Chapter 1: Understanding NACA Airfoils

This article aims to elucidate the essential features of NACA airfoils, particularly for students new to aerodynamics. Initially, we will explore the fundamental principles underlying airfoil geometry. Subsequently, we will look into how these equations can be implemented in Python to compute numerical attributes for visualizing a NACA 4-Series 2D wing profile using Matplotlib.

Airfoils represent the cross-sectional shapes of wings. The National Advisory Committee for Aeronautics (NACA) created and assessed a range of airfoils known as NACA airfoils. The following figure illustrates various samples of these wing sections.

The four-digit and five-digit series are commonly studied in introductory aerodynamics courses, although six-digit models also exist. This article concentrates on the four-digit series, specifically the NACA 4415 airfoil.

Airfoil Geometry

Below is a diagram of a symmetrical airfoil, highlighting its key geometric parameters:

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

A cambered airfoil is depicted in the next figure, where camber refers to the curvature of the airfoil.

• Mean Camber Line: This line lies midway between the upper and lower surfaces and serves as the geometric centerline.
• Thickness (t): This refers to the distribution of height along the length of the airfoil.

From these illustrations, it is clear that the two main variables defining the geometric profile of the airfoil surface are camber and thickness.

A significant aspect of the design is that the 4-Series airfoil shapes are derived from analytical equations that describe the mean camber line and the thickness distribution of the section. In contrast, later families, such as the 6-Series, are established using more complex theoretical methods.

4-Series Equations

The NACA 4415 is an example of the 4-Series family. The digits 4415 represent the two-dimensional profile.

• Equation 1: The first digit indicates the maximum camber (m) as a percentage of the chord. For the 4415 airfoil, this maximum camber is 4% of the chord length.

• Equation 2: The second digit, 4, specifies the distance (p) of maximum camber from the leading edge in tenths of the chord. For the 4415 airfoil, the maximum camber is located at 40% of the chord length.

• Equation 3: The last two digits (15) denote the maximum thickness (t) as a percentage of the chord. Thus, the thickness of the 4415 airfoil is 15% of the chord length.

The following Python code (Gist 1) defines three functions to extract numerical characteristics based on the four-digit NACA code.

For symmetric airfoils, the first two digits are zero, for example, 0015 implies m = 0 and 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 illustrated in Equation 4.

It is crucial to note that the equations presented here are analytical, developed by NACA through extensive research and experimentation.

Gist 2 provides the Python code for implementing Equation 4.

The variable yₜ represents the thickness distribution. The thickness values above (+) and below (-) the mean camber line are determined by Equation 4.

The x⁴ coefficient changes based on whether the trailing edge is open or closed. For a closed surface, the coefficient is -0.1036, while for a finite thickness trailing edge, it changes to -0.1015. The thickness values are computed in Python using Gist 3.

To compute the upper and lower coordinates of the airfoil surfaces, use Equations 8-11, where θ represents the angle obtained from the inverse tangent of the derivative of the mean camber line.

Gist 6 contains the code for determining the final (x, y) coordinates for the upper and lower surfaces of the airfoil.

Plotting Results

The final wing profile can be visualized using the resulting (xᵤ, yᵤ) and (xₗ, yₗ) values. Below is a plot of the NACA 4415 generated using Matplotlib.

Another example of a cambered 4-Series airfoil, the NACA 2412, is illustrated below. By visually comparing the 4415 and 2412, one can note the differences in their geometric properties, particularly in relation to the y-axis scale.

As previously mentioned, these analytical equations apply to symmetrical airfoils. Both the mean camber line and thickness distribution are perfectly aligned with the chord, as evident in the plot of a 0015 shown below.

Conclusion

Each equation is versatile and can be parameterized with any four integers to visualize any member of the 4-Series NACA family. This article has outlined basic properties of airfoils and demonstrated how to implement the geometric expressions to visualize the 2D surface profile of a wing.

Thank you for reading! If you're interested in more articles related to aerodynamics, please let me know.

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

The first video illustrates a tutorial on Python scripting in FreeCAD, focusing on modeling NACA airfoils, which can help deepen your understanding of airfoil geometry.

The second video demonstrates how to plot a NACA 4-digit aerofoil, offering visual insights into the plotting process discussed in this article.