -0.09094314 0.4793 -0.09114315 0.4839 -0.09124316 0.4886 -0.09134317 0.4933 -0.09144318 0.498 -0.09154319 0.5027 -0.09164320 0.5073 -0.0917

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

1 2 3 4 5 6 7 8 9 10

-0.4-0.2

00.20.40.60.8

11.2

Variable Maximum Camber

ClCm

Value of camber

Coef

fici

ent

5

1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

Variable Maximum Camber Po-sition

ClCm

Value of max camber position

Coef

fici

ent

2 4 6 8 10 12 14 16 18 20 22

-0.2-0.1

00.10.20.30.40.50.6

Variable Thickness

ClCm

Value of thickness

Coef

fici

ent

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 it’s the camber position. Optimum is the maximum value. In this case the maximum coefficient of lift is 1.1911.

3 General Airfoil Library

3.1

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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.

AirfoilCl Cm Xcp

N001035 0.0000 0.0000 0.00RAE2822 0.2530 -0.0736 0.54FOIL31 0.6503 -0.1528 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.

3.2

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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.

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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 NACA 4312 ONERAM6Angle of attack (degrees) Coefficient of lift Coefficient of lift Coefficient of lift-10 -0.4612 -0.7125 -1.1599-8 -0.2151 -0.4778 -0.9305-6 0.0321 -0.2418 -0.6994-4 0.28 -0.005 -0.467-2 0.5283 0.2324 -0.23380 0.7763 0.4699 02 1.0239 0.7071 0.23374 1.2704 0.9435 0.46696 1.5154 1.1787 0.69938 1.7586 1.4122 0.930410 1.9995 1.6437 1.1598

-15 -10 -5 0 5 10 15

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

f(x) = 0.116243181818182 x − 4.54545454545405E-05

f(x) = 0.118064545454545 x + 0.468218181818182f(x) = 0.123293636363636 x + 0.773472727272727

Thin, Medium and Thick airfoil

VEZCAN

Linear (VEZCAN)

NACA4312

Linear (NACA4312)

ONERAM6

Linear (ONERAM6)

Alpha

Coef

fici

ent

of li

ft

The slopes for the above plots was found and the angles in radians can be put in the following table:

Airfoil Angles in radiansVEZCAN 0.0022NACA4312 0.0021ONERAM6 0.0020

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

5 Effect of the type of singularity

5.1

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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 coefficient of pressure curve for a symmetric airfoil.

Non-Symmetric Airfoil:

NACA23024

Figure 17: The figure above shows the Airfoil NACA23024 with 58 panels. It has a lift coefficient of 0 when alpha is 0. The constant source method is used.

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Figure 18: The figure above shows the Airfoil NACA23024 with 58 panels. It has a lift coefficient of 0.1609 when alpha is 0. The constant doublet method is used.

Figure 19: The figure above shows the Airfoil NACA23024 with 58 panels. It has a lift coefficient of 0.1606 when alpha is 0. The linear vortex method is used.

The symmetric airfoils all the methods produce a very similar pressure coefficient curve, while for a non-symmetric airfoil, in this case NACA23024 the results are different. When the constant doublet method is used the curve tends to behave very differently, it seems like the upper curve doesn’t meet the lower one and instead gets more negative and the lower curve becomes more positive.

5.2

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NACA 8812

Figure 20: The figure above shows the Airfoil NACA8812 with 50 panels. It has a lift coefficient of 1.105 when alpha is 0. The constant doublet method is used.

Figure 21: The figure above shows the Airfoil NACA8812 with 250 panels. It has a lift coefficient of 1.5738 when alpha is 0. The linear vortex method is used.

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Figure 22: The figure above shows the Airfoil NACA8812 with 50 panels. It has a lift coefficient of 0 when alpha is 0. The constant source method is used.

Figure 23: The figure above shows the Airfoil NACA8812 with 250 panels. It has a lift coefficient of 1.105 when alpha is 0. The constant doublet method is used.

The constant doublet curve of the 200 panels is very similar to the linear vortex and constant doublet of 50 panels. This is because a linear vortex distribution is equivalent to a quadratic doublet distribution. The constant doublet for the 200 panels method and the 50 panels method gives a lift coefficient of 1.105.

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REFRENCES:

Abbott., Doenhoff, v., Hill, M., York, N., 1949., & airfoils, i. w. (n.d.). Wing Design - CONVENTIONAL AIRFOILS and LAMINAR FLOW AIRFOILS. Aeronautics Learning Laboratory for Science Technology, and Research (ALLSTAR) Network. Retrieved November 28, 2011, from http://www.allstar.fiu.edu/aero/wing31.htm

Airfoil Geometry. (n.d.). Applied Aerodynamics: A Digital Textbook. Retrieved November 28, 2011, from www.desktop.aero/appliedaero/airfoils1/airfoilgeometry.html

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