Aerodynamics of Airfoil Sections

J. Gordon Leishman

doi:https://doi.org/10.15394/eaglepub.2022.1066.n23

30 Aerodynamics of Airfoil Sections

Introduction

Understanding the aerodynamic behavior of airfoils and wings (often referred to as lifting surfaces) is a significant part of the practice of aerospace engineering, and this understanding is critical to the successful design of all aircraft. Any lifting surface that is moved through a fluid will create some form of fluid-dynamic force upon it. By definition, the component of this force that acts on the surface in a direction perpendicular to the relative free-stream velocity, $V_{\infty}$ , or “relative wind direction”^[1] is called the lift, as shown in the figure below. The force component on the surface in a direction parallel to the relative wind direction is called the drag. The magnitude of the lift and drag forces depends on many factors, including the size and shape of the lifting surface and its orientation to the flow, as well as the flight Reynolds number (based on a characteristic length) and free-stream Mach number. Moments can also be produced on the lifting surface about each of the longitudinal, lateral, and vertical axes.

A lifting surface such as a wing will experience both forces and moments, depending on its shape, size, angle of attack, Mach number, and Reynolds number.

However, before examining the characteristics of finite wings, i.e., three-dimensional wings with finite span and perhaps with twist, planform taper, and thickness variations, it is prudent to investigate the aerodynamic characteristics of two-dimensional airfoil sections. Such two-dimensional airfoils are equivalent to wings of infinite span and aspect ratio, the aspect ratio indicating the slenderness of the wing. While the concept of a “two-dimensional” wing section may initially sound somewhat artificial, it is possible to mimic a wing of infinite aspect ratio, both experimentally and theoretically, and obtain aerodynamic results that pertain only to the shape of the airfoil section itself. Furthermore, this approach makes it possible to isolate the other more complicated and interrelated effects associated with the finite span of a wing, including the impact of the wing tip vortices and other aerodynamic effects caused by sweepback, twist, planform (chord) variations, and potentially other things such as winglets.

Learning Objectives

Be conversant with the various definitions of aerodynamic forces and moments, as well as lift coefficient, drag coefficient, lift-curve slope, maximum lift coefficient, aerodynamic center, and center of pressure.
Know about the aerodynamic characteristics of airfoil sections, both in attached flow and with flow separation, and how these characteristics change at different Reynolds and Mach numbers.
Appreciate the effects of flaps and other high-lift devices.
Understand how to calculate the lift and other integrated quantities from the pressure and shear stress distributions about a body.
Know the differences between subsonic, transonic, and supersonic airfoil sections.

Origin of Aerodynamic Forces

The origin of the net aerodynamic forces on an airfoil or wing, such as lift and drag, comes from the integrated effects of the pressure and the boundary layer shear stress distributions acting over its surface, as shown in the figure below. These distributions, which are produced as the flow moves over the upper and lower surfaces, are not uniform and can be positive or negative, e.g., a higher pressure pushing inward toward the surface (as shown in red) or a lower pressure pulling outward away from the surface (shown in green). In addition, boundary layer-induced shear stresses can be positive (flow moving downstream) or negative (e.g., reversed flow).

It can be deduced from the figure above that the shear stresses, in the aggregate, will act in the downstream direction primarily parallel to the chord line so that the net shear will contribute significantly to the drag force on the airfoil section. Likewise, the differences in the pressure distribution between the upper and lower surfaces will contribute primarily to the lift force and pitching moment on the airfoil, the shear stresses having a minor net contribution in the vertical direction.

In practice, the net forces and moments can be measured with a balance (e.g., a scale) or obtained by suitable integration of the effects of the pressures and stresses that act around the surfaces. In the meantime, it is possible to proceed under the assumption that either measurement or calculation can be used to obtain these integrated results, a particular approach being detailed at the end of this chapter.

Two-Dimensional Flow

It should be appreciated that the flow over any wing of finite span will be inherently three-dimensional and further complicated by the effects of the vortices that trail behind the wing, as shown in the figure below. The presence of these vortices produces a downwash flow over the wing, affecting the local angles of attack over the entire wing and, therefore, its lift and overall aerodynamic characteristics. The flow can be assumed nominally two-dimensional only at sections well away from the wing tip vortices.

The aerodynamics of a finite wing are very three-dimensional, in part because of the effects of the wing tip vortices. The flow can be assumed nominally two-dimensional only at sections well away from the wing tip vortices.

In the first instance, it seems obvious to understand the aerodynamic behavior of a wing without the influence of the tip vortices. To that end, it is possible to think of a case where the wing span and corresponding aspect ratio become infinitely large. Under these conditions, the effects of the tip vortices are moved so far away from the central part of the wing that they will have a negligible impact, the principle being shown in the figure below. Of course, in a practical sense, a wing of infinite span is impossible. Still, with higher aspect ratios, the effects of the wing tip vortices are sufficiently far removed that an airfoil at mid-span behaves effectively as if it were in a two-dimensional flow. Therefore, it is possible to mimic two-dimensional wings theoretically and experimentally, deriving a better understanding of the aerodynamics of the airfoil section by itself.

An airfoil section can be considered a slice of a three-dimensional wing of infinite aspect ratio. The effects of the wing tip vortices are sufficiently far removed that the flow at mid-span behaves as if it were two-dimensional.

Without the effects of the wing tip vortices, a single cross-section of the wing, especially near its mid-span, will behave in a nominally two-dimensional manner, which now forms the basis for further discussion. The word “nominally” must be used in this context because it must be recognized that an actual flow can never be strictly two-dimensional, no matter how large the span or aspect ratio can reach in theory or practice. Nevertheless, the literal meaning is that all three-dimensional effects, for all practical purposes, will have been reduced to the point that they can be considered negligible.

It is also possible to mimic an infinite span and aspect ratio in the wind tunnel. One approach is to span the wing from wall to wall, thereby eliminating the tip vortices. Another common practice is to test a short-span wing between two “false” walls, as shown in the photograph below. In wind tunnel terminology, this is called a “two-dimensional insert.” This particular approach has seen widespread use for testing two-dimensional airfoils, and flow visualization has confirmed the validity of the nominally two-dimensional flow developments over the wing section at the mid-span, at least up to the onset of the stall. Post-stall, the flow always tends to become inherently more three-dimensional on any lifting surface, regardless of how it is being tested.

A “two-dimensional insert” in a wind tunnel is used to measure representative two-dimensional aerodynamics of an airfoil.

Dynamic Pressure

The aerodynamic forces acting on a body are directly proportional to dynamic pressure in the flow, which is the pressure associated with the “dynamics” or kinetic energy of fluid movement. Given the symbol $q$ , dynamic pressure comes into most (if not all) aerodynamic problems. In general, the dynamic pressure is given by

(1) $\begin{equation*} q = \frac{1}{2} \varrho V^{2} \end{equation*}$

where $\varrho$ is the flow density, and $V$ is its velocity. Notice that dynamic pressure depends on the squared value of the flow speed. It can be easily confirmed that dynamic pressure has units of pressure, i.e., force per unit area so that it will be expressed in base units of Nm $^{-2}$ (Pa) in SI or lb ft $^{-2}$ in USC.

In particular, the free-stream dynamic pressure is defined as

(2) $\begin{equation*} q_{\infty} = \frac{1}{2} \varrho_{\infty} V_{\infty}^{2} \end{equation*}$

where $\infty$ means conditions at “infinity” or just far away from the wing where there is an undisturbed “free-stream” flow. In practice, this point will be at least one chord length upstream of the airfoil. As will be shown, the free-stream dynamic pressure $q_{\infty}$ is often used as the reference pressure in the definition of most of the non-dimensional coefficients used in airfoil and wing aerodynamics.

Airfoil Forces and Moments

The resulting forces acting on an airfoil section from the integrated effects of the pressure and shear can be resolved into a wind-axis system (i.e., in terms of lift $L'$ and drag $D'$ ) or a chord-axis system (i.e., in terms of normal force $N'$ and a chord force $A'$ ), as shown in the figure below. As previously discussed, an airfoil section can be considered a “two-dimensional” wing. Therefore, for airfoils, dimensional force and moment quantities per unit span are used, i.e., the lift per unit span would be written as $L$ /unit span = $L'$ , i.e., the lift force acting over a span of one unit. Furthermore, when force and moment coefficients are defined for airfoils, a reference length is needed, which is usually the chord length. In some cases, the semi-chord, $c/2$ , may be used as a reference length, but this is less common other than in some advanced applications.

The resultant forces on an airfoil at an angle of attack $\alpha$ . Two statically equivalent axes systems may be used.

By definition, the lift force $L$ or lift force per unit length, $L'$ , acts in a direction that is perpendicular to the free-stream velocity, $V_{\infty}$ . The corresponding drag $D$ or drag per unit span, $D'$ , is in a direction parallel to $V_{\infty}$ . Alternatively, the forces can be decomposed into the sum of two other forces, i.e., the normal force per unit span, $N'$ , which acts normal (perpendicular) to the airfoil chord, and the leading-edge suction force or axial chord force per unit span, $A'$ , which points toward the trailing-edge and acts parallel to the chord. Notice, however, that in some cases, the axial or chord force may be defined as positive when pointing toward the leading edge, so it is essential to know the sign convention being used.

Both of these latter force systems are useful in various analyses, although there is usually a preference for using lift and drag. It will be apparent from the preceding that the wind axis and body axis systems are statically equivalent, and one force system can be derived from the other through resolution using the angle of attack $\alpha$ . For example, to transform from a body axis system ( $N'$ and $A'$ ) to a wind axis system ( $L'$ and $D'$ ), then

(3) $\begin{eqnarray*} L' & = & N' \cos \alpha - A' \sin \alpha \\[8pt] D' & = & N' \sin \alpha + A' \cos \alpha \end{eqnarray*}$

Alternatively, with the use of some trigonometry, transforming from a wind axis system ( $L'$ and $D'$ ) to a body axis system ( $N'$ and $A'$ ) gives

(4) $\begin{eqnarray*} N' & = & L' \cos \alpha + D'' \sin \alpha \\[8pt] A' & = & D' \cos \alpha - L ' \sin \alpha \end{eqnarray*}$

Force and Moment Coefficients

Engineers must become familiar with how non-dimensional force and moment quantities are defined and used for airfoils, wings, and other body shapes and what particular values of these coefficients mean. While dimensional values forces (lb or N) and moments (ft-lb or N-m) are helpful in many forms of analysis, it is far more convenient to work with non-dimensional aerodynamic quantities, such as lift and drag coefficients.

For example, suppose the lift on a given wing is 900 N, and the corresponding drag is 30 N; these values are undoubtedly helpful to know. However, consider if the wing’s corresponding lift coefficient is 0.6 and its drag coefficient is 0.02. In that case, these values reveal much about the aerodynamic operating state of the airfoil, and the coefficients also allow comparisons of the effects of wings of different shapes and sizes. For this reason, airfoil data (measured or computed) are generally presented in terms of non-dimensional force and moment coefficients.

Physical interpretation of the sectional aerodynamics in terms of per unit span.

The airfoil section’s non-dimensional or dimensionless force coefficients are defined using the free-stream dynamic pressure $q_{\infty}$ and chord $c$ as a reference length. The span of a two-dimensional wing is infinite, so the aerodynamic forces and moments are referred to as per unit length or unit span of the wing, i.e., span = 1 unit, which can be interpreted from the figure below. A force is pressure times area, so the reference area of a force per unit span will be $S = c \times b (= 1) = c$ .

Therefore, the two-dimensional lift coefficient can be defined as

(5) $\begin{equation*} C_{l} = \displaystyle{\frac{L}{q_{\infty} \, c \, (1) }} = \displaystyle{\frac{L/ \mbox{\small unit~span}}{q_{\infty} c}} = \displaystyle{\frac{L'}{q_{\infty} c}} = \displaystyle{\frac{L'}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 c}} \end{equation*}$

where $c$ is the chord of the wing section or the airfoil, which is the distance from its leading edge to its trailing edge. Furthermore, it is assumed that the spanwise lift is uniform from tip to tip, consistent with the two-dimensional assumption.

In summary, the two-dimensional force and moment coefficients can be written as:

Lift coefficient, $C_{l} = \displaystyle{\frac{L/ \mbox{\small unit~span}}{q_{\infty} c}} = \displaystyle{\frac{L'}{q_{\infty} c}} = \displaystyle{\frac{L'}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 c}}$

$\vspace*{1pt}$ Drag coefficient, $C_{d} = \displaystyle{\frac{D/ \mbox{\small unit~span}}{q_{\infty} c}} = \displaystyle{\frac{D'}{q_{\infty} c}} = \displaystyle{\frac{D'}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 c}}$

$\vspace*{1pt}$ Normal force coefficient, $C_{n} = \displaystyle{\frac{N/ \mbox{\small unit~span}}{q_{\infty} c}} = \displaystyle{\frac{N'}{q_{\infty} c}} = \displaystyle{\frac{N'}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 c}}$

$\vspace*{1pt}$ Axial (chord) force coefficient, $C_{a} = \displaystyle{\frac{A/ \mbox{ \small unit~span}}{q_{\infty} c}} = \displaystyle{\frac{A'}{q_{\infty} c}} = \displaystyle{\frac{A'}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 c}}$

Pitching moments are defined as positive when the moment tends to increase the angle of attack of the airfoil section, i.e., positive pitching moments are equivalent to nose-up moments. Again, the convention is that a moment $M'$ would represent a moment per unit span. Also, there are several convenient points about which the moments can be conveniently calculated, namely the leading-edge ( $x=0$ ), 1/4-chord ( $x = c/4$ ), and the center of pressure $x = x_{\rm cp}$ , as shown in the figure below. Notice that when the reference point is moved to different locations on the chord, the values of the moments will change, but the forces acting at that point remain the same. i.e., there is no change in static force equilibrium.

Moments on an airfoil can be taken about any convenient point; the transformation needs the application of statics.

The dimensionless moment coefficients for an airfoil section are defined as:

Moment coefficient at the leading-edge, $C_{m_{\rm LE}} = \displaystyle{\frac{M_{\rm LE}/ \mbox{\small unit~span}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_{\rm LE}}{q_{\infty} c^2}} = \displaystyle{\frac{M'}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 c^2}}$

$\vspace*{1pt}$ Moment coefficient at some point $a$ , $C_{m_{a}} = \displaystyle{\frac{M_a/ \mbox{\small unit~span}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_a}{q_{\infty} c^2}} = \displaystyle{\frac{M'_a}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 c^2}}$

$\vspace*{1pt}$ Moment coefficient at 1/4-chord, $C_{m_{1/4}} = \displaystyle{\frac{M_{1/4}/ \mbox{\small unit~span}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_{1/4}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_{1/4}}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 c^2}}$

$\vspace*{1pt}$ Moment coefficient at center of pressure, $C_{m_{\rm cp}} = \displaystyle{\frac{M_{\rm cp}/ \mbox{\small unit~span}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_{\rm cp}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_{\rm cp}}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 c^2}}$

Notice the use of lower-case subscripts on all coefficients (e.g., $C_l$ , not $C_L$ and $C_d$ , not $C_D$ ) when applied to a “2-dimensional” airfoil section. When applied to a finite span wing, capital letter subscripts are used, as discussed later; this standard notation distinguishes the finite-span aerodynamic coefficients from the two-dimensional ones.

Representative Force & Moment Coefficients

The data shown in the plots below (indicated by the symbols) show measured values of the lift coefficient, $C_l$ , the drag coefficient, $C_d$ , and the 1/4-chord moment coefficient $C_{m_{1/4}}$ as functions of the angle of attack for a NACA 23012 airfoil section in a low-speed flow. In this latter regard, a low-speed flow one at low Mach numbers where compressibility effects are relatively small. These results cover an angle of attack range from fully attached flow into the stalled flow condition. They are typical of airfoil characteristics found in various standard catalogs, e.g., Abbott & Von Doenhoff. The results are often shown with respect to variations of the chord-based Reynolds number and/or the free-stream Mach number.

Variations of $C_l$ and 1/4-chord moment coefficient, $C_{m_{1/4}}$ . versus angle of attack for a NACA 23012 airfoil, including the effects of surface roughness. The drag coefficient measurements, $C_d$ , are shown as a “polar” with respect to the values of the lift coefficient, $C_l$ .

Quantities such as maximum lift coefficient, minimum drag coefficient, maximum lift-to-drag ratio, pitching moments, and other metrics can all be significant in quantifying the aerodynamic characteristics of airfoils. These parameters can also be a basis for airfoil selection. However, interpreting the results on these preceding graphs and finding relevant quantities takes some practice because many points and curves almost overlap, which can be confusing. To this end, a more straightforward presentation is shown in the figure below, in which the lift coefficient $C_l$ , the moment coefficient about the 1/4-chord, $C_{m_{1/4}}$ , and the drag coefficient, $C_d$ , versus the angle of attack of the airfoil to the free-stream flow, are more clearly delineated.

Representative variations of the lift coefficient $C_l$ , the moment coefficient about the 1/4-chord, $C_{m_{1/4}}$ , and the drag coefficient, $C_d$ , versus the angle of attack.

At low angles of attack, the lifting characteristics of an airfoil are not substantially influenced by viscosity and the presence of the boundary layers. Recall that boundary layers are thin, viscous-dominated regions near the surface. Notice that in this region, the lift coefficient increases almost proportionally with the angle of attack. The parameter $\alpha_0$ is the angle of attack for zero lift or what is usually known as the zero-lift angle of attack. For a symmetric airfoil, $\alpha_0 = 0$ ; for positively cambered airfoils, $\alpha_0$ is usually a small negative angle.

As the angle of attack increases further, the boundary layer thickens, and the aerodynamic characteristics start to become non-linear with respect to the angle of attack. The airfoil will soon reach its maximum lift coefficient, $C_{l_{\rm max}}$ . A further increase in the angle of attack is followed by the onset of flow separation on the upper surface, a sudden loss of lift, and a significant increase in drag; the process is called a stall.

The essential flow physics of stall has been previously discussed and is illustrated schematically in the figure below. At low subsonic Mach numbers, the onset of stall usually occurs at an angle of attack between 12 $^{\circ}$ and 15 $^{\circ}$ depending on the airfoil section and the Reynolds number. Higher Reynolds numbers inevitably delay the onset of flow separation and stall, increasing the value of $C_{l_{\rm max}}$ . Higher Mach numbers generally reduce the values of $C_{l_{\rm max}}$ because of the higher adverse pressure gradients over the upper surface.

The development of a thickening boundary layer and the onset of flow separation will eventually limit the lift production on an airfoil and will also increase its drag, which is called “stall.”

Pitching moments about the 1/4-chord are usually relatively low on most airfoils, but for a significantly cambered airfoil, as in the case shown, the moments will be non-zero. Often, the moment curve has a shallow positive slope because the aerodynamic center (defined later) is close to but just forward of the 1/4-chord.

Check Your Understanding #1 – Obtaining airfoil characteristics from graphs

The graphs below show the aerodynamic characteristics of a NACA 2412 airfoil section directly from Abbott & Von Doenhoff., i.e., the lift coefficient $C_l$ , the drag coefficient $C_d$ , and the pitching moment coefficient about the 1/4-chord axis $C_{m_{1/4}}$ . Use these graphs to find for a Reynolds number of 5.7 x 106 for both the smooth and rough surface cases: 1. The zero-lift angle of attack, $\alpha_0$ . 2. The maximum lift coefficient, $C_{l_{\rm max}}$ , 3. The stall angle of attack, $\alpha_s$ , 4. The minimum drag coefficient $C_{d_{0}}$ and the lift coefficient at which it occurs.

Measured values of lift and drag coefficient shown plotted on a graph.

Show solution/hide solution.

For the smooth airfoil at Reynolds number of 5.7 x 10⁶:

The legend on the graphs shows that for a Reynolds number of 5.7 x 10⁶, the results are marked by square symbols. Referring to the left side plot, this zero-lift angle $\alpha_0$ for the smooth airfoil is about -2.1^o.
The maximum lift coefficient $C_{l_{\rm max}}$ is about 1.68 for the smooth airfoil.
For the smooth airfoil, the corresponding stall angle of attack $\alpha_s$ is about 17.6^o.
These values can be found from the right side plot, the minimum drag coefficient $C_{d_{0}}$ for the smooth airfoil being about 0.06 at a lift coefficient of 0.4.

For the rough airfoil at Reynolds number of 5.7 x 10⁶:

The legend shows that triangular symbols mark the results in this case, so referring to the left side plot, this zero-lift angle $\alpha_0$ for the rough airfoil is unchanged at about -2.1^o.
The maximum lift coefficient $C_{l_{\rm max}}$ is less and is about 1.21 for the rough airfoil.
The corresponding stall angle of attack $\alpha_s$ is about 15.1^o for the rough airfoil.
These values can be found from the right side plot, the minimum drag coefficient $C_{d_{0}}$ being higher than for the smooth airfoil and is about 0.098 at a lift coefficient of 0.18.

The drag curves show that an airfoil’s drag coefficient remains relatively low and reasonably constant below the onset of the stall. However, the effects of surface roughness, in which the airfoil’s leading edge is roughened up to represent the in-service degradation of a wing’s surface, are different. Roughness always causes a boundary layer to become turbulent almost immediately after it is formed, thereby creating a higher average drag and a more rapid increase in drag with an increasing angle of attack.

Lift Characteristics

At lower angles of attack below the stall, it has been pointed out that the lift coefficient on the airfoil section is almost proportional to its angle of attack and the local dynamic pressure. This linear relationship is exact within the framework of what is known as linearized inviscid aerodynamic theory. The lift per unit span can be written as

(6) $\begin{equation*} L' = \frac{1}{2} \varrho_{\infty} V_{\infty}^{2} c C_l = \frac{1}{2} \varrho_{\infty} V_{\infty}^{2} c C_{l_{\alpha}} \left( \alpha - \alpha_0 \right) \end{equation*}$

where $C_{l_{\alpha}}$ is called the lift-curve slope (i.e., of the slope of the lift curve in the linear part of the graph) and is measured in units per degree or radian angle of attack. It is a common mistake to assume that the value of $C_{l_{\alpha}}$ is dimensionless, but remember, it has units per angle of attack, i.e., $/^{\circ}$ or $/$ rad. At low Mach numbers $C_{l_{\alpha}} \approx 2\pi$ per radian or 0.11 per degree.

In coefficient form, then

(7) $\begin{equation*} C_l = C_{l_{\alpha}} \left( \alpha - \alpha_0 \right) \end{equation*}$

Remember that the preceding relationship in Eq. 7 comes under the category of linearized aerodynamics. In fact, for most airfoils, it is found that the lift coefficient varies linearly to within about 10% of that given in Eq. 7 up to an angle of about 10 to 12 degrees, depending on the Mach number and Reynolds number, i.e., up to the point of stall.

Check Your Understanding #2 – Calculating lift coefficients

Assuming a constant lift-curve slope of 6.1 per radian angle of attack for a two-dimensional airfoil, calculate the lift coefficients at 5, 10, and 15 degrees angle of attack for: 1. A symmetric airfoil; 2. A positively cambered airfoil with a zero lift angle of attack of -1.2 degrees.

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The relevant equation for the lift coefficient is

$C_l = C_{l_{\alpha}} \left( \alpha - \alpha_0 \right)$

The lift-curve slope $C_{l_{\alpha}}$ is given as 6.1 per radian angle of attack. Notice that 360 $^{\circ} = 2\pi$ radians so 1 $^{\circ} = \pi/180$ radians, i.e., $C_{l_{\alpha}}$ = 6.1 per radian is equal to 0.1065 per degree.

$C_l = C_{l_{\alpha}} \left( \alpha - \alpha_0 \right) = 0.1065\left( \alpha - 0^{\circ} \right) = 0.1065 \alpha$ . For 5 $^{\circ}$ then $C_l$ = 0.532, for 10 $^{\circ}$ then $C_l$ = 1.065, and for 15 $^{\circ}$ then $C_l$ = 1.597.
$C_l = C_{l_{\alpha}} \left( \alpha - \alpha_0 \right) = 0.1065\left( \alpha - (-1.2^{\circ})\right) = 0.1065 \left( \alpha + 1.2^{\circ} \right)$ . For 5 $^{\circ}$ then $C_l$ = 0.660, for 10 $^{\circ}$ , then $C_l$ = 1.193, and for 15 $^{\circ}$ then $C_l$ = 1.725.

The simplicity of the relationship in Eq. 7 is advantageous in various engineering analyses, although its limitations to airfoil operation below stall must be recognized. However, it should be appreciated that no simple equations can be used to represent the variation of the aerodynamic coefficients in the post-stall regime because they vary non-linearly with the angle of attack.

Effect of Flaps

The figure below shows the effects of a flap on the lift characteristics of an airfoil. First, notice that the flap deflection introduces a form of trailing-edge camber, so the lift curve shifts to the left, giving a much lower zero lift angle of attack, changing from about -2^o to nearly -16^o. Second, notice the significant increase in the maximum lift coefficient from about 1.6 to 2.6 before the airfoil stalls. Such substantial increases before the onset of stall are beneficial because they will allow a wing to fly at a lower airspeed. Therefore, flaps will significantly decrease an aircraft’s takeoff and landing distances. Notice, however, that there is also a decrease in the stall angle of attack with flaps.

The effects of a flap on the lift coefficient produced by an airfoil section.

Further increases in maximum lift coefficient may be possible using different types of flaps, such as slotted flaps, Fowler flaps, and double-slotted or triple-slotted flaps. In conjunction with leading edge slats, significant stall speed reductions can be achieved with larger airliner types of airplanes. For an airplane, however, the penalties for using more complicated flap systems are increased airframe weight, costs, and maintenance.

Drag Characteristics

The variation of the drag with the angle of attack and/or with the lift coefficient is of great interest because of its essential effects on aircraft performance, i.e., the drag generally always acts to diminish the aircraft’s performance. Some representative results are shown in the figure below in terms of drag coefficient $C_d$ versus lift coefficient $C_l$ for both a conventional airfoil (in this case, the NACA 23012) and a so-called “laminar flow” airfoil (a NACA 63-series airfoil). The variation of the drag coefficient for the NACA 23012 airfoil is typical. The drag stays low until the angle of attack and the corresponding lift coefficient have increased to the point that significant boundary layer thickening occurs. Then, the drag grows rapidly when a stall occurs.

Representative variations of drag coefficient of conventional and laminar flow airfoils as a function of lift coefficient.

Notice that the effects of surface roughness substantially increase the values of drag. Although these particular results are from wind tunnel tests, the effects of roughness, in this case, can simulate the typical “in-service” abrasion of the leading edge of wings compared to when they were new from the factory (and so were initially smoother), the effects of roughness are significant. In airplane design, it is necessary to consider surface roughness effects so that the aircraft’s potential performance is not overestimated by assuming a perfectly smooth airfoil shape that is unlikely to be unrealizable in practical flight operations.

In the lower angle of attack regime, the drag coefficient on an airfoil can be represented by the equation

(8) $\begin{equation*} C_d = C_{d_{0}} + C_{d_{1}} \, \left( \alpha - \alpha_0 \right) + C_{d_{2}} \, \left( \alpha - \alpha_0 \right)^2 \end{equation*}$

where $C_{d_{0}}$ , $C_{d_{1}}$ , and $C_{d_{2}}$ are empirically derived coefficients (obtained through curve fitting) to drag measurements for a specific airfoil at a given Mach number and Reynolds number. Because in the linear regime then $C_l = C_{l_{\alpha}} \left( \alpha - \alpha_0 \right)$ , then it is also possible to write that

(9) $\begin{equation*} C_d = C_{d_{0}} + d_1 \, C_l + d_2 \, {C_l}^2 \end{equation*}$

Again, it is essential to recognize that this preceding equation can only represent airfoil characteristics below stall; they are invalid with any significant amounts of flow separation or in the stalled flow regime. In the case of a symmetric airfoil (where $\alpha_0 = 0)$ then $C_{d_{1}}$ and $d_1$ will be zero.

Also shown in the figure above are results for a NACA 63-series laminar flow airfoil, which is a design that maximizes the extent of the laminar boundary layer over the leading edge of the airfoil, hence reducing skin friction drag. These airfoils (with very smooth surfaces) tend to produce low drag “buckets” where the drag is relatively lower, but typically only over a limited range when the airfoil operates at low angles of attack and low lift coefficients.

The geometric shapes of these laminar flow airfoils tend to have a point of maximum thickness much closer to the 1/2-chord than more conventional airfoils, which will have a maximum thickness nearer to the 1/4-chord, as shown in the figure below. This geometric feature produces a favorable pressure gradient over more of the leading edge, thereby encouraging the boundary layer to be laminar for longer. The downside is that such airfoils typically produce lower values of maximum lift coefficient, i.e., stall occurs at lower angles of attack.

Shapes of two NACA low-drag “laminar flow” airfoil sections compared with the NACA 23012 airfoil as a baseline.

It can also be inferred from the previous discussion that surface roughness encourages turbulent mixing, destroying the laminar boundary layer in short order. The resulting increase in skin friction drag makes the airfoil perform comparably (or sometimes worse) than a conventional airfoil. This latter reason is why laminar flow airfoil sections are challenging to use successfully in practice because some surface roughness on the wing is inevitable.

However, such airfoils have seen better success for us on the smooth, almost glass-like wings of sailplanes, although the wings must be polished and always be kept completely clean and free of bugs, etc., for laminar flow to prevail. Nevertheless, obtaining an extensive laminar flow region on an airfoil can profoundly reduce its drag, at least over some smaller range of angles of attack and lift coefficients. Unfortunately, a practical and robust means for achieving extensive laminar flow regions on an airplane wing remains a research challenge, although certain surface coatings have been found to help.

Drag Polars

Types of graphical presentations of $C_l$ versus $C_d$ or $C_d$ versus $C_l$ are often called a “drag polar” or just a”polar,” an example being shown in the figure below. Polars are a helpful way of presenting airfoil section characteristics because results for different operating conditions (e.g., different Reynolds numbers of Mach numbers) or different airfoils can be readily compared and contrasted on a single plot.

An example of a “drag polar” for a two-dimensional airfoil.

One advantage of the $C_l$ versus $C_d$ form of the presentation shown above is that the slope of a straight line running from the origin (0, 0) of the graph to any point on the polar is a measure of the lift-to-drag ratio and hence is a quantitative measure of the airfoil’s aerodynamic efficiency. Notice that the tangent to the polar gives the highest slope. So this point will be the best lift-to-drag ratio, i.e., the operating conditions when the airfoil reaches its highest aerodynamic efficiency. Notice also that there is only one angle of attack and operating lift coefficient for best efficiency, which depends on the particular airfoil shape and its operating Reynolds number and Mach number.

Effects of Reynolds Number

The classic NACA airfoil measurements also indicate some effect of Reynolds number, at least over a limited range, but all the results are higher than one million. Remember that for an airfoil, the Reynolds number will be based on the chord, $c$ , so

(10) $\begin{equation*} Re_c = \frac{\varrho_{\infty} \, V_{\infty} \, c}{\mu_{\infty}} \end{equation*}$

The “c” subscript on the Reynolds number is often dropped, so just $Re$ is used. Flight vehicles typically experience higher chord Reynolds numbers, often as high as $10^{8}$ or “one-hundred million.” Notice that the Reynolds number is frequently quoted in terms of millions. For example, for $Re = 3 \times 10^6$ , the value of the Reynolds number is usually referred to as “a Reynolds number of three million.”

As the Reynolds number decreases below one million ( $Re = 10^6$ ) and the effects of viscosity begin to manifest more significantly compared to inertia effects in the flow, the lift and drag characteristics begin to change profoundly. The aerodynamic characteristics of airfoils at lower chord Reynolds numbers are particularly important as they affect the performance of many smaller-scale flight vehicles, such as UAVs and drones, which may have chord-based Reynolds numbers well below a million. The data in the figure below shows the profound effects of operating airfoils at lower Reynolds numbers, which in this case range from as low as 20,000 up to 3,000,000 (three million).

Variations of lift coefficient $C_l$ with the angle of attack for a NACA 4412 airfoil when operating at different Reynolds numbers.

At the highest Reynolds number of $3 \times 10^6$ , the lift coefficient varies almost linearly with the angle of attack, and the drag is nominally constant, which is typical of airfoil behavior at higher Reynolds numbers, as previously discussed. However, as the Reynolds number decreases below a million, the lift and drag curves become much more “rounded” and eventually become significantly nonlinear for variations in the angle of attack. Indeed, these effects start to become pronounced even at Reynolds numbers of 500,000.

This latter behavior directly results from the relatively thicker boundary layers on the airfoil surfaces at these lower Reynolds numbers. Furthermore, the development of a long laminar separation bubble (LSB) on the airfoil is often a source of these non-linear characteristics, as shown in the figure below. In these cases, the airfoil’s lift (and drag) behavior becomes much more challenging to generalize as a function of the angle of attack, i.e., the relationship in Eq. 7, for example, is no longer applicable. In general, at lower Reynolds numbers where $R\!e < 10^5$ , the lower momentum in the boundary layer flow leads to thicker boundary layers and higher drag. Naturally, this type of flow physics presents challenges in measuring lift and drag in the wind tunnel, where careful ensemble averaging should be performed.

The formation of long laminar separation bubbles on the suction surface is a characteristic of airfoil flows at lower Reynolds numbers, limiting lift generation and increasing drag.

The corresponding drag polar for these conditions is shown in the plot below, which is another helpful way of summarizing the effects of the Reynolds number on the aerodynamic characteristics. Notice again the profound effects of reducing the Reynolds number below 500,000, which deleteriously affects the lift-to-drag ratio, especially below values of 50,000. For extremely low Reynolds numbers of less than $10^4$ , lift-to-drag ratios of less than 5 are expected, at least with “conventional” airfoil sections.

Drag polars for a NACA 4412 airfoil for a NACA 4412 airfoil when operating at different Reynolds numbers.

At this point, a natural question is whether there is a better airfoil shape for use at low Reynolds numbers. The answer is “yes,” but the best shapes require a keen understanding of the surface boundary layer developments at these low Reynolds numbers. Michael Selig discusses these aerodynamic issues in his low Reynolds number design notes. More recently, the use of CFD has allowed the confident prediction of aerodynamic characteristics of airfoils at low Reynolds numbers. Measurements at this scale are incredibly difficult because of the low forces and moments, so CFD offers a tool to help fill in many details and establish systematic trends that cannot easily be determined by wind tunnel testing alone.

As shown in the figure below, one notable difference is the significant increase in drag at lower lift coefficients with decreasing Reynolds number, as previously discussed. Another observation is that the NACA 0012 and Clark-Y^[2] airfoils are much more sensitive to the Reynolds number than plate-type airfoils. On the one hand, the maximum lift coefficient of the 6% cambered plate increases from approximately 1.15 at a Reynolds number of 200,000 to approximately 1.32 at a Reynolds number of 10⁶. On the other hand, the peak lift coefficient of the flat plate is virtually unchanged. Over the same Reynolds number range, the peak lift coefficient more than doubles for the NACA 0012 and Clark-Y. Clearly, at the lowest Reynolds numbers, the thinner airfoils outperform the thicker airfoils in terms of aerodynamic efficiency, which is contrary to conventional aerodynamic wisdom. For example, the cambered plate shows a maximum lift-to-drag ratio of about 23 at a lift coefficient of 0.8, whereas the Clark-Y has a lift-to-drag ratio of about half of that at a much lower lift coefficient.

Effects of Reynolds number on the drag polars of a 2% thick flat plate, 6% cambered plate, and NACA 0012 and Clark-Y airfoils. (Adapted from Winslow et al.)

Effects of Mach Number

During the 1950s, aeronautics continued to advance quickly, and with the advent of the turbojet engine, airplanes began to fly much faster. Soon, the effects of compressibility on their performance became increasingly apparent, including the production of higher drag and phenomena such as wave drag, shock wave-induced flow separation, and buffeting from flow separation. The significant increase in drag and buffeting on an aircraft as it approached the speed of sound (Mach 1) was initially referred to as the sound barrier. Still, it is now known there is no intrinsic aerodynamic barrier to attaining supersonic flight if sufficient thrust is available to overcome the associated aerodynamic drag.

Much research has been devoted to understanding compressibility effects on airfoils and wings and developing suitable airfoil sections and wing shapes to reduce the adverse impact of compressibility at higher flight Mach numbers. The figure below shows schlieren images of the compressible flow about a 10% thickness-to-chord ratio symmetric airfoil set at an angle of attack of 2 degrees in the wind tunnel. The refraction of light rays produces the schlieren effect as they pass through the regions of different flow densities. (Note: The lower dark vertical line in these images is a model support in the optical path but not in the flow.) Furthermore, it should be noted that these circular images arise because the schlieren flow visualization system uses spherical or parabolic concave mirrors.

Schlieren images of the compressible flow about a 10% thick symmetric airfoil at an angle of attack of 2^o. These are classic photographs from the book “Mechanics of Fluids” by Duncan, Thom, and Young.

This sequence of images shows what happens to the flow about the airfoil as the free-stream Mach number $M_{\infty}$ (notice that $M_1$ is used to denote the free-stream Mach number in the captions to these images) is gradually increased from a subsonic flow just below the critical Mach number through transonic conditions and until the flow becomes supersonic. The differences delineate the developments of the shock waves and other compressible flow patterns; the shock waves are the darker zones produced by the schlieren effects.

No strong shock waves appear on the airfoil at $M_{\infty} = 0.7$ . Nevertheless, a close inspection of the image shows a small dark line near the 25% chord, suggesting a weak shock wave and that the critical Mach number, $M^*$ , is being reached, i.e., the flow about the airfoil becomes locally supersonic. By $M_{\infty} = 0.75$ , a series of small shocklets (which are mild shock waves) can be seen between 10% and 30% of the chord, confirming the existence of a supersonic pocket, and by $M_{\infty} = 0.775$ , a weak shock wave has formed. However, no shocks have yet formed on the lower surface.

By $M_{\infty} = 0.82$ , a more substantial shock wave has formed on the upper surface and moved aft on the chord, and by $M_{\infty} = 0.84$ , a shock develops on the lower surface almost directly below the one on the upper surface. In addition, there is clear evidence of boundary layer thickening at the bottom of the upper shock at the airfoil’s surface, resulting from the adverse pressure gradient produced there (see also the pressure distributions about airfoils discussed later in this chapter). Recall that an adverse pressure gradient is one in which the pressure increases with downstream distance, thereby slowing or retarding the flow’s development.

At $M_{\infty} = 0.88$ , both the upper and lower surface shock waves continue to grow in strength, the upper surface shock becoming bifurcated at the airfoil surface as it interacts with the boundary layer, and the flow visualization suggests that the downstream boundary layer is now relatively thick. By $M_{\infty} = 0.90$ , the boundary layer has separated downstream of the shock on the upper surface, a phenomenon known as shock wave-induced flow separation, usually called shock-induced separation. By $M_{\infty} = 0.95$ , both shocks have reached the trailing edge and become significantly bifurcated from their interaction with the relatively thick boundary layer. After that, the flow becomes entirely supersonic. A summary of these preceding observations is shown below as a schematic for better clarity.

As the free-stream Mach number increases from subsonic to supersonic, there are profound changes to the flow fields, emphasizing the importance of Mach number in understanding airfoil characteristics.

While the preceding flow visualization images are interesting and of much value, the resulting forces and moments on the airfoil are also important. The effects of the Mach number and, hence, the compressibility of the flow on the lift curve are shown in the figure below. Typically, flow with a Mach number lower than 0.3 is considered incompressible, and an airfoil will have a lift-curve slope of about $2\pi$ per radian or 0.11 per degree. Nevertheless, it can be seen that the lift-curve slope increases quickly at the higher Mach numbers because of the effects of compressibility. Notice also that the maximum lift of the airfoil decreases with increasing Mach number. As the Mach number approaches the critical Mach number and the onset of the transonic region (about 0.7), the attainable maximum lift coefficient without producing flow separation and stall can be seen to be relatively low.

Increasing the free-stream Mach number generally produces two primary effects on an airfoil: An increase in its lift-curve slope and a decrease in its maximum lift coefficient.

The theoretical relationship representing this latter behavior on the lift-curve slope is called the Glauert rule, summarized in the figure below and compared to measurements made on three NACA symmetric airfoil sections. The Glauert rule states that the lift-curve slope increases according to

(11) $\begin{equation*} C_{l_{\alpha}}(M_{\infty}) = \frac{2 \pi}{\sqrt{1 - M_{\infty}^2}} = \frac{2 \pi}{\beta} \end{equation*}$

which is theoretically exact for a thin airfoil in subsonic flow. A generalization of this result is to correct the low Mach number value of the lift coefficient using

(12) $\begin{equation*} C_{l_{\alpha}}(M_{\infty}) = \frac{C_{l_{\alpha}} \mbox{(\small for $M_{\infty} < 0.3$)}}{\sqrt{1 - M_{\infty}^2}} \end{equation*}$

These latter relationships hold for many airfoils of practical interest. Still, there are exceptions, especially for airfoils with large thickness-to-chord ratios or large amounts of nose camber. The validity of the Glauert rule generally extends up to the critical Mach number for the airfoil section, $M^*$ , i.e., the onset of supercritical flow.

Experimental verification of the “Glauert rule,” sometimes called the “Prandtl-Glauert” correction, has its theoretical origin in linearized subsonic aerodynamics.

Further examples of the effects of compressibility on the lift and drag characteristics of an airfoil are shown in the two figures below. Notice the increase in the lift coefficient for a given angle of attack with increasing Mach number but to a point. This outcome means that the lift is higher for a given dynamic pressure. However, also notice that for a given angle of attack, reductions in lift coefficient and increases in the drag coefficient will occur when the Mach number increases beyond a critical value, i.e., the stall angle of attack of the airfoil decreases with increasing Mach number.

Summary of the effects of free-stream Mach number as it increases from subsonic through transonic to supersonic conditions.

Drag coefficient characteristics of the NACA 23015 airfoil section as a function of Mach number for several angles of attack.

What airfoil section to use?

Selecting an airfoil for a specific purpose is a deliberate process that requires much care and usually a lot of time. If done properly, the process will involve the use of both computational methods (for iterative design) and wind tunnel testing (for verification of the final airfoil), but this is only ideal. Today, many airfoils can be confidently designed using computational methods alone, although this does leave some element of risk. Airfoils tend to be “point” designs, so they often perform optimally only at one specific combination of angle of attack (or lift coefficient), Reynolds number, and Mach number. Trade studies will inevitably focus on the minimum drag coefficient, maximum lift-to-drag ratio and lift-to-drag ratio at the design point, maximum lift coefficient, and overall pitching moment behavior. The most important aircraft performance criteria will inevitably define the most suitable airfoil shape. However, the success of the airfoil section also depends on its ability to perform well in off-design conditions. It is generally well advised to allow for sufficient robustness in the design to allow for acceptable levels of operation throughout the operational flight envelope. The consideration of maneuvers and gusts may also dictate the aerodynamic margins that need to be built into the airfoil design to prevent premature stall or other adverse aerodynamic behavior.

Pitching Moments

The behavior of the pitching moment on an airfoil is also important. It is always convenient to place the integrated forces at some convenient point on the airfoil, but the question is: At what point? The answer is that the forces can be located at any point if the corresponding moment about that same point is also defined.

In many aerodynamic applications, the 1/4-chord point is used as a reference point, i.e., $a = c/4$ . The 1/4-chord has theoretical significance, this being the aerodynamic center for a thin airfoil in an incompressible flow. However, even if another reference point were selected, converting from one reference point to another is easy because it is just the application of the rules of statics, as shown in the figure below.

Calculating moments about any point on an airfoil section is simply the application of statics.

For example, assume the lift force (or normal force) and pitching moment per unit span are known at a point at a distance $x_a$ from the leading edge of the airfoil, and it is desired to find the pitching moment about another point, say at a distance $x$ behind the leading-edge. Taking moments about the leading edge in each case gives

(13) $\begin{equation*} M_{x} = M_{a} + L \left( x - x_a \right) \end{equation*}$

Converting to coefficient form by dividing by $\frac{1}{2} \varrho_{\infty} V_{\infty}^{2} c^2$ gives

(14) $\begin{equation*} C_{m_{x}} = C_{m_{a}} + C_{l} \left( \overline{x} - \overline{x}_a \right) \end{equation*}$

where the overbar means that the length scale has now been non-dimensionalized with respect to the chord, i.e., $\overline{x} = x/c$ . For example, if the known pitching moment is about the leading edge, $C_{m_{\rm LE}}$ , then
$\overline{x}_a = 0$ and the above equation becomes

(15) $\begin{equation*} C_{m_{x}} = C_{m_{\rm LE}} + \overline{x} \, C_{l} \end{equation*}$

Center of Pressure & Aerodynamic Center

The concepts of center of pressure and aerodynamic center are used routinely in aerodynamic analysis, and it is essential to understand their differences. They are frequently confused in practice, though they are quite different.

Center of Pressure

By definition, the center of pressure is a point about which the moments are zero, i.e., a point where the resultant forces can be assumed to act. The principle is illustrated in the figure below, the center of pressure location $x_{\rm cp}$ effectively being the balance point (or fulcrum) of aerodynamic forces.

The center and pressure (as well as the aerodynamic center) can be determined if the lift and moment coefficients are known about any other point, the 1/4-chord often being used as a reference point, i.e., the values of $C_l$ and $C_{m_{1/4}}$ are available. The best way for the student to understand the process is to work through an example using actual airfoil measurements, which are given in the table below for a NACA 0012 (symmetric) airfoil.

Angle of attack	Lift coefficient	Drag coefficient	Moment coefficient
0	0	0.00662	0
1	0.1096	0.0067	0.0006
2	0.2182	0.00693	0.0013
3	0.3254	0.00736	0.0024
4	0.4309	0.008	0.0038
5	0.5365	0.00881	0.0054
6	0.6509	0.00976	0.0057
7	0.7743	0.01085	0.0057
8	0.9006	0.01203	0.0041
9	0.9957	0.01328	0.0046
10	1.0836	0.01466	0.0046
11	1.1729	0.01627	0.0079
12	1.2585	0.01817	0.0113
13	1.3343	0.02057	0.0157
14	1.3928	0.02328	0.0221
15	1.4322	0.02739	0.0284
16	1.4511	0.0345	0.0315
17	1.4508	0.04615	0.0287
18	1.4004	0.06732	0.0186
19	1.2739	0.10324	0.0001

These data are shown in the figure below in the form of lift coefficient, $C_l$ , versus angle of attack, $\alpha$ , the pitching moment coefficient about the 1/4-chord versus $\alpha$ , and the center of pressure location versus $C_l$ . The slope of the lift curve (i.e., the lift-curve slope, $C_{l_{\alpha}}$ , and the slope of the moment curve, i.e., $C_{l_{\alpha}}$ .

Measurements of two-dimensional airfoil characteristics can be used to find the variations in the center of pressure and the location of the aerodynamic center.

To find the position of the center of pressure, say $x_{\rm cp}$ downstream of the leading edge (LE) where the pitching moment would be zero, this is done by first taking moments about the leading edge, i.e., the application of statics gives

(16) $\begin{equation*} M_{\rm LE} = M_{c/4} - L \frac{c}{4} = -L x_{\rm cp} \end{equation*}$

and in coefficient form (divide by $\frac{1}{2} \varrho_{\infty} V_{\infty}^2$ ) this becomes

(17) $\begin{equation*} C_{m_{c/4}} = C_{l} \left( \frac{1}{4} - \frac{x_{\rm cp}}{c} \right) \end{equation*}$

so that the center of pressure (as a fraction of the chord) is given by

(18) $\begin{equation*} \frac{x_{\rm cp}}{c} = \frac{1}{4} - \frac{C_{m_{c/4}}}{C_{l}} \end{equation*}$

For most airfoils with positive camber, the value of $C_{m_{c/4}}$ is negative, so the center of pressure is generally behind the 1/4-chord. In particular, notice that the center of pressure will be a function of the lift coefficient (and hence also the angle of attack), so it is not a fixed point, as shown in the above figure. Because the center of pressure is a moving point and may not even be on the airfoil’s chord, the center of pressure is only sometimes a convenient concept to use in aerodynamics. Hence, the center of pressure to resolve the forces and moments is used sparingly in practice, even though the pitching moment here is zero, by definition.

Aerodynamic Center

By definition, the aerodynamic center is a point where the moment is constant and independent of the angle of attack. The procedure for finding the aerodynamic center, like that for the center of pressure, requires values of the lift and moment coefficient about any other point, say a distance $a$ from the leading edge.

If the aerodynamic center is at a distance $x_{ac}$ behind the leading edge, then the application of statics (as explained previously) gives

(19) $\begin{equation*} C_{m_{a}} = C_{m_{\rm ac}} - C_{l} \left( \frac{x_{ac}}{c} - \frac{a}{c} \right) \end{equation*}$

The objective is to find the location of $x_{ac}$ such that the value of the pitching moment at that point is constant. Differentiating the above equation with respect to $C_{l}$ gives

(20) $\begin{equation*} \frac{d C_{m_{a}}}{dC_{l}} = \frac{d C_{m_{\rm ac}}}{dC_{l}} - \left( \frac{x_{\rm ac}}{c} - \frac{a}{c} \right) \end{equation*}$

The value of $dC_{m_{\alpha}}/d C_l$ can be obtained by using

(21) $\begin{equation*} \frac{d C_{m_{\alpha}} }{ d C_l} = \left( \frac{ d C_{m_{\alpha}}}{d \alpha}\right) \left( \frac{d \alpha}{d C_{l}}\right) \end{equation*}$

Therefore, to calculate the location of the aerodynamic center, the slope of the lift curve (in the linear range) and the slope of the moment curve (also in the linear range) are needed. Remember that the linear range corresponds to the conditions where the flow would be fully attached to the airfoil surface. This process is performed by finding the slopes of the best straight line fit to the values of $C_l$ versus $\alpha$ and then $C_{m_{1/4}}$ versus $\alpha$ in the linear range (attached flow).

Following on with the previous example, a least-squares fit to the measurements below the stall gives

(22) $\begin{equation*} C_{l_{\alpha}} = \frac{dC_l}{d \alpha} = 0.0916\mbox{\small /deg.} \end{equation*}$

and

(23) $\begin{equation*} C_{m_{\alpha}} = \frac{dC_{m_{1/4}}}{d \alpha} = 0.000683\mbox{\small /deg.} \end{equation*}$

Therefore, because the 1/4-chord is being used as a moment reference, the aerodynamic center, in this case, is
given by

(24) $\begin{equation*} \frac{x_{ac}}{c} = \overline{x}_{ac} = \dfrac{1}{4} - \left( \dfrac{\dfrac{d C_{m_{1/4}}}{d \alpha}}{ \dfrac{d C_l}{d \alpha }} \right) = \dfrac{1}{4} - \dfrac{0.000683}{0.0916} = 0.2425 \end{equation*}$

It can be seen that for the particular airfoil used here, the aerodynamic center location is at 24.25% chord. The location of the aerodynamic center depends on the airfoil section and the Mach number at which it operates. Because the value of the lift-curve slope is always positive, the slope of the moment curve defines the sign of position of the aerodynamic center relative to 1/4-chord. So, if this slope is positive, the aerodynamic center is in front of the 1/4-chord; if it is negative, the aerodynamic center is behind the 1/4-chord. For thin airfoils, the value of $C_{m_{1/4}}$ is almost constant, so the aerodynamic center is generally always close to 1/4-chord at low free-stream Mach numbers.

It can be concluded that the advantage of using the aerodynamic center to resolve the forces and moments is convenience. It is a fixed point that does not change with the angle of attack. However, finding the aerodynamic center location takes more work because the slope of the lift curve and the moment curve must be determined.

What is a least-squares linear fit?

The linear least-squares method is a numerical process of finding a straight line that best represents the trends shown in a particular data set. The main objective is to reduce as much as possible the sum of the squares of the differences or residuals between the fitted straight line and all of the data points, as shown in the figure below.

It can be assumed that the given data points are $(x_1, y_1)$ , $(x_2, y_2)$ , $(x_3, y_3)$ , …., $(x_N, y_N)$ , where $N$ is the number of data points. A straight line is $y = m x + b$ , where $m$ is the slope, and $b$ is the intercept on the $y$ axis. The formulas to calculate the slope and intercept of the best straight-line fit are given by

$m = \frac{N \sum x y - \sum x \sum y}{N \sum x^2 - (\sum x)^2}$

and

$b = \frac{\sum y - m \sum x}{N}$

The process can also be performed in MATLAB by using the polyfit function.

Pressure & Shear Stress Coefficients

Pressure coefficients (i.e., non-dimensional pressures) are usually employed when presenting pressure distributions around body shapes and airfoil sections. The pressure coefficient, $C_p$ , is defined as

(25) $\begin{equation*} C_{p} = \frac{p - p_{\infty}}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2} = \frac{p - p_{\infty}}{q_{\infty}} \end{equation*}$

where $p$ is the local static pressure and $p_{\infty}$ is the free-stream static pressure. Again, notice the use of free-stream dynamic pressure as a reference pressure.

Suppose the local pressure equals the free-stream static pressure $p_{\infty}$ , then $C_p = 0$ . If the local pressure is less than the free-stream static pressure $p_{\infty}$ , then $C_p < 0$ . At a stagnation point (where the flow velocity is brought to zero), then $C_p = 1$ , at least if the flow is assumed incompressible and the Bernoulli equation is being applied.

The shear stress or local skin friction coefficient is defined as

(26) $\begin{equation*} c_f = \frac{\tau_w}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2} = \frac{\tau_w}{q_{\infty}} \end{equation*}$

where $\tau_w$ is the boundary layer shear stress, which is proportional to the velocity gradient at the surface.

The total skin friction coefficient, denoted by $C_f$ , is obtained by integrating the local skin friction coefficient over the surface. For example, if the surface exposed to the flow has a length $L$ , then the total shear stress coefficient is

(27) $\begin{equation*} C_f = \frac{1}{L} \int_{x=0}^{x=L} c_f \, dx \end{equation*}$

If two surfaces are exposed to the flow, such as the upper (u) and lower (l) surfaces of an airfoil, then the total skin friction coefficient would be equivalent to the skin friction drag coefficient, i.e.,

(28) $\begin{equation*} C_{d_{f}} = \frac{1}{L_{u}} \int_{u} c_{f_{\rm{up}}} \, dx_{\rm {u}} + \frac{1}{L_{l}} \int_{l} c_{f_{l}} \, dx_{l} \end{equation*}$

Of course, these principles can be applied to any body shape and extended to three dimensions.

Integration of Pressures & Shear Stresses

Calculating or measuring pressure distributions over body shapes, such as on airfoil sections, is possible. The process of integration around the body contour can then be used to find quantities such as lift, drag, and pitching moment coefficients. The pressure values are typically two orders of magnitude greater than the shear stress, so adequate results for the lift can often (but not always) be obtained by considering only the pressures. However, for the drag, it is necessary to account for the shear stress because this component is the dominant contribution to the drag at low angles of attack.

The pressure distributions around airfoil sections are non-uniform, with higher and lower pressure regions relative to ambient pressure, typical variations being shown in the figure below. While pressure is a scalar, in this presentation, the local pressure, $p$ , is plotted as a vector directed perpendicular to the local surface slope of the airfoil, i.e., in terms of $p \,\vec{n}$ where $\vec{n}$ is the unit normal vector. The green zones represent lower than ambient pressure, and the red zones are higher than ambient pressure. Airfoil sections also produce substantial pressure gradients, especially near the leading edge at higher angles of attack. Therefore, the challenge is to determine the integrated quantities, which requires a knowledge of the values of the pressures and their detailed distributions.

Representative static pressure distributions around symmetric and cambered airfoils. The local pressure is plotted as a vector perpendicular to the local surface slope of the airfoil.

The principle behind the integration process to find the lift can be explained with reference to the figure below. On the upper surface, the per unit span force components acting on an elemental area of width $ds_u$ are

(29) $\begin{eqnarray*} dN'_u & = & (p_u \cos \theta_u - \tau_u \sin \theta_u ) ds_u \\[10pt] dA'_u & = & (p_u \sin \theta_u + \tau_u \cos \theta_u ) ds_u \end{eqnarray*}$

and on the lower surface, they are

(30) $\begin{eqnarray*} dN'_l & = & (p_l \cos \theta_l - \sin \theta_l ) ds_l \\[10pt] dA'_l & = & (p_l \sin \theta_l + \tau_l \cos \theta_l ) ds_l \end{eqnarray*}$

Method used to integrate the pressure and shear stress distribution over an airfoil section.

Integration from the leading edge to the trailing edge produces the total per unit span forces, i.e.,

(31) $\begin{eqnarray*} N' & = & \int_{\rm LE}^{\rm TE} dN'_u + \int_{\rm LE}^{\rm TE} dN'_l \\[10pt] A' & = & \int_{\rm LE}^{\rm TE} dA'_u + \int_{\rm LE}^{\rm TE} dA'_l \end{eqnarray*}$

After $N'$ and $A'$ have been determined then $L'$ and $D'$ can be found using

(32) $\begin{eqnarray*} L' & = & N' \cos \alpha - A' \sin \alpha \\[10pt] D' & = & N' \sin \alpha + A' \cos \alpha \end{eqnarray*}$

The pitching moment about the leading edge is the integral of these forces weighted by their moment arms $x$ and $y$ with appropriate signs remembering the moments are positive nose-up, i.e.,

(33) $\begin{equation*} M'_{\rm LE} = \int_{\rm LE}^{\rm TE} -x dN'_u + \int_{\rm LE}^{\rm TE} -x dN'_l + \int_{\rm LE}^{\rm TE} y dA'_u + \int_{\rm LE}^{\rm TE} y dA'_u \end{equation*}$

From the geometry, then $ds \cos \theta = dx$ , $ds \sin \theta = dy = (dy/dx) dx$ , which allows all the above integrals to be performed in terms of $x$ by using the upper and lower shapes of the airfoil, i.e., $y_u(x)$ and $y_l(x)$ as well as the slopes $\theta_u$ and $\theta_l$ , respectively.

Typical Pressure Distributions

The standard practice is to calculate (or measure) the pressure distributions around airfoils and plot the results in terms of pressure coefficient, $C_p$ , as a function of a non-dimensional distance or chord, $x/c$ . Remember that to determine the pressure coefficient, the value of free-stream dynamic pressure is needed, as discussed previously, which requires that the density of the air be obtained from measurements of ambient pressure and temperature.

An example of a measured (discrete) pressure distribution around an airfoil at subsonic conditions is shown below. Notice that the $C_p$ values are plotted by convention in such a way that the upper surface of the airfoil is on the upper surface of the plot and the lower surface of the airfoil is on the lower surface of the plot, i.e., the negative $C_p$ values are on the top half of the plot.

Representative chordwise pressure distributions on the NACA 0012 airfoil at a subsonic Mach number.

The NACA 0012 is a symmetric airfoil, so at zero angle of attack (or almost at $-0.1$ degrees in the case), the upper surface pressure distribution is identical to the lower surface. Increasing the angle of attack produces significant suction (negative) pressures on the upper surface, the lowest at the leading edge. At an angle of attack of 8.1 $^{\circ}$ , the negative pressures reach a peak value of about $C_p = -4$ .

A representative pressure coefficient distribution around an airfoil in transonic flow is shown in the figure below. Notice in this case that the region of supersonic flow over the leading edge region produces a more uniform low pressure terminated by an abrupt pressure recovery, a classic signature indicative of the presence of a shock wave. This outcome contrasts with the subsonic flow, where the lowest pressure value is much closer to the leading edge, giving an adverse pressure gradient over most of the chord. The steep adverse pressure gradient near the shock wave makes the boundary layer much more prone to local thickening and often produces flow separation, i.e., the onset of shock-induced separation.

Representative chordwise pressure distributions on the NACA 0012 airfoil at transonic Mach numbers.

Check Your Understanding #3 – Integrating a pressure distribution

Calculate the lift coefficient on a two-dimensional body where the pressure coefficient $C_p$ distribution at low angles of attack is described by

$C_{p_{u}} = 1 - 300 \left( \frac{x}{c} \right)^2 \quad \mbox{for $0 \le x/c \le 0.1$}$

$C_{p_{u}} = -2.2277 + 2.2777 \left( \frac{x}{c} \right) \quad \mbox{for $0.1 < x/c \le 1.0$}$

$C_{p_{l}} = 1 - 0.95 \left( \frac{x}{c} \right) \quad \mbox{for $0.0 \le x/c \le 1.0$}$

Show solution/hide solution.

The lift coefficient $C_l$ is given by

$C_l = \int_0^1 \Delta C_p \, d(x/c) = \int_0^1 (C_{p_{l}} - C_{p_{u}}) \, d \overline{x}$

For this problem, it can be split into two integrals, i.e.,

$C_l = \int_0^{0.1} (C_{p_{l}} - C_{p_{u}}) d \overline{x} + \int_{0.1}^1 (C_{p_{l}} - C_{p_{u}}) d \overline{x} = C_{l_{1}} + C_{l_{2}}$

Taking the first integral gives

$C_{l_{1}} = \int_0^{0.1} (C_{p_{l}} - C_{p_{u}}) d \overline{x} = \int_0^{0.1} \big( \left( 1 - 0.95 \overline{x} \right) - \left( 1 - 300 \overline{x} ^2 \right) \big) d \overline{x}$

and so

$C_{l_{1}} = \int_0^{0.1} \left( 300 \overline{x} ^2 - 0.95 \overline{x} \right) d \overline{x} = \Bigg[ 100 \overline{x}^3 - 0.475 \overline{x}^2 \Bigg]_{0}^{0.1} = 0.09525$

Taking the second integral gives

$C_{l_{2}} = \int_{0.1}^{1} (C_{p_{l}} - C_{p_{u}}) d \overline{x} = \int_{0.1}^{1} \big( \left( 1 - 0.95 \overline{x} \right) - \left( -2.2277 + 2.2777 \overline{x} \right) \big) d \overline{x}$

and so

$C_{l_{2}} = \int_{0.1}^{1} 3.2277 \left( 1- \overline{x} \right) d \overline{x} = \Bigg[ 3.2277 \left( \overline{x} - \frac{\overline{x}^2}{2} \right) \Bigg]_{0.1}^{1} = 1.307$

Therefore, adding the two parts of the integral gives

$C_{l} = C_{l_{1}} + C_{l_{2}} = 0.09525 + 1.307 = 1.4025$

Numerical Integration of Pressure

Rarely are analytic (continuous) distributions of pressure available about a body as in the preceding example. The pressure values are usually known at discrete points, e.g., from measurements made on a wing in the wind tunnel or from CFD calculations. Therefore, if the lift and moment coefficients are to be obtained from the pressure distributions, then a numerical integration approach must be adopted. It is generally impossible to find the drag this way, however, at least using measurements, and usually, a momentum deficiency approach is used.

The lift force coefficient, $C_l$ , can be determined by evaluating the integral

(34) $\begin{eqnarray*} C_{l} & = & \frac{1}{c} \int_{0}^{c} ( C_{p_{l}} - C_{p_{u}}) \; dx = \frac{1}{c} \int_{0}^{c} \Delta C_p \; dx \nonumber \\ & = & \int_{0}^{1} \Delta C_p \, d\left(\frac{x}{c}\right) = \int_{0}^{1} \Delta C_p \, d\overline{x} \end{eqnarray*}$

which will be apparent as just the area under the $C_p$ versus the $x/c$ curve.

The easiest numerical integration method to find an area (which also has good accuracy) is to use the trapezoidal rule where the integral is written as a numerical summation, i.e.,

(35) $\begin{equation*} C_l = \int_{0}^{1} \Delta C_p \, d\overline{x} \approx \sum_{i=1}^N \left( \frac{ \Delta C_{p}^{i} + \Delta C_{p}^{i+1}}{2} \right) \Delta \overline{x}^i \end{equation*}$

where $N$ is the number of discrete points for which $C_p$ or $\Delta C_p$ is known. The idea is shown in the figure below. Notice that $i$ is an index, i.e., $i = 1, 2, 3, ... , N$ , and that for $N$ points, there would be $N-1$ incremental area contributions to the integral. The more points available with pressure values, the more accurate the numerical integration will be.

The principle of numerical integration of the differential pressure distribution over a surface to find the lift, etc.

The preceding is a satisfactory approach if the values of $\Delta C_{p}$ are known along the chord at all discrete points (both on the upper and lower surfaces). The problem is that pressure points are available at different values of $x/c$ on the upper and lower surfaces. This means that the integral (area under the curve) must be found in two parts, i.e., a numerical summation of the like

(36) $\begin{equation*} C_l \approx \sum_{i=1}^{N_u} \left( \frac{ C_{p_{u}}^{i} + C_{p_{u}}^{i+1}}{2} \right) \Delta \overline{x}_u^i + \sum_{i=1}^{N_l} \left( \frac{ C_{p_{I}}^{i} + C_{p_{l}}^{i+1}}{2} \right) \Delta \overline{x}_l^i \end{equation*}$

where $N_u$ would be the number of points on the upper surface and $N_l$ would be the number of points on the lower surface, the idea being shown in the figure below.

The principle of numerical integration of the chordwise pressure distribution over the upper and lower surfaces to find the lift, etc.

The corresponding pitching moment about the leading edge of the airfoil can be determined from

(37) $\begin{eqnarray*} C_{m_{\rm LE}} & = & -\frac{1}{c^{2}} \int_{0}^{c} ( C_{p_{l}} - C_{p_{u}}) \; x \; dx = -\frac{1}{c^2} \int_{0}^{c} \Delta C_p \; x \; dx \nonumber \\ & = & -\int_{0}^{1} \Delta C_p \left( \frac{x}{c} \right) d \left( \frac{x}{c} \right) = -\int_{0}^{1} \Delta C_p \, \overline{x} \, d \overline{x} \end{eqnarray*}$

An extra $\overline{x}$ term appears in the equation, which is a moment arm. Notice the sign on the pitching moment, which can be recalled as being positive in a nose-up sense. Again, this latter integral would be converted to a numerical summation where the moment arms will be included in the summation, which is calculated at the mid-point of each trapezoid, i.e., giving an equation of the form

(38) $\begin{eqnarray*} C_{m_{\rm LE}} & \approx & -\sum_{i=1}^{N_u} \left( \frac{ C_{p_{u}}^{i} + C_{p_{u}}^{i+1}}{2} \right) \left( \frac{\overline{x}_u^i + \overline{x}_u^{i+1}}{2} \right) \Delta \overline{x}_u^i \nonumber \\ & & + \sum_{i=1}^{N_l} \left( \frac{ C_{p_{I}}^{i} + C_{p_{l}}^{i+1}}{2} \right) \left( \frac{\overline{x}_l^i + \overline{x}_l^{i+1}}{2} \right) \Delta \overline{x}_l^i \end{eqnarray*}$

The pitching moment about 1/4-chord can be determined from

(39) $\begin{equation*} C_{m_{1/4}} = -\int_{0}^{1} \Delta C_p \left( \overline{x} - \frac{1}{4}\right) d \overline{x} \end{equation*}$

After that, the pitching moment about any other point can be found by using simple statics from the values of $C_l$ and $C_{m_{\rm LE}}$ , i.e., for the moment about a point that is a non-dimensional distance $a$ from the leading-edge then

(40) $\begin{equation*} C_{m_{a}} = C_{m_{\rm LE}} + a \, C_l \end{equation*}$

Finally, the axial force (or chord force) coefficient can be determined by evaluating

(41) $\begin{equation*} C_{a} = -\frac{1}{c} \int_{0}^{1} \left(\displaystyle{\frac{dy_l}{dx} } C_{p_{l}} - \displaystyle{ \frac{dy_u}{dx}} C_{p_{u}}\right) \; dx = -\int_{0}^{1} \left(\displaystyle{ \frac{d\overline{y}_l}{d\overline{x}}} C_{p_{l}} - \displaystyle{ \frac{d\overline{y}_u}{d\overline{x}}} C_{p_{u}}\right) \; d\overline{x} \end{equation*}$

where $\overline{y} = y/c$ . Again, this integral will be evaluated numerically using the trapezoidal rule, but this is not an easy one to integrate because not only is the shape of the airfoil needed (i.e., $y$ values and the slope of the surface) but a lot of points is needed around the nose of the airfoil (where there are steep pressure gradients and relatively high values of surface curvature) to get good accuracy, as shown in the figure below. Caution should also be exercised to ensure the correct signs (positive or negative) are used.

Representative subsonic pressure distribution around an airfoil plotted with respect to chord and thickness.

Check Your Understanding #4 – Numerically integrating a pressure distribution

Write some MATLAB code to plot the chordwise form of the pressure distribution in Example #3 and perform the integration numerically to find the lift. Hint: MATLAB has a handy function called “trapz” to do this.

Show solution/hide solution

Some MATLAB code is given below, which can be copied and pasted into your version of MATLAB. Running the code gives $C_l$ = 1.4025, which agrees with the value from the analytic integration given in Worked Example #3. Of interest is to vary the number of points used for the numerical integration in the linspace function.

clc
figure
axis([0.0 1.0 1.0 5.0])
x1 = linspace(0,0.1,500);
x2 = linspace(0.1,1,500);
cpu_1 = 1.-300*x1.^2;
cpu_2 = -2.2277 + 2.2777.*x2;
x = [x1 x2];
cpu = [cpu_1 cpu_2];
cpl = (1.0-0.95*x);
dcp = cpl-cpu;
trapz(x,dcp) % to find the section cl using the trapezoidal rule
plot(x,cpu);hold on
plot(x,cpl)
xlabel(‘x/c’)
ylabel(‘C_P’)
legend(‘Upper surface’, ‘Lower surface’)

Supersonic Airfoils

A supersonic airfoil shape is designed to operate at Mach numbers greater than unity. Unlike subsonic airfoils, supersonic airfoils produce strong shock waves. These shock waves create significant pressure changes, affecting the airflow and aerodynamic performance of the airfoil.

The well-rounded, cambered airfoil sections that are 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 constructed of angled planes called double-wedge airfoils or opposed circular arcs called biconvex airfoils, as shown in the figure below.

Three types of supersonic airfoil shapes: biconvex circular arcs, double-wedge or diamond, and biconvex with a drooped leading edge.

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. For example, in the schlieren image shown below, the supersonic flow passes smoothly over the sharp nose of the airfoil profile, thereby creating minimum drag. For a supersonic airfoil, the thickness and camber shapes are designed to minimize energy losses associated with the compression and expansion of the flow. However, this goal can often only be achieved over a small range of operating angles of attack.

Visualization of the shockwave patterns over a double-wedge-shaped airfoil at a supersonic free-stream Mach number of 1.6.

Consider the flow about an airfoil in the form of a double-wedge or diamond shape experiencing a supersonic flow, as shown below. Unlike a subsonic airfoil with smooth surface curvature, a double-wedge airfoil is ideal for supersonic flow. Notice that the upstream flow gets no warning of the approaching airfoil in a supersonic flow, so the streamlines have no curvature.

Representative flow pattern and pressure distribution about a claasic supersonic double-wedge (diamond) airfoil shape.

Oblique compression shock waves occur at the leading edge of the airfoil. The Mach number across the shocks decreases (but remains supersonic), and the static pressure increases over the free-stream value. Notice the more significant pressure increase on the lower surface, which contributes significantly to the lift. At the points of maximum thickness, expansion waves appear, which causes the Mach number to increase and the pressure to decrease after the expansion is complete. The upper surface of the airfoil now contributes to the lift. Rarefaction shock waves form at the trailing edge, increasing the Mach number and returning the pressure to the free-stream value.

The linearized supersonic airfoil theory shows that the lift coefficient is independent of the airfoil shape but depends on the angle of attack, $\alpha$ , as given by

(42) $\begin{equation*} C_l = \frac{4 \, \alpha }{\sqrt{M_{\infty}^2 - 1}} = \left( \frac{4 }{\sqrt{M_{\infty}^2 - 1}} \right) \alpha = C_{l_{\alpha}} ( M_{\infty} ) \, \alpha \end{equation*}$

This latter result shows that the lift-curve slope is lower in supersonic than subsonic flight and decreases with increasing Mach number.

The results in Eq. 11 (subsonic) Eq. 42 (supersonics) are classical solutions in the airfoil theory and allow the determination of the lift coefficients on airfoils at small angles of attack. These thin airfoil approximations agree well with measurements of the lift-curve slope made on thin two-dimensional airfoils, as shown in the figure below. The linearized supersonic theory also has good validity when extended to three dimensions, such as predicting the lift-curve slope of a finite wing with a higher aspect ratio.

Linearized subsonic and supersonic thin airfoil theory largely agrees with measurements, at least up to a Mach number of 3.0.

The linearized theory shows that the lift coefficient is independent of the airfoil shape. For example, the lift coefficient produced by a supersonic airfoil at a given angle of attack is the same for a flat plate, a diamond-wedge airfoil, or a biconvex airfoil. The drag, however, is different because the dominant source of drag in supersonic flight is wave drag, which depends strongly on the airfoil shape.

For example, for a diamond-wedge airfoil, the drag coefficient is

(43) $\begin{equation*} C_{d_{w}} = \frac{4}{\sqrt{M_{\infty}^2 - 1}} \bigg( \alpha^2 + \left( \frac{t}{c} \right)^2 \bigg) \end{equation*}$

which shows that wave drag increases with the square of the airfoil thickness-to-chord ratio, $t/c$ . In general, the wave drag of a supersonic airfoil can be expressed as

(44) $\begin{equation*} C_{d_{w}} = \frac{4}{\sqrt{M_{\infty}^2 - 1}} \bigg( \alpha^2 + k_t \left( \frac{t}{c} \right)^2 + k_c \, \beta^2 \bigg) \end{equation*}$

where $t/c$ is its maximum thickness-to-chord ratio and $\beta$ is its maximum camber. The values of the constants $k_t$ and $k_c$ depend on the exact shape of the airfoil.

Therefore, it becomes clear why the airfoils used on the wings of supersonic airplanes must be very thin and mildly cambered compared to those used on subsonic airplanes. You can read more about supersonic airfoils and wings in a later chapter of this book.

Check Your Understanding #5 – Calculating the lift & drag coefficients on supersonic Airfoil

A supersonic double-wedge airfoil with a thickness-to-chord ratio of 8% is operated at an angle of attack of 5.0 degrees at a Mach number of 3.5. Calculate the lift-to-drag ratio.

Show solution/hide solution.

In this case, the values given lead to

$\sqrt{M_{\infty}^2 - 1} = \sqrt{3.5^2 - 1} = 3.35$

and

$\alpha = 5.0 \times \frac{\pi}{180} = 0.0873~\mbox{rads}$

The lift coefficient is

$C_l = \frac{4 \, \alpha }{\sqrt{M_{\infty}^2 - 1}} = \frac{ 4 \times 0.0873}{3.35} = 0.104$

The wave drag coefficient is

$C_{d_{w}} = \frac{4}{\sqrt{M_{\infty}^2 - 1}} \bigg( \alpha^2 + \left( \frac{t}{c} \right)^2 \bigg) = \frac{4.0}{3.35} \left( 0.0873^2 + 0.08^2 \right) = 0.0167$

Therefore, the lift-to-drag ratio is

$\frac{C_l}{C_{d_{w}} } = \frac{0.104}{0.0167} = 6.23$

A significant problem with thin supersonic airfoils is that at low (subsonic) flow flight speeds, they tend to produce leading-edge flow separation and stall even at moderate angles of attack. Therefore, many aircraft that employ supersonic airfoils must use high-lift devices for takeoff and landing, such as large leading-edge slats along the entire wing. Another method is drooping the leading edge using camber, which maintains the supersonic performance while markedly improving low-speed characteristics. Nevertheless, supersonic aircraft have relatively high takeoff and landing speeds and may need drogue parachutes to reduce their landing distances.

Supercritical Airfoils

Because commercial jet airliners need 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 unique wing shapes called supercritical wings. A supercritical wing also uses a supercritical airfoil to reduce the strength of shock waves, thereby reducing wave drag and increasing the drag divergence Mach number, the principle of which is shown in the figure below.

The classic approach to increasing the drag divergence Mach number is reducing wing thickness, i.e., thickness-to-chord ratio. However, this has several other disadvantages, including less fuel tank capacity and higher structural weight for the same strength and stiffness. Supercritical airfoils were initially studied in the 1950s by Herbert Pearcey at NPL in England and refined during the 1960s by Richard Whitcomb at NACA. Today, supercritical airfoils are used on nearly all commercial jet aircraft, allowing them to cruise at higher flight Mach numbers between 08 and 0.85.

A supercritical airfoil shape is notably distinctive. It has a point of maximum thickness fairly aft on the chord, with a relatively flat upper surface with a slight camber. The aerodynamic principle used in transonic wing and airfoil design is to control the expansion of the flow to supersonic speed and its subsequent recompression, making the leading edge pressure distribution less “peaky” and more uniform. The small amount of leading-edge camber helps limit the strength of the shock waves at the expense of some loss of lift at a given angle of attack. However, such transonic airfoils have significant camber at their trailing edges, which helps to recover the lost lift on the airfoil.

The net effect is a significant increase in the critical Mach number and drag divergence Mach number, as shown in the figure below. Increasing the critical Mach number allows the aircraft to operate at higher speeds without encountering excessive drag or other aerodynamic issues associated with transonic flight. Notice, however, that drag divergence Mach number can vary depending on various design factors, including its thickness-to-chord ratio, $t/c$ . Using a supercritical airfoil generally limits the onset of drag divergence to a higher free-stream Mach number. It improves the aerodynamic performance of aircraft that operate near the speed of sound.

A supercritical airfoil delays the onset of drag divergence to a higher flight Mach number.

Inverse Airfoil Design and Shape Optimization

Inverse airfoil design aims to determine the shape of an airfoil that achieves the desired level of aerodynamic performance. By using inverse design,^[3] engineers can tailor airfoil shapes to achieve specific aerodynamic goals, such as minimizing drag, maximizing lift, minimizing pitching moments, maximizing lift-to-drag ratios, achieving a desired pressure distribution, or maximizing the critical Mach number. This optimization process of the lifting surface(s) may have multiple aerodynamic design goals and can lead to more efficient aircraft or other aerodynamic component(s).

The aerodynamic shape optimization process requires defining an objective function, parameterizing the airfoil’s shape mathematically, applying formal optimization algorithms, and then iterating (carefully!) until an optimal airfoil design shape is achieved. The process combines aerodynamic solution methods, optimization techniques, and sensitivity analysis to refine the airfoil shape to meet the specified performance goals. Airfoil shape optimization is often conducted at a given angle of attack or lift coefficient. Airfoil sections are very much “point designs,” but shape optimization must also consider off-design aerodynamic operation. In this regard, off-design operation is often referred to as “robustness.” Some level of robustness is desired because an aircraft can never operate exactly at the design point.

The shape optimation process involves:

Establish the desired aerodynamic characteristics, such as the pressure distribution $p_{\text{desired}}(x)$ , lift coefficient, and/or drag coefficient, and/or pitching moment.
Specify the boundary and flow conditions, including free-stream velocity, angle of attack, or Mach number and Reynolds number. These parameters affect the aerodynamic performance and need to be included in the design process.
To solve for the aerodynamics, one can use a panel method in which the airfoil shape is discretized into panels and integral equations are solved to find the resulting pressure distribution or by using computational fluid dynamics (CFD).
Implement optimization algorithms to adjust the airfoil shape based on performance goals. This typically involves iterative algorithms that adjust shape parameters based on performance criteria.
Adjust the airfoil shape to minimize the difference between the desired and actual performance.
Validate the design using wind tunnel testing to ensure the airfoil meets the desired performance levels.

Shape optimization involves finding the airfoil shape that meets the performance goals. In this regard, an objective function $J$ is used to quantify the difference between the desired and actual performance. For a pressure distribution, say $p(x)$ , this can be expressed as

(45) $\begin{equation*} J = \frac{1}{2} \int_{x_{\text{min}}}^{x_{\text{max}}} \bigg( p_{\text{des}}(x) - p_{\text{act}}(x) \bigg)^2 \, dx \end{equation*}$

where $p_{\text{des}}(x)$ is the desired or “design” pressure distribution, and $p_{\text{act}}(x)$ is the actual or current pressure distribution for the given shape. Any constraints the shape must satisfy, such as geometric and aerodynamic constraints, can now be introduced. For example, constraints may be placed on the maximum or minimum thickness-to-chord ratio, e.g., for structural reasons, the amount of camber, or both. The feasibility of manufacturing the shape may also be included as a design constraint, especially when different airfoil may be used along the span of the wing or other lifting surface, e.g., as would be used on a propeller, helicopter rotor or wind turbine blade.

Usually, the airfoil’s shape must be parameterized mathematically by some shape variables. If control points or coefficients describe the shape, it can be expressed in the form

(46) $\begin{equation*} (x, y) = (x_0, y_0) + \sum_{i=1}^N \delta_i \, \phi_i \end{equation*}$

where $(x, y)$ is the modified shape of the airfoil, $x_0, y_0$ is the initial or baseline shape of the airfoil, $\delta_i$ are shape parameters that control the contribution of each basis function, and $\phi_i$ are basis functions that describe how the shape is modified. For example, in the case of using a standard Fourier series

(47) $\begin{equation*} y(x) = a_0 + \sum_{n=1}^{N} \bigg( a_n \cos(n \pi x) + b_n \sin(n \pi x) \bigg) \end{equation*}$

where $y(x)$ represents the location of the upper and lower surfaces of the airfoil, $a_0$ , $a_n$ , and $b_n$ are coefficients of the Fourier modes of frequency $n$ that can be used to adjust the shape of the airfoil, and $\cos(n \pi x)$ and $\sin(n \pi x)$ are referred to as the basis functions used to represent the shape.

Gradient-based optimization methods, which are well established, can then be used to find the gradient of the objective function with respect to the shape parameters describing the airfoil. The shape parameters can then be updated using

(48) $\begin{equation*} \delta_i^{(k+1)} = \delta_i^{(k)} - K_r \left( \frac{\partial J}{\partial \delta_i} \right) \end{equation*}$

where $K_r$ is a coefficient, often called a relaxation parameter, which is used to define the new shape of the airfoil to meet the specific aerodynamic requirements. Usually, the relaxation parameter must be kept relatively small in value, e.g., less than 0.05, so that large shape changes to the airfoil shape do not occur in any design iteration, which may cause the method to fail. As part of the process, changes in shape parameters can be explored as they affect the objective function, with the sensitivity being determined by $\dfrac{\partial J}{\partial \delta_i}$ .

Finally, the shape parameters are updated based on the optimization algorithm to be able to recalculate the objective function. This process can be continued until convergence criteria are met, i.e., $\| \nabla J \| < \epsilon$ , where $\epsilon$ is some acceptable tolerance level based on the design goals. If the process runs smoothly, and to this end, the numerical approach and shape changes must be carefully monitored, then the new shape of the airfoil is finally obtained.

The figure below shows representative outcomes of this process for a transonic airfoil section. The initial “conventional” airfoil shape has a strong shock wave and a relatively low drag divergence Mach number. The goal of the optimization is to maximize the drag divergence Mach number while maintaining a thickness-to-chord ratio above 13% for structural purposes. The optimization process results in an airfoil with a flatter upper surface. This design modification leads to a weaker shock wave and a higher drag divergence Mach number, thereby improving the airfoil’s performance at transonic speeds. Additionally, the trailing edge of the optimized airfoil shows increased camber, which enhances aerodynamic efficiency while meeting the structural constraint.

Inverse airfoil design can determine the shape of an airfoil to reach a desired level of aerodynamic performance.

Successful outcomes for the optimized airfoil shape depend on the fidelity of the optimization process, the accuracy of the aerodynamic simulations, and the practical feasibility of the optimized airfoil design. Wind tunnel testing, if available, can verify the aerodynamic outcomes by placing a scaled or full-sized model of the wing section-controlled airflow environment. Pressure measurements over the surface of the airfoil can be used to compare with the predictions that were made, and any aerodynamic improvements can be quantified against the original baseline airfoil. While the method to optimize the airfoil shape is well-established and relatively quick, the verification process in the wind tunnel may be much more lengthy.

Summary & Closure

Understanding the aerodynamic characteristics of two-dimensional airfoil sections is a prerequisite to understanding the characteristics of finite wings. While airfoil characteristics can be predicted, the most reliable results still come from measurements made in the wind tunnel, especially near the maximum lift or into the stall. Naturally, the importance of testing techniques here must be noticed, and airfoil measurements must be made carefully using established methods. Nevertheless, most of the understanding of airfoil section characteristics, including the effects of geometric shape, has come from measurements.

It is essential for engineers how to interpret and use two-dimensional airfoil characteristics. To this end, the ability to access, analyze, and use graphs from catalogs of airfoil characteristics is critical. For example, in a design problem, it may be necessary to select an airfoil shape to meet a set of aerodynamic requirements. Quantities such as maximum lift coefficient, minimum drag coefficient, maximum lift-to-drag ratio, etc., can all be necessary for quantifying the aerodynamic characteristics of airfoils and can also be a basis for airfoil selection. In some cases, the characteristics of candidate airfoils may need to be compared, which should always be done on a non-dimensional basis in terms of force coefficients, such as in the form of a polar.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Research how lift and drag measurements of a “two-dimensional” airfoil section could be best performed in a wind tunnel environment.
Explain why the aerodynamic center for an airfoil moves aft on the airfoil with increasing free-stream Mach number. Hint: Consider the nature of the pressure distributions in subsonic, transonic, and supersonic flow.
What is the role of the angle of attack in determining the characteristics of an airfoil section?
Can you explain the terms “stall” and “critical angle of attack” in relation to airfoil sections?
How do airfoil sections contribute to the efficiency and performance of aircraft wings?
What factors affect the lift-to-drag ratio of an airfoil section?
Describe the phenomenon of boundary layer separation and explain its impact on airfoil performance.
Research shadowgraph and schlieren flow visualization systems for use in the wind tunnel. What are their relative advantages, and which one may be preferred?
Integration of the pressures to find the lift is usually a numerically accurate and successful process, but what is the most reliable method of finding the drag?

Other Useful Online Resources

Check out some of these additional online resources about the aerodynamics of airfoil sections:

Smoke flow visualization video of wing stall.
Aerodynamics – pressure profile around airfoil – a video by the NACA.
Video: Wind tunnel pressure data for a NACA 0012 symmetric airfoil
Video on understanding aerodynamic lift.
With this simulation, you can investigate how a wing produces lift and drag.
Prandtl-Glauert rule video from GaTech.
Video: Wind tunnel pressure data for a NACA 0012 symmetric airfoil.

Where $\infty$ means conditions at "infinity" or just far away from the wing where there is an undisturbed "free-stream" flow. ↵
Virginius E. Clark designed the Clark Y airfoil in 1922, which was derived from the Gottingen 398 airfoil. The airfoil has a thickness-to-chord ratio of 11.7% and is flat on the lower surface aft of 30% chord. The Clark Y gives reasonable overall aerodynamic performance but is rarely used in modern designs. ↵
See: "Robust Aerodynamic Shape Optimization—From a Circle to an Airfoil," X. He, J. Li, C. A. Mader, A. Yildirim, and J. R. R. A. Martins, Aerospace Science and Technology, 8748–61, 2019, doi:10.1016/j.ast.2019.01.051 ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2024 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n23