Introduction

This article aims to clarify the essential features of NACA airfoils, particularly for beginners studying aerodynamics. It starts with the basic principles of airfoil geometry and then discusses how to implement these concepts in Python to compute important numerical characteristics for visualizing a NACA 4-Series 2D wing profile with Matplotlib.

Airfoils represent the cross-sectional shape of wings. The National Advisory Committee for Aeronautics (NACA) developed a variety of airfoil designs, known as NACA Airfoils. Figure 1 showcases different examples of these wing sections.

The most frequently examined series in introductory aerodynamics include four-digit and five-digit designs, with some six-digit variants as well. This article specifically addresses the four-digit series, such as the NACA 4415 airfoil.

Airfoil Geometry

Figure 2 illustrates a sample symmetrical airfoil, highlighting key geometric parameters.

• Leading and trailing edges: The foremost and rearmost points of an airfoil, respectively.
• Chord: A straight line connecting the leading and trailing edges of the airfoil.
• x: The horizontal distance along the chord, starting from zero at the leading edge.
• y: The vertical height relative to the horizontal x-axis.

Figure 3 shows a cambered airfoil. Camber is essentially the curvature of the airfoil.

• Mean camber line: This line is located halfway between the upper and lower surfaces, corresponding to the geometric centerline.
• Thickness (t): The height distribution along the length of the airfoil.

From the diagrams presented, it is clear that two main variables describe the geometric profile of the airfoil surface: camber and thickness.

A crucial element of the design is that the 4-Series airfoil shapes are derived from analytical equations that define the mean camber line and the thickness distribution of the section. Later families, such as the 6-Series, are created using more complex theoretical methods.

4-Series Equations

A NACA 4415 exemplifies a member of the 4-Series family, with the digits 4415 characterizing the 2D profile.

Equation 1 represents digit 1, indicating the maximum camber (m) as a percentage of the chord. Thus, the maximum camber for the 4415 airfoil is 4% of the chord length.

Digit 2 is used in Equation 2 to find the maximum camber (p) location from the leading edge in tenths of the chord. Therefore, for a 4415 airfoil, the maximum camber occurs at 40% along the chord length.

Equation 3 employs digits 3 and 4 to determine the maximum thickness (t) as a percentage of the chord. Thus, the thickness of the 4415 airfoil is 15% of the chord length.

Gist 1 includes Python code that defines three functions to extract the numerical airfoil characteristics based on the NACA four digits.

For a symmetric airfoil, the first two digits are zero, e.g., 0015 ? m = 0, p = 0.

Two equations specify the mean camber line, depending on whether the x coordinate is less than or greater than the maximum camber position (p), as shown in Equation 4.

It’s vital to emphasize that the equations provided are purely analytical, established by NACA through extensive research and experimentation.

Gist 2 presents the Python implementation of Equation 4.

y corresponds to the thickness distribution. The thickness values above (+) and below (-) the mean camber line depend on Equation 4.

The x coefficient varies depending on whether the trailing edge is open or closed; for instance, -0.1036 corresponds to a closed surface, while for a finite thickness trailing edge, use -0.1015 instead.

y is calculated in Python using Gist 3.

Thickness values are added perpendicular to the mean camber line, necessitating an angle to indicate the offset of the addition.

Taking the derivative of the mean camber line yields the slope of the tangent line to the curve. Differentiating Equation 4 with respect to x results in Equation 6.

Obtain the derivative of the mean camber line using the Python code in Gist 4.

Calculating the inverse tangent of this derivative provides the angle ? to offset the thickness from vertical. Refer to Figure 3 to understand the significance of ?.

Finally, to determine the upper and lower airfoil surface coordinates, use Equations 8–11, where ? represents the inverse tangent of the derivative of the MCL at x, as derived from Equation 7.

Gist 6 provides the code to calculate the final (x, y) values for the airfoil’s upper and lower surfaces.

Plotting Results

Utilize the resulting values of (x, y) to plot the final wing profile. Figure 4 demonstrates the NACA 4415 plotted using Matplotlib.

Figure 5 shows another example of a cambered 4-Series airfoil, the NACA 2412. Visually compare the geometric characteristics of the 4415 and 2412, paying attention to the y-axis scale.

As mentioned, these analytical expressions are applicable to symmetrical airfoils.

Both the mean camber line and thickness distribution align perfectly with the chord, as evidenced in the plot of a 0015 in Figure 6.

Conclusion

Each equation is generic and can be parameterized with any four digits to visualize any member of the NACA 4-Series family.

This article has outlined fundamental airfoil properties and demonstrated a method to implement the geometric expressions for plotting the 2D surface profile of a wing.

Thank you for reading! Please let me know if you would be interested in other aerodynamics related articles.

References

[1] Fundamentals of Aerodynamics. Sixth Edition. John D. Anderson, Jr. Curator of Aerodynamics. National Air and Space Museum. Smithsonian Institution. [2] NACA Airfoils — NASA. Last Updated: Aug 7, 2017, Editor: Bob Allen [3] The NACA airfoil series (AA200_Course_Material) — Stanford [4] Explained: NACA 4-Digit Airfoil [Airplanes] — Josh The Engineer