# 23 Airfoil Geometries

# Introduction

Aerospace engineers must know how to select or design suitable cross-sectional wing shapes (often called airfoil *profiles *or *airfoils*) for use on a diverse range of flight vehicles such as airplanes, various types of space launch and re-entry vehicles, as well as helicopter rotors, propeller blades, wind turbines, UAVs, etc. To this end, not all airfoils are created equally, and different airfoil shapes will be better suited for one application versus another.

For example, airfoils for use on the wings of low-speed airplanes are generally thicker (in terms of their thickness-to-chord ratio) and have more curvature or *camber.* Airfoils for high-speed aircraft, especially for supersonic flight, are much thinner with more pointed leading edges and much less camber. Historically, the design of airfoil shapes for specific applications has proceeded evolutionarily, with wind tunnel experiments, theoretical analysis, and flight testing all being used synergistically to develop the best airfoil shapes for application to specific types of flight vehicles.

Learning Objectives

- Appreciate the historical evolution of airfoil sections for aircraft applications.
- Be able to identify and explain the significance of the critical geometric parameters that define the shape of an airfoil.
- Know how to geometrically construct a NACA airfoil profile using a camberline shape and a thickness envelope.
- Understand the differences in the shapes between subsonic, transonic, and supersonic airfoil sections.

Some of the earliest known airfoil sections for aircraft concepts were patented in the 1880s by Horatio Phillips, as shown in the figure below, which were inspired by the wings of birds. Taking inspiration from nature is nothing new in the field of engineering. Notice the thin, highly cambered profile shapes compared to most modern airfoils. Phillips also tested these airfoils in one of the very first wind tunnels.

There are 1000s of airfoils in current use, most selected or otherwise adapted to optimize their performance for their specific aircraft application(s). Typical design requirements for airfoil sections include:

- Obtaining high values of the maximum attainable lift coefficient before flow separation and stall occur.
- The minimization of drag over a broad range of operating conditions.
- The attainment of a particular value of nose-up or nose-down pitching moment.
- The ability to reach high values of the lift-to-drag ratio, perhaps also at specific angles of attack.
- A high critical Mach number, i.e., the free-stream Mach number when supersonic flow first develops over the airfoil.

There has recently been much interest in designing efficient airfoils for use at the very low flow speeds and low Reynolds numbers found on UAV systems, which require detailed knowledge of boundary layer developments, including the process of laminar to turbulent boundary layer transition. Unfortunately, airfoil characteristics at low Reynolds numbers are usually quite different from those at higher Reynolds numbers, often showing remarkably low aerodynamic efficiencies.

# Shape of an Airfoil

The basic geometry of an airfoil is described in terms of a profile *shape* or *envelope* that defines the curvature of its upper and lower surfaces. As shown in the figure below, airfoils can be *symmetric*, which is an airfoil with the same shape and curvature on the upper and lower surfaces, or *cambered*, which has a different upper and lower surface shape. In addition, some airfoils have camber in which the trailing edge region has an upward or negative camber, called *reflex camber*, often used on flying wings, helicopters, and autogiros.

As shown in the following figure below, the critical length dimension of an airfoil profile is defined in terms of its chordline; the *chord* is defined as the distance measured from the leading edge of the airfoil profile to its trailing edge. However, in the geometric construction of airfoil profiles, it is necessary to be more precise about how exactly the profile shape is defined, including the value and position of the maximum thickness (thickness-to-chord ratio), the value and position of the maximum camber, as well as the nose shape or radius.

In most geometric constructions of airfoil profiles, the airfoil’s thickness envelope is defined so that the upper and lower camber surfaces of the envelope evolve if the thickness is plotted perpendicular to the slope of a defined *mean* camberline. The mean camber of the airfoil profile is a measure of its average curvature, and the shape and amount of the mean camber will also affect the shapes and curvature of the airfoil’s upper and lower surfaces. There is a formalized geometric process to trace out the envelope in terms of the coordinates of the upper and lower profile shapes, which can also be tabulated for various purposes such as plotting, the creation of a CFD grid, or for CAD/CAM. In addition, the leading-edge shape of the airfoil is often defined geometrically in terms of a nose radius, which also affects the airfoil’s aerodynamic characteristics.

Other geometric parameters of interest for airfoils are the maximum thickness and maximum camber, usually defined, again, as a ratio relative to the chord, i.e., the maximum thickness-to-chord ratio and the maximum camber ratio. The chordwise position of these latter parameters may also be defined and used to describe the shape of the airfoil profile, especially as they subsequently relate to the effects on the aerodynamic characteristics of the airfoil. For example, it is known that increasing the camber at the leading edge of an airfoil can increase its maximum lift coefficient to a point, but camber will also increase pitching moments.

# Early History of Airfoils

Historically, the most suitable airfoils for most practical engineering applications were obtained through an evolutionary process. In this regard, both theory and experimentation (e.g., wind tunnel testing) have been used to design airfoils to meet specific operating requirements for different aircraft types, including low-speed airplanes, high-speed airplanes, helicopters, propellers, wind turbines, etc. The computational tools to help design airfoils that produce specific aerodynamic characteristics first became available in the 1920s. The development of the thin-airfoil theory by Max Munk (in the U.S.) and Hermann Glauert (in the U.K.) during the 1920s led to a better understanding of how the camber affected an airfoil’s lift and pitching moments.

The problem of defining the airfoil pressure distribution for an airfoil with thickness and arbitrary shape was tackled by Theodorsen & Garrick in the early 1930s. The design of practical airfoil profiles was further aided by methods such as the conformal transformation first developed by Prandtl & Tietjens. This latter approach made it possible to compute pressure distributions and the resulting lift and pitching moment characteristics of some specially shaped *“Joukowski” airfoils*. The aerodynamic properties of Joukowski airfoils were measured in wind tunnel tests starting in the late 1920s at Gottingen in Germany and by the NACA in the U.S.A. from 1930 onward. The figure below shows a relatively rapid evolution of airfoil shapes tailored to aircraft applications between 1908 and 1944, with the thin and highly cambered airfoil sections used on early airplanes being relegated to history.

This experimental work to measure airfoil characteristics was soon followed by the first development of validated numerical methods to predict chordwise pressure distributions and airfoil characteristics without making as many measurements in the wind tunnel. Today, it is possible to predict the aerodynamic characteristics of airfoils with a high confidence level, with several popular computer codes such as XFoil being available. Even today, however, measurements of airfoil characteristics in wind tunnels have proven to be much more reliable than results from calculations, mainly when the airfoils operate at higher angles of attack, higher subsonic and transonic Mach numbers, or lower Reynolds numbers.

# NACA Method of Drawing Airfoil Shapes

As early as 1920, research institutions in Europe and the U.S.A. had embarked on the systematic measurement of the aerodynamic characteristics of airfoils that were already in practical use. This work led to the organization of the results into *families* of airfoils known to produce specific aerodynamic characteristics. By having a catalog of airfoils with measured aerodynamic characteristics, aircraft designers could quickly choose the most appropriate airfoil profile from the catalog for a given application.

The National Advisory Committee for Aeronautics, or the NACA, which is always pronounced as “N-A-C-A” and *never* “NACA,” conducted the most comprehensive and systematic study of the effect of airfoil shape on aerodynamic characteristics. Existing cambered airfoils, such as the Clark-Y and Gottingen sections, were known from the earliest wind tunnel experiments to have good aerodynamic characteristics. Therefore, the NACA used these airfoils as a basis; these two airfoils had geometrically similar profiles when the camber was removed, and the airfoils were reduced to the same thickness-to-chord ratio. A polynomial curve fit defined the resulting thickness shape, which became fundamental to many of the subsequent NACA airfoil families, i.e., what has become known as the classic NACA 00-series symmetric airfoils.

In the NACA method of defining the shape of an airfoil, a coordinate system is placed at the nose of the airfoil and is defined in terms of the and distances, as shown in the figure below. The airfoil profile is then constructed of a series of upper and lower points by using a thickness shape distributed around a camber line and by plotting the thickness perpendicular to the slope of the camberline, as detailed in the lower part of the figure below.

The basic steps of the process are straightforward geometry and easily programmed on the computer. If the slope of the camberline makes an angle with the chord line, as shown in the figure above, then the upper coordinate and lower coordinates of the airfoil can be obtained by using the equations

(1)

where and , i.e., coordinates non-dimensionalized with respect to the chord, . The slope angle, , of the camberline is given by

(2)

where or expresses the shape of the camberline. For airfoils with small camber, i.e., small values of surface slope angle , applying the thickness along the axis is a reasonable approximation.

The various upper and lower surface points can be exported to a data file, so the airfoil’s final shape is then obtained by connecting the points. Generally, more points will be needed in the nose region of the airfoil because of the higher curvature of the section. To give a reasonable approximation of the shape, 100 points should be used. 500 or 1,000 points across the chord may be needed for a CFD grid or a CNC machine file.

The nose curvature or radius must also be formally located on the profile and is obtained with an inscribed circle. The center for the leading edge radius (defined by a circle) is found by drawing a straight line through the end of the chord at the origin of the axes, but with a slope equal to the slope of the camberline at , and then marking off a distance along this line that is equal to the leading edge radius. This point then becomes the origin location for the leading-edge nose circle. For a symmetric airfoil, the center of this circle lies on the axis. The nose circle is then drawn and geometrically blended into the upper and lower surface coordinates.

Another way of drawing an airfoil section graphically is to draw a series of circles of radius (or ) if done non-dimensionally as a fraction of chord) centered on the camberline, as shown in the figure below. The upper and lower surfaces of the airfoil are then formed from curves drawn tangential to all of the circles. While this method offers a simple way of graphical construction, which can be a helpful approach in visualizing the overall process of drawing an airfoil section, it is best to construct the shapes of the surfaces and tabulate the data points, as mentioned previously.

The forgoing NACA approach also allowed the systematic construction of several families of airfoil sections differing only by a single geometric parameter, such as the location of maximum camber. The various families of airfoils developed by NACA were then tested in the wind tunnel to measure the effects of varying the important geometrical parameters on the lift, drag, and pitching moment characteristics as a function of angle of attack, as well as in some cases, the chord Reynolds number and Mach number. The primary geometric characteristics that affect the airfoil characteristics include the maximum camber and its distance aft of the leading edge and the leading-edge nose curvature (nose radius) airfoil. A summary of the results is documented in considerable detail in the NACA report (later a book) “Theory of Wing Sections: Including a Summary of Airfoil Data,” by Ira H. Abbott and A. E. von Doenhoff.

# Symmetric NACA Airfoils

Symmetric airfoil sections are often selected for horizontal and vertical tail surfaces on airplanes and other aircraft. The upper and lower surfaces of the NACA 00-series or four-digit symmetrical sections are described by the polynomial

(3)

where , , and , i.e., the geometry of the airfoil and the coordinates are expressed as a fraction of its chord or for . The coefficients through were obtained by a curve fit to the best-known airfoils when they were all reduced to the same thickness-to-chord ratio. They are given by

(4)

The shape of the airfoil is then obtained by plotting as a function of and for any number of points, at least 50 points and, more typically, 100 points will be required to define the airfoil shape to good fidelity. The corresponding leading-edge radius of the airfoil is , which is smoothly blended into the upper and lower surfaces as previously described. Examples of the NACA 00-series symmetric airfoils are shown in the figure below. The number denotes the thickness-to-chord ratio in percent of the chord; e.g., a NACA 0015 has a 15% thickness-to-chord ratio.

# Cambered NACA Airfoils

As previously discussed, cambered airfoils are constructed by distributing the thickness envelope (as defined above) around a mean camberline shape. The camberline can be specified as . Precisely, the thickness envelope is plotted perpendicular to the camberline to trace out the profiles of the upper and lower surfaces. There are many camberline profiles in the NACA portfolio, including the two-digit and three-digit camberlines, some examples of which are shown in the figure below. The first two digits define the amount of camber, and chordwise location of maximum camber, e.g., the NACA 2408 has a 2% camber, the maximum camber location is at 40% of the chord length, and the airfoil is 8% thick.

## NACA Two-Digit Camberlines

The simplest cambered airfoils are used to form the NACA 4-digit series, which is comprised of the standard NACA four-digit thickness envelopes and the following camberline based on two coefficients, i.e.,

(5)

where is the maximum camber (100 is the first of the four digits), is the location of the maximum camber, with 10 being the second digit in the NACA 4-digit airfoil description. The slopes of the camberline are

(6)

## NACA Three-Digit Camberlines

The NACA three-digit mean (camber) lines are also very popular and are given in this case in terms of three coefficients, i.e.,

(7)

where the coefficients of the camberline are given in the table below.

Mean Line | |||

210 | 0.05 | 0.0580 | 361.4 |

220 | 0.10 | 0.1260 | 51.64 |

230 | 0.15 | 0.2025 | 15.957 |

240 | 0.20 | 0.2900 | 6.643 |

250 | 0.25 | 0.3910 | 2.230 |

# Other NACA Airfoils

Modifications to the NACA four-digit and five-digit series of airfoil sections include reflex camber to produce zero pitching moment and changes in the nose radius and position of thickness to improve the maximum lift capability. The latter sections are denoted by a two-digit suffix, such as the NACA 0012-64 and NACA 23012-64. After the dash, the first integer indicates the relative magnitude of the nose radius, with a standard nose radius denoted by the number 6 and a sharp radius by 0. The second digit indicates the position of the maximum thickness in tenths of the chord.

The camberline for the NACA 3-digit 231-series reflexed airfoils are of some interest because they are designed to give zero-pitching moment about the 1/4-chord axis. In this regard, they are considered suitable for rotor blades (e.g., for a helicopter or an autogiro) because they need to keep torsional twisting moments on the blades to a minimum. The camberline of these airfoils is defined by

(8)

and

(9)

where , , , and .

Another set of NACA airfoils that have seen some use on various aircraft is the six-digit series. These airfoils were designed to achieve lower drag, higher drag divergence Mach numbers, and higher maximum lift coefficients. Their profiles are such that they are conducive to maintaining an extensive run of laminar flow over the leading-edge region, thereby lowering skin friction drag, at least over a range of angle of attack limited to low lift coefficients.

This latter goal is achieved using camberlines that produce a more uniform pressure loading from the leading edge to a distance . After that, the loading decreases linearly to zero at the trailing edge. The favorable pressure gradients tend to give the airfoils lower drag than other airfoils, at least over a limited range of attack angles. Unfortunately, surface contaminants or other transition-causing disturbances quickly spoil the characteristics of laminar flow types of airfoils, sometimes resulting in significant adverse characteristics.

Many designator combinations are used in the six-digit airfoil number system, which tends to become rather complicated. For example, consider the NACA 64-215 section. In this case, the number 6 denotes the airfoil series, and the number 4 represents the position of minimum pressure in tenths of the chord for the basic symmetric section. The number 3 denotes the range of lift coefficient in tenths above and below the design lift coefficient for which low drag may be obtained. The number 2 after the dash indicates a design lift coefficient of 0.2, and the number 15 denotes a 15% thickness-to-chord ratio.

# Grid Generation for CFD

The numerical generation of airfoil coordinates can also be used to generate input points, grids, or meshes to calculate their aerodynamic characteristics using programs like XFoil or other methods such as computational fluid dynamics (CFD). CFD grids are composed of discrete cells over which the conservation laws of fluid mechanics can be applied. An example of a grid about an airfoil section is shown in the figure below.

The resulting flow solution can then be used to calculate various properties around the airfoil, including local pressure, local Mach number, etc. CFD methods can also be used to design the shape of an airfoil to obtain a specified level of performance. However, this tends to be a lengthy process because of its iterative nature and slow numerical convergence. Nevertheless, the ability to design airfoil shapes on the computer, to a point, is much quicker than the repetitive testing of many prospective shapes in the wind tunnel.

Numerical grid generation for CFD solutions can take on a variety of types, including structured and unstructured. Structured grids are geometrically regular, whereas unstructured grids have more randomly generated points, which is a valuable approach that can reduce the computational time needed to find a flow solution. Several software tools are available to engineers to help create grids about particular airfoil shapes. The fidelity of the resulting aerodynamic solution strongly depends on the grid, especially the number of grid points, which can reach many millions. Of course, the numerical cost (and time) to obtain a solution increases commensurately with the number of grid points.

# Supersonic Airfoils

The well-rounded, cambered airfoil sections well-suited to subsonic flight speed are generally inappropriate for high-speed and supersonic flight. Supersonic airfoils are distinctive in their geometric shapes in that they are thin (i.e., have a low thickness-to-chord ratio) with sharp leading edges. Supersonic airfoils generally have thinner sections formed of either angled planes called *double-wedge airfoils* or opposed circular arcs called *biconvex airfoils*, as shown below.

The sharp leading edges on supersonic airfoils prevent the formation of a detached bow shock in front of the airfoil, which is a high source of drag called *wave drag*.

# Transonic Airfoils

Because commercial airliners have been designed to reach higher and higher cruise speeds approaching the speed of sound, i.e., for flight at transonic Mach numbers, this requirement has led to the design of a unique wing shape called a *supercritical* wing. A supercritical wing also uses a supercritical airfoil to reduce the strength of shock waves, thereby reducing wave drag. The principle used in transonic wing and airfoil design is to control the expansion of the flow to supersonic speed and its subsequent recompression. As shown in the figure below, a supercritical airfoil shape is distinctive. It has a point of maximum thickness fairly aft on the chord, with a relatively flat upper surface with a slight camber. However, such airfoils also tend to have significant camber at their trailing edges. Supercritical airfoils were extensively studied and refined during the 1960s by Richard Whitcomb and the NACA.

# Examples to Try

There are many airfoils to choose from, but for the student, it is valuable to understand the NACA method of geometric airfoil construction; it is very systematic, relatively easy in terms of the mathematics, and the algorithm lends itself naturally to the being programmed on the computer.

For example, the shape of a NACA 0018 airfoil takes no more than to plot the shape of the upper and lower surfaces using

(10)

where, in this case, .The corresponding leading-edge radius of the airfoil is .

The shape of the airfoil is obtained by plotting as a function of ; the results can be tabulated for any number of specified discrete points along the chord line, but 50 to 100 is usually enough, as shown in the two figures below. For practical reasons, all NACA airfoils have a finite thickness at the trailing edges.

The NACA 23018 airfoil section is a cambered airfoil comprised of the NACA 0018 thickness envelope (described above) wrapped around the NACA 230 mean camberline, the equations again being

(11)

where the slope of the camberline is .

The resulting NACA 23018 airfoil is shown in the two figures below. It is apparent on the enlarged plot of the nose region that the leading-edge part of the nose radius protrudes very slightly forward of the origin at ; this is an artifact of the construction technique and is of no practical significance when building a wing with such an airfoil.

A MATLAB code to draw these airfoil shapes is given below. The student is encouraged to try using other NACA camberlines better to understand the drawing process of NACA airfoil shapes.

MATLAB Code to Draw a Cambered NACA 230-Series Airfoil

# Summary & Closure

Using the most suitable airfoil section or sections is fundamental to the success of the design of a wing as a whole or for other applications such as rotary-wing concepts and UAVs. To this end, many different types of airfoil sections have been geometrically tailored to give the best aerodynamic performance in the conditions of flight. For example, thicker and more cambered airfoils with rounded nose shapes are more suitable for slower flight speeds and low Mach numbers. In contrast, very thin airfoils with sharp leading edges are more suitable for high speeds and supersonic Mach numbers. In addition, special supercritical airfoil sections have been developed for transonic flight conditions, where many airliners fly, to reduce wave drag and prevent boundary layer separation behind the shock wave, allowing airliners to cruise closer to the speed of sound.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

- Do some research into laminar flow airfoil sections. What are particular geometric features incorporated into these airfoils to produce laminar flow over the surfaces?
- What kinds of airfoil shapes are likely to be used for supersonic flight? Are there any NACA supersonic airfoil sections?
- Research the types of airfoils used on propeller blades. Why are different airfoils with different thicknesses along the blade span used on propellers?
- What airfoils are likely to be used on wind turbines, and why?
- What are the main parameters used to describe the geometry of an airfoil section?
- Can you describe the concept of the chord line and camber of an airfoil section?

Other Useful Online Resources

There are many more resources on airfoils to explore:

- All about airfoils on Wikipedia.
- The University of Illinois at Urbana-Champagne has complied airfoil coordinates and related links. Here is the guide to many uses from the UIUC Airfoil Data site.
- Go here for some interesting airfoil drawing tools.
- A NASA site as to how the shape of an airfoil affects lift.
- YouTube video on how airfoil shapes can affect lift generation.
- Airfoil shape basics video on YouTube.
- Information about grids and CFD calculations about airfoils.
- Learn about Richard Witcomb’s contributions to transonic airfoil design here.