Vortex Panel Paper

AA 302Vortex Panel Mini-Project

Introduction

Flow simulation is a common instrument which can describe the flow around an airfoil. There are many types of programs which can simulate the flow around an airfoil. The program used in generating the plots in the Discussion section of this report were generated from MATLAB output which had input data from a CFD program called “xfoil.” From these plots, the lift, drag, moment, and pressure coefficients were determined.

Theory

Many types of forces act on an airfoil during flight. The two main forces considered in a 2-D wing analysis are the pressure and shear along the airfoil. Pressure acts normal to the surface of the airfoil and shear acts tangential to the surface. These forces are then integrated to find a net force and moment on the airfoil [3]. The net resultant force is caused by the pressure distribution of the airfoil, where the force is resolved into normal and axial forces with respect to the chord line of the airfoil as shown in Figure 1 below.

Figure 1: Free Body Diagram of Forces Acting on Arbitrary Airfoil

Flows around an object can be analyzed in the terms of common aerodynamic coefficients which are often expressed in terms of the applied force, effective surface area, and dynamic pressure. Three types of aerodynamic coefficients are of particular interest in this experiment: lift, drag, and pressure. The lift, drag, moment, and pressure coefficients per unit length are defined in Equations 1, 2, 3, and 4, respectively.

c l=Lq∞ c

(1)

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cd=Dq∞ c

(2)

c p=ps−p∞q∞

(3)

cm=M

q∞ c2(4)

A previous experimental distribution of the pressure with respect to the chord fraction coefficient for the airfoil being tested (NACA 0016) in this experiment is shown in Figure 2 below.

Figure 2: Pressure Coefficient vs. Chord Fraction

From Figure 2 above, the pressure coefficient is positive for the lower surface of the airfoil and negative for the upper surface. This difference in pressure causes the lifting force for aircraft.An ideal qualitative graph for the lift coefficient for a cambered airfoil is shown in Figure 3 below.

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Figure 3: Ideal Lift Coefficient Curve for a Cambered Airfoil

As indicated on Figure 3 on the previous page, the maximum lift corresponds to a stall angle from flow separation. In reality, the aircraft starts to stall before the lift coefficient becomes a maximum. This occurs when the lift slope transitions from a linear behavior to a non-linear regime. The force-moment system is used to determine the moment at any point along the airfoil as long as the corresponding moment about a point due to a distributed load is given [3]. The corresponding moment equation is given as Equation 5 below:

M ¿'=−c

4L'+M c/4

' =−xcp L' (5)

Figure 4 below shows pictures representing force-moment systems of interest in this report.

Figure 4: Resultant Forces and Moments around Points of Interest

For thin airfoils, several simplifications and approximation can be made to certain coefficients and parameters for low angles of attack. These approximations are tabulated in Table 1 below for symmetric airfoils.

Table 1: Approximate Values for Specified Parameters for a Symmetric Airfoil

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Term Symmetric Airfoil

c l 2πα

cm, c/4 0

d cldα

2π

The airfoils being tested are NACA airfoils. They are numbered either in a four digit or five digit numerical system. For a four digit system, the 1st digit represents the maximum camber in 100th of a chord. The second digit represents the location of maximum camber on chord in 10th of chord. The last two digits represent the maximum thickness in 100th of a chord. For a five digit system, the 1st digit represents 2/3 of the design lift coefficient. The second and third digits divided by two represents the location of maximum camber along the chord in 100th of a chord. The last digits represent the maximum thickness in 100th of a chord.

Parameters

Four Types of airfoils were used in the “xfoil” program. These airfoils were:1) NACA 230122) NACA 24123) NACA 00184) NACA 4418

The angle of attack was varied from -5° to 20°. For the purposes in generating valid results, the Mach number was limited to 0.25. A Reynolds number of 8*106 was used with 200 iterations. Viscous forces were also included to generate more realistic plots.

Results

Figure 5 below shows plots of CL vs. CD for the NACA 23012, 2412, 0018, and 4418 airfoils.

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Figure 5: CL vs. CD for Tested Airfoils

From Figure 5 above, the behavior of the CL vs. CD graphs were very similar for each tested airfoil. As the lift coefficient increased, the drag was slightly parabolic in behavior with little increase. Then, when the lift coefficient becomes a maximum, the drag coefficient greatly increases, due to flow separation from going past the stall angle. The drag coefficient significantly increased after the maximum lift coefficient for the NACA 2412 and 23012 airfoils vs. the NACA 0018 and 4418 airfoils. The stall angle can be more clearly shown with theCL vs. α and CD vs. α plots to follow.

Figure 6 below shows a graph of CD vs. α for the airfoils mentioned previously.

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Figure 6: CD vs. α

From Figure 6 above, the drag coefficient was slightly parabolic but relatively constant for angles of attack less than 15°. The NACA 4418 started to show an increase in CD at around

12.5°. The other airfoils had a dramatic increase in CD of around 17°. This means that the stall angle for all the airfoils except the NACA 4418 was before 17°. This sudden increase in drag is due to flow separation (and thus pressure drag) from the stall. The stall angle is more clearly shown in the CL vs. α graph on the next page. Figure 6 suggests that there is not a relation between the camber of the airfoil and the drag coefficient.

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Figure 7 below shows plots of CL vs. α for the tested airfoils.

Figure 7: CL vs. α

From Figure 7 above, the stall angle occurs where the flow starts becoming detached from the

flow. This is indicated on the CL graph when the linear behavior for low angles of attack becomes non-linear. These plots matched relatively closely from Figure 3 from Theory. Furthermore, the lift slope for each graph also matched closely with thin airfoil theory. For all the airfoils except the NACA 4418, the stall angle occurs at roughly 12°. For the NACA 4418, this occurs at around 7°, much sooner than the other airfoils. The lift coefficient was overall the highest for the NACA 4418 and the lowest for the NACA 0018. The maximum lift coefficient for all the airfoils ranged from 1.75 for the NACA 0018 to 1.8 for the NACA 4418 and the NACA 2412. After the maximum CL was achieved, the NACA 23012 had a sudden drop in CL

but the other airfoils had a more gradual decrease inCL. Figure 7 above also suggests that a more cambered airfoil will generate more lift compared to more symmetric airfoils. Additionally, a more cambered airfoil has a gentler stall regime compared to more symmetric airfoils. The optimal angle of attack for cruise for the non-cambered airfoils would be zero degrees and slightly less than zero for cambered airfoils. For the NACA 0018 airfoil, the optimal angle would be zero degrees; the NACA 4418 airfoil would be at around -2°, the NACA 2412 would

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be at about -1°, and the NACA 23012 airfoil would be approximated at -1°. Any further decrease in the angle of attack would mean that the aircraft would be descending in altitude.

Figure 8 below shows plots of CM vs. α for the tested airfoils. The moment plots are taken from the leading edge.

Figure 8: CM vs. α

The moment coefficients for each of the airfoils remained relatively constant until the transition from a linear behavior to a non-linear behavior occurred as shown on Figure 7 on Page 7. Afterwards, CM increased positively until the lift coefficient from Figure 7 reached a maximum. After this maximum was reached, the moment coefficient sharply increased negatively with increasing angles of attack.

The moment coefficient for the NACA 0018 was zero for low angles of attack. This matches the thin airfoil theory for a symmetric airfoil. The NACA 4418 had the largest negatively valued moment and behaved the least linearly. From Figure 8 above, the plots suggest that an airfoil with larger camber will have a more negative moment coefficient, which implies more lift on the airfoil.

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Figure 9 below shows a graph of CP vs. xc

forα=0 ° ,5° ,10 °.

Figure 9: CP vs. x /c for α=0 ° ,5° ,10 °

From Figure 9 above, the pressure coefficient distribution was the largest for the NACA 4418 compared to other airfoils. The NACA 2412 had the largest magnitude for the pressure coefficient at an α=10 °. For all plots, the distribution of the pressure coefficient was the largest towards the leading edge and eventually reached a value close to zero towards the trailing edge. The pressure distribution for the NACA 4418 was more triangular in behavior compared to other airfoils. This is likely because the NACA 4418 airfoil has more camber compared to other airfoils. Additionally, the plot from Figure 9 suggests that the larger the maximum thickness in the 100th chord, the finer the pressure distribution.

References

[1] Anderson, John D. Fundamentals of Aerodynamics, 5th ed. New York: McGraw Hill, 2011.

[2] Anderson, John D. Introduction to Flight, 6th ed. New York: McGraw Hill, 2008.

[3] Bruckner, Adam A. April 12, 2011. Guggenheim Hall, University of Washington. Class

Lecture. <https://catalyst.uw.edu/workspace/bruckap/18018/113394>.

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[4] Dabiri, Dana. May 2, 2011. Guggenheim Hall, University of Washington. Class Lecture.

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