Aerodynamics of Airfoil Sections

J. Gordon Leishman

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

33 Aerodynamics of Airfoil Sections

Introduction

Understanding the aerodynamic behavior of airfoils and wings, often referred to as lifting surfaces, is crucial to aerospace engineering practice and essential for the successful design of most airplanes. Any lifting surface that moves through a fluid experiences a form of fluid-dynamic force. By definition, the component of this force that acts on the surface in a direction perpendicular to the relative freestream velocity, ${V_{\infty}}$ , or “relative wind direction”^[1] is called the lift, as shown in Figure 1. The component of the force acting on the surface in a direction parallel to the relative wind is called drag. The magnitudes of the lift and drag forces depend on several factors, including the size and shape of the lifting surface, its orientation relative to the flow, the flight Reynolds number (based on a characteristic length), and the freestream 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 or “2-D” airfoils are equivalent to wings with infinite span and aspect ratio, where the aspect ratio indicates the slenderness of the wing. While the concept of a 2-D 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 enables the isolation of other, more complex and interrelated effects associated with the finite span of a wing, including the impact of wing tip vortices and other aerodynamic effects caused by sweepback, twist, planform (chord) variations, and potentially other factors such as winglets.

Learning Objectives

Be aware of 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.
Understand how to model airfoil characteristics, both in attached and separated flow.
Appreciate the effects of flaps and other high-lift devices on airfoil performance.
Understand how to calculate the lift and other integrated quantities from the pressure and shear stress distributions about an airfoil.
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.

Origin of Aerodynamic Forces

The origin of the net aerodynamic forces on an airfoil or wing, such as lift and drag, stems from the integrated effects of the pressure and boundary layer shear stress distributions acting over its surface, as illustrated in Figure 2. These distributions, produced as the flow moves over the upper and lower surfaces, are nonuniform and can be either positive or negative. For example, higher pressure produces a larger inward normal force on the surface, whereas lower pressure produces a smaller inward normal force. The resulting pressure difference between the upper and lower surfaces produces the net aerodynamic force. Additionally, boundary-layer-induced shear stresses can be either positive (when the flow moves downstream) or negative (e.g., when the flow reverses).

Figure 2 shows that the shear stresses in the aggregate act downstream, primarily parallel to the chord line. Hence, the net shear contributes significantly to the drag force on the airfoil section. Likewise, differences in the pressure distribution between the upper and lower surfaces primarily contribute to the lift force and pitching moment on the airfoil. In contrast, shear stresses make 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 calculated by integrating the effects of the pressures and stresses acting on the surfaces. In the meantime, it is possible to proceed on the assumption that either measurement or calculation can be used to obtain these integrated results, with a particular approach 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 Figure 3. 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 to be nominally two-dimensional (2-D) only at sections well away from the wingtip vortices, which would be at the mid-span.

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

Initially, it is conceptually straightforward to understand the aerodynamic behavior of a wing by imagining the span and corresponding aspect ratio increasing without bound. In this idealized limit, the wingtip vortices are displaced so far from the central region that their influence becomes negligible, as shown in Figure 4. Consequently, the flow over the mid-span can be treated as nominally two-dimensional. In this manner, it is possible to theoretically and experimentally mimic the behavior of 2-D wings, thereby gaining insight into the aerodynamics of the airfoil section in isolation. The term “nominally” should be used with caution, as a real flow cannot be perfectly 2-D, regardless of how large the aspect ratio becomes. Still, the residual three-dimensional effects are reduced to a level that, for all practical purposes, can be regarded as insignificant. This approximation provides a sound foundation for further analysis of isolated airfoil aerodynamics.

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

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 preventing wake rollup at the tips and effectively eliminating the tip vortices. However, this approach not only requires a large wing but also promotes three-dimensional flow separation and the local formation of “stall cells” before stall onset. Another common practice is to test a short-span wing between two “false” walls, as shown in Figure 5. In wind tunnel terminology, this is called a “2-D insert.” This approach has been widely used to test 2-D airfoils, and flow visualization has confirmed the validity of nominally 2-D flow development over the wing section at mid-span, at least up to the onset of stall. Post-stall, the flow always tends to become inherently more three-dimensional on any lifting surface, regardless of the test method used.

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

A key metric in two-dimensional airfoil performance is the maximum attainable lift coefficient, $C_{l_{\text{max}}}$ . Despite advances in computational fluid dynamics (CFD), accurately predicting $C_{l_{\text{max}}}$ remains challenging, making wind-tunnel measurements indispensable. However, 2-D values of $C_{l_{\text{max}}}$ for the same airfoil section can vary significantly from one wind tunnel to another because of differences in their design, turbulence intensity, model aspect ratio, and wall interference effects. As previously mentioned, near the stall, three-dimensional flow phenomena frequently occur even in nominally 2-D tests, especially with longer-span airfoils. Hysteresis in stall behavior is also typical and often depends on whether the angle of attack is set before or after the wind tunnel is turned on. These factors underscore the importance of consistency in test methodology and adherence to standardized procedures when measuring 2-D airfoil characteristics.

Dynamic Pressure

The aerodynamic forces acting on a body are directly proportional to the dynamic pressure in the flow. 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. Note that dynamic pressure is proportional to the square 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 N m $^{-2}$ (Pa) in the SI system or lb ft $^{-2}$ in the USC system.

In particular, the freestream 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 as far away from the wing as possible, where there is an undisturbed “freestream” flow. This reference condition must be taken sufficiently far from the airfoil that the flow is essentially undisturbed by its presence, which in practice is at least a chord length and ideally more. As will be shown, the freestream dynamic pressure ${q_{\infty}}$ is often used as the reference pressure in the definitions of most non-dimensional coefficients used in airfoil and wing aerodynamics.

Pressure Coefficient

Pressure distributions on airfoils are usually presented in nondimensional form using the pressure coefficient, denoted by ${C_p}$ . The pressure coefficient compares the local static pressure at a point on the airfoil surface with the freestream static pressure, using the freestream dynamic pressure as the reference pressure. Therefore,

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

where $p$ is the local static pressure on the airfoil surface, $p_\infty$ is the freestream static pressure, and $q_\infty$ is the freestream dynamic pressure.

The sign of ${C_p}$ is important. If the local pressure equals the freestream static pressure, then $C_p = 0$ . If the local pressure exceeds the freestream static pressure, then $C_p > 0$ . If the local pressure is less than the freestream static pressure, then $C_p < 0$ , which is often referred to as a suction pressure. On the upper surface of a lifting airfoil, much of the pressure distribution usually has negative values of ${C_p}$ , especially near the leading edge, where the flow accelerates, and the static pressure decreases.

For incompressible flow, Bernoulli’s equation provides a useful relationship between local velocity and the pressure coefficient. Along a streamline outside the boundary layer, the Bernoulli equation can be written as

(4) $\begin{equation*} p + \frac{1}{2}\varrho_\infty V^2 = p_\infty + \frac{1}{2}\varrho_\infty V_\infty^2 \end{equation*}$

where $V$ is the local flow speed just outside the boundary layer. This equation can then be rearranged as

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

Dividing by the freestream dynamic pressure, $q_\infty = \frac{1}{2}\varrho_\infty V_\infty^2$ , gives

(6) $\begin{equation*} C_p = \frac{p - p_\infty}{q_\infty} = 1 - \left( \frac{V}{V_\infty} \right)^2 \end{equation*}$

This result shows that the pressure coefficient responds to changes in the flow velocity relative to the freestream. If $V = V_\infty$ , then $C_p = 0$ . If the flow accelerates over the airfoil, then $V > V_\infty$ and $C_p < 0$ , meaning that the local static pressure is below the freestream value. These negative values are often referred to as suction pressures. If the flow decelerates, then $V < V_\infty$ and $C_p > 0$ . At a stagnation point, where the local velocity is zero, the incompressible relation gives $C_p = 1$ .

Plots such as the one shown in Figure 6 are widely used to interpret airfoil behavior, including leading-edge suction peaks, stagnation points, pressure recovery, flow separation, shock waves in transonic flow, and the effects of camber and angle of attack. Notice that pressure coefficient plots are usually drawn with negative values of ${C_p}$ upward on the vertical axis. This convention may seem unusual at first, but it makes the plot resemble the physical suction loading on the airfoil, i.e., stronger suction on the upper surface appears higher on the graph. On a lifting airfoil, the flow usually accelerates around the leading edge and over the upper surface, producing a region of strongly negative ${C_p}$ known as a suction peak. Downstream of this peak, the pressure usually recovers as the flow slows. If this pressure recovery is too rapid, the boundary layer may thicken or separate, producing a stall. Therefore, a plot of ${C_p}$ along the chord helps indicate where the airfoil produces lift and where flow separation may begin.

Typical pressure coefficient distribution around an airfoil section.

Airfoil Forces and Pitching Moments

Pressure coefficient distributions like that shown in Figure 6 are especially useful because they show how an airfoil produces lift and pitching moments. The difference between the lower-surface and upper-surface pressure coefficients gives the net pressure loading on the airfoil section. The pressure contribution to the section lift coefficient, denoted by $C_l$ , and the corresponding pitching moment coefficient about a reference point $x_{\rm ref}$ on the chord, denoted by $C_{m,{\rm ref}}$ , are approximately

(7) $\begin{equation*} C_l \simeq \int_0^1 \left( C_{p_l} - C_{p_u} \right) \, d(x/c) \qquad \text{and} \qquad C_{m,{\rm ref}} \simeq - \int_0^1 \left( C_{p_l} - C_{p_u} \right) \left( \frac{x}{c} - \frac{x_{\rm ref}}{c} \right) \, d(x/c) \end{equation*}$

where $C_{p_u}$ and $C_{p_l}$ are the pressure coefficient distributions on the upper and lower surfaces, respectively. The term $\left(x/c - x_{\rm ref}/c\right)$ is the nondimensional moment arm from the reference point to the pressure point on the chord. The minus sign in the pitching moment expression follows from the usual convention that a positive pitching moment is nose-up. Therefore, an upward pressure loading acting behind the reference point gives a nose-down contribution to the pitching moment. For example, for the pitching moment about the quarter-chord point, $x_{\rm ref}/c = 1/4$ . These expressions neglect the small contribution of shear stress to lift and pitching moment, but they capture the primary physical mechanism by which an airfoil produces lift and pitching moment, which is by differential pressure.

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 Figure 7. As previously discussed, an airfoil section can be considered a “2-D” wing. Therefore, for airfoils, dimensional force and moment quantities per unit span are used; i.e., the lift per unit span is denoted by $L'$ = $L$ /unit span. This represents the lift force per unit span. Similarly, the drag per unit span is written as $D'$ , and the pitching moment per unit span about a specified reference point is written as $M'_{\rm ref}$ . The corresponding nondimensional section coefficients are obtained by using the dynamic pressure and chord as reference quantities, so that

(8) $\begin{equation*} C_l = \frac{L'}{\frac{1}{2}\varrho_\infty V_\infty^2 c}, \qquad C_d = \frac{D'}{\frac{1}{2}\varrho_\infty V_\infty^2 c}, \qquad \text{and} \qquad C_{m,{\rm ref}} = \frac{M'_{\rm ref}}{\frac{1}{2}\varrho_\infty V_\infty^2 c^2} \end{equation*}$

These coefficients are defined more formally in the next section, but they are introduced here to emphasize the connection between the dimensional section forces and moments, $L'$ , $D'$ , and $M'_{\rm ref}$ , and the corresponding airfoil coefficients, $C_l$ , $C_d$ , and $C_{m,{\rm ref}}$ . Furthermore, when force and moment coefficients are defined for airfoils, a reference length is needed, which is usually the chord length, denoted by $c$ .

The resultant forces on an airfoil at an angle of attack ${\alpha}$ showing two statically equivalent axis systems.

By definition, the lift force $L$ or lift force per unit length, $L'$ , acts in a direction that is perpendicular to the freestream 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 axial chord force per unit span, $A'$ , which is taken here as positive 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 lift and drag are usually preferred. 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

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

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

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

Aerodynamic Coefficients

Engineers must become familiar with how non-dimensional force and moment quantities are defined and applied to airfoils, wings, and other body shapes, as well as the specific values of these coefficients and their meanings. While dimensional forces (lb or N) and moments (ft-lb or N-m) are useful in many analyses, it is far more convenient to work with nondimensional 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.

The airfoil section’s non-dimensional or dimensionless force coefficients are defined using the freestream dynamic pressure ${q_{\infty}}$ and chord ${c}$ as a reference length. The span of a 2-D 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, as illustrated in Figure 8. A force is pressure times area, so the reference area of a force per unit span will be ${S = c \times b (= 1) = c}$ .

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

Therefore, the 2-D lift coefficient can be defined as

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

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 across the span, consistent with the 2-D assumption.

In summary, the 2-D 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 they tend to increase the angle of attack of the airfoil section; thus, 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 Figure 9. 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 about any convenient point on an airfoil can be taken; the transformation requires the application of statics. The 1/4-chord is the most common reference axis.

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

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

$\vspace*{1pt}$ 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, $C_{m_{\rm cp}} = \displaystyle{\frac{M_{\rm cp}/ \mbox{\small unit~span}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_{\rm cp}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_{\rm cp}}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 c^2}} = 0$

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 finite-span aerodynamic coefficients from those of 2-D airfoils.

Representative Aerodynamic Coefficients

The data shown in Figure 10 (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 is one for which the Mach number is low enough that compressibility effects are relatively small. These results span an angle-of-attack range from fully attached flow to the stalled-flow condition. They are typical of the airfoil characteristics found in various standard catalogs, such as those from Abbott & Von Doenhoff. The results are often presented in relation to variations of the chord-based Reynolds number and/or the freestream 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 lift coefficient, $C_l$ .

Parameters such as the maximum lift coefficient, minimum drag coefficient, maximum lift-to-drag ratio, pitching moments, and other metrics are significant for quantifying the aerodynamic characteristics of airfoils. These parameters can also be a basis for airfoil selection. However, interpreting the results from the preceding graphs and identifying relevant quantities requires practice, as the overlap among many points and curves can be confusing. To this end, a more straightforward presentation is shown in Figure 11, 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 freestream 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 lift-curve slope of many conventional airfoils is not strongly influenced by viscosity, provided the flow remains attached and the boundary layer is thin. Recall that boundary layers are thin, viscous-dominated regions near the surface. Notice that in this region, the lift coefficient increases almost in proportion to 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 slight negative angle.

As the angle of attack increases, the boundary layer thickens, and the aerodynamic characteristics begin to exhibit nonlinear behavior. The airfoil will soon reach its maximum lift coefficient, $C_{l_{\rm max}}$ . A further increase in angle of attack leads to flow separation on the upper surface, a sudden loss of lift, and a significant increase in drag; this process is known as a stall. The flow also becomes unsteady at stall, as turbulence and vortices are shed from the airfoil’s upper surface, resulting in fluctuating aerodynamic forces.

The essential flow physics of the stall have been discussed previously and are illustrated schematically in Figure 12. 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 usually 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.

A stall occurs when a thickening boundary layer causes flow separation, thereby reducing lift and increasing drag.

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, indicating that the effective aerodynamic center is slightly forward of the 1/4-chord over this range of angle of attack. This follows from the moment transfer relation

(12) $\begin{equation*} \frac{dC_{m,c/4}}{dC_l} = \frac{x_{c/4} - x_{\rm ac}}{c} \end{equation*}$

so a positive slope of $C_{m,c/4}$ versus $C_l$ implies $x_{\rm ac} < x_{c/4}$ .

Check Your Understanding

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:

The zero-lift angle of attack, $\alpha_0$ .
The maximum lift coefficient, $C_{l_{\rm max}}$ .
The stall angle of attack, $\alpha_s$ .
The minimum drag coefficient $C_{d_{\rm min}}$ 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 a Reynolds number of 5.7 x 10⁶:

The legend on the graphs indicates that, for a Reynolds number of 5.7 × 10^6, the results are shown with square symbols. Referring to the left-side plot, the zero-lift angle $\alpha_0$ for the smooth airfoil is approximately -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 obtained from the right-hand plot; the minimum drag coefficient, $C_{d_{\rm min}}$ , for the smooth airfoil is approximately 0.006 at a lift coefficient of 0.4.

For the rough airfoil at a 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 lower for the rough airfoil, at approximately 1.21.
The corresponding stall angle of attack $\alpha_s$ is about 15.1^o for the rough airfoil.
These values are shown in the right-hand plot, where the minimum drag coefficient, $C_{d_{\rm min}}$ , is higher than for the smooth airfoil and is approximately 0.0098 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 are different when the airfoil’s leading edge is roughened up to represent the in-service degradation of a wing’s surface. Roughness causes a boundary layer to become turbulent almost immediately after it forms, resulting in higher average drag and a more rapid increase in drag with increasing angle of attack.

Lift Characteristics

At angles of attack below the stall, the lift on the airfoil section is approximately proportional to the angle of attack and the local dynamic pressure. This linear relationship is exact within the framework of linearized inviscid aerodynamic theory. The lift per unit span can be written as

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

where $C_{l_{\alpha}}$ is called the lift-curve slope, i.e., 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 is measured in terms of per unit of angle of attack, i.e., per degree ( $/^{\circ}$ ) or per radian ( $/$ rad). At low Mach numbers, $C_{l_{\alpha}} \approx 2\pi$ per radian or 0.11 per degree. In coefficient form, then

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

Remember that the preceding relationship in Eq. 14 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. 14 up to an angle of about 10^o to 12^o, 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^o, 10^o and 12^o angle of attack for:

A symmetric airfoil.
A positively cambered airfoil with a zero-lift angle of attack of -1.2^o.

Show solution/hide solution.

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 12 $^{\circ}$ then $C_l$ = 1.278.
$C_l = C_{l_{\alpha}} \left( \alpha - \alpha_0 \right) = 0.1065\left( \alpha - (-1.2^{\circ})\right) = 0.1065 \left( \alpha + 1.2^{\circ} \right)$ . For 5 $^{\circ}$ then $C_l$ = 0.660, for 10 $^{\circ}$ then $C_l$ = 1.193, and for 12 $^{\circ}$ then $C_l$ = 1.406.

The simplicity of the relationship in Eq. 14 is advantageous in various engineering analyses, although its limitations to airfoil operation below stall must be recognized. However, no universal linear relationship can represent the variation of aerodynamic coefficients in the post-stall regime. In this regime, $C_l$ , $C_d$ , and $C_m$ vary nonlinearly with angle of attack and depend strongly on airfoil geometry, Reynolds number, Mach number, surface condition, and the extent of flow separation.

Maximum Lift & Stall Types

As previously discussed, as the angle of attack increases, the adverse pressure gradient over the upper surface intensifies, thickening the boundary layer and disrupting the otherwise linear relationship between lift and angle of attack. As a critical angle is approached, flow separation occurs, and the airfoil stalls. The lift coefficient at which the airfoil stalls is called the maximum lift coefficient, denoted by $C_{l_{\rm max}}$ .

The stall process and the achievable value of $C_{l_{\rm max}}$ depend on both the airfoil shape and the operating conditions, i.e., Reynolds and Mach numbers. Some airfoils, particularly those with greater thickness and camber, exhibit a gradual onset of stall, whereas others with sharp leading edges stall abruptly. Under subsonic flow conditions, airfoils generally fall into three primary categories of stall: thin-airfoil stall, leading-edge stall, and trailing-edge stall. These classifications are based on the location and movement of the point of flow separation on the airfoil’s upper surface, as well as on its effects on lift and pitching moment.

Representative lift and moment curves illustrating the three types of stall are shown in Figure 13 for a NACA 63-series airfoil with varying thickness-to-chord ratios. As the name implies, thin-airfoil stall occurs on airfoils with low thickness-to-chord ratios and sharp leading edges. A strong adverse pressure gradient near the nose leads to laminar separation and the formation of a long separation bubble. As the angle of attack increases, the reattachment point moves rearward until reattachment fails. The lift curve displays a gradual break, and the moment curve shows a nose-down trend at relatively low angles. In the stalled condition, the upper surface experiences nearly constant static pressure, leading to reduced lift and a significant increase in pressure drag. The center of pressure shifts aft, resulting in a more decisive nose-down pitching moment about the quarter-chord. Airfoils that exhibit thin-airfoil stall tend to produce relatively low values of $C_{l_{\text{max}}}$ .

Lift and pitching moment characteristics of the NACA 63-series airfoil section with different values of the thickness-to-chord ratio.

Leading-edge stall occurs on airfoils with moderate thickness and pronounced camber. A small laminar separation bubble typically forms near the leading edge of the airfoil section. As the angle of attack increases, this bubble may either burst when the boundary layer fails to reattach, or the flow may undergo turbulent reseparation just downstream of the bubble. The latter is the more common mechanism at higher Reynolds numbers. Both processes lead to an abrupt stall onset, marked by a sharp loss of lift and a rapid change in pitching moment. Airfoils that exhibit leading-edge stall can achieve high values of $C_{l_{\text{max}}}$ , but the attainable value depends strongly on airfoil geometry, Reynolds number, surface condition, and whether the laminar separation bubble remains stable before bursting.

Trailing-edge stall is more gradual and typically occurs on thicker or more highly cambered airfoils. The turbulent boundary layer separates near the trailing edge and shifts forward as the angle of attack increases. This results in a rounded peak in the lift curve and a relatively modest nose-down change in pitching moment. Trailing-edge stall is generally less sensitive to airfoil shape and Reynolds number and is more predictable. However, at higher Reynolds numbers, it can still occur abruptly and may resemble a leading-edge stall if detailed flow diagnostics are unavailable. Airfoils that exhibit trailing-edge stall generally achieve high values of $C_{l_{\text{max}}}$ . A schematic summary of the three stall mechanisms is shown in Figure 14.

The three stall mechanisms that occur on airfoils under subsonic flow conditions are: thin-airfoil stall, leading-edge stall, and trailing-edge stall.

Airfoil geometry, including thickness-to-chord ratio and camber, strongly influences the maximum lift coefficient, $C_{l_{\text{max}}}$ . A summary of these effects is shown in Figure 15, with the ordinate at 25% representing a combined geometric metric of the thickness-to-chord ratio ( $t/c$ ) and camber. Increasing the $t/c$ of the airfoil up to about 15% generally improves the value of $C_{l_{\text{max}}}$ , although further increases yield diminishing returns unless needed for structural considerations. Forward camber increases $C_{l_{\text{max}}}$ to a point by moderating the leading-edge suction peak and the associated adverse pressure recovery, thereby delaying flow separation. Cambered airfoils typically achieve higher values of $C_{l_{\text{max}}}$ than symmetric ones. However, this often comes with a more negative nose-down pitching moment, which may be a consideration in specific applications.

Effect of airfoil thickness and camber on the maximum lift coefficient of subsonic airfoil sections.

Effect of Flaps

Figure 16 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 for this airfoil and flap configuration, there is also a decrease in the stall angle of attack with flap deflection.

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

The increase in lift coefficient on an airfoil from a trailing-edge flap deflected by a small angle $\delta_f$ is often written in the approximate form

(15) $\begin{equation*} \Delta C_l = C_{l_\alpha} \, \kappa_f \, \delta_f \end{equation*}$

where ${C_{l_\alpha}}$ is the lift-curve slope of the airfoil section, $\delta_f$ is the flap deflection in radians, and $\kappa_f$ is an empirical flap lift-increment factor. The value of $\kappa_f$ depends on the flap type, flap chord ratio, hinge location, gap sealing, Reynolds number, and whether flow separation occurs. For a plain trailing-edge flap, typical values of $\kappa_f$ are less than unity and must normally be obtained from experiment, handbook data, or a more detailed aerodynamic calculation.

For example, if a plain flap has a representative lift-increment factor of $\kappa_f = 0.5$ and the lift-curve slope is approximated as $C_{l_\alpha} = 2\pi$ per radian, then

(16) $\begin{equation*} \Delta C_l = 2\pi \times 0.5 \, \delta_f = 3.14 \, \delta_f \end{equation*}$

where $\delta_f$ is in radians. For a flap deflection of $10^\circ$ , or $\delta_f = 0.1745$ radians, then

(17) $\begin{equation*} \Delta C_l = 3.14 \times 0.1745 = 0.55 \end{equation*}$

This value should be regarded as an approximate two-dimensional estimate. Actual flap increments depend on the airfoil shape, flap geometry, Reynolds number, gap details, and the onset of flow separation.

Further increases in the maximum lift coefficient may be achievable with different types of flaps, such as slotted, Fowler, or double- or triple-slotted. When $\kappa_f$ is interpreted as an effective lift-increment factor in the simple relation above, typical approximate values for common flap types are shown below.

Flap type

Approximate $\kappa_f$

Plain flap

0.4 to 0.7

Split flap

0.8

Slotted flap

1.2 to 1.3

Fowler flap

1.3 to 1.4

In conjunction with leading-edge slats, significant stall-speed reductions can be achieved on larger airliner types. For an airplane, however, the penalties for using more complex flap systems include increased airframe weight, higher costs, and increased maintenance requirements.

Drag Characteristics

The variation in drag with angle of attack and/or lift coefficient is of great interest because of its significant effects on aircraft performance. Specifically, drag generally reduces the aircraft’s performance; drag must be overcome by thrust, so it directly affects speed, climb, range, and endurance. Some representative results are shown in Figure 17, illustrating the drag coefficient $C_d$ versus the 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 remains low until the angle of attack and the corresponding lift coefficient increase to the point at which significant boundary-layer thickening occurs. Then, the drag grows rapidly when a stall occurs.

Representative variations of the 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 results are from wind tunnel tests, the effects of roughness can simulate the typical “in-service” abrasion of wing leading edges compared with when they were new from the factory (and thus initially smoother). The effects of roughness are significant. In aircraft design, it is necessary to account for surface roughness to avoid overestimating aircraft performance by assuming a perfectly smooth airfoil, which is unlikely to be achieved in practice.

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

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

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

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

Again, the preceding equation applies only to airfoil characteristics below stall and is invalid in the presence of significant flow separation or in the stalled-flow regime. In the case of a symmetric airfoil, where $\alpha_0 = 0$ , $C_{d_{1}}$ and ${d_1}$ will be zero.

Figure 17 also shows results for a NACA 63-series laminar-flow airfoil. This design maximizes the extent of the laminar boundary layer over the forward portion of the airfoil, thereby reducing skin-friction drag. These airfoils, with very smooth surfaces, tend to produce low-drag “buckets” in which drag is relatively low, but typically only over a limited range of lift coefficients near the design operating condition.

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 Figure 18. This geometric feature produces a favorable pressure gradient over a larger forward portion of the airfoil, thereby promoting laminar boundary-layer flow for a greater downstream distance. The downside is that such airfoils typically produce lower maximum lift coefficients, 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.

The previous discussion also suggests that surface roughness promotes turbulent mixing, thereby rapidly destroying the laminar boundary layer. The resulting increase in skin-friction drag makes the airfoil perform comparably to, or sometimes worse than, a conventional airfoil. This latter reason is why laminar-flow airfoil sections are challenging to use effectively in practice: some surface roughness on the wing is inevitable.

However, such airfoils have seen greater success on the smooth, almost glass-like wings of sailplanes, though the wings must be polished and kept completely clean and free of debris for laminar flow to prevail. Nevertheless, obtaining an extensive laminar flow region on an airfoil can significantly reduce its drag, at least over a specific range of angles of attack and lift coefficients. Unfortunately, a practical and robust means of achieving extensive laminar-flow regions on an airplane wing remains a research challenge, although specific surface coatings can be helpful.

Drag Polars

Graphical presentations of $C_l$ versus $C_d$ or $C_d$ versus $C_l$ are often referred to as a “drag polar” or simply a “polar,” as shown in Figure 19. Polar plots are a valuable way to present airfoil-section characteristics because results for different operating conditions (e.g., Reynolds or 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, with $C_l$ plotted on the vertical axis and $C_d$ on the horizontal axis, is that the slope of a straight line running from the origin (0, 0) of the graph to any point on the polar is the lift-to-drag ratio and hence is a quantitative measure of the airfoil’s aerodynamic efficiency.

The best lift-to-drag ratio occurs where a straight line drawn from the origin is tangent to the polar curve. This point represents the best lift-to-drag ratio, i.e., the operating conditions at which the airfoil achieves 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, its operating Reynolds number, and Mach number.

Effects of Reynolds Number

The classic NACA airfoil measurements also indicate Reynolds-number effects, at least over a limited range, but all results exceed 1 million. Remember that for an airfoil, the Reynolds number will be based on the chord, ${c}$ , so

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

The “c” subscript on the Reynolds number is often dropped, so $Re$ is used instead. Flight vehicles typically experience higher chord Reynolds numbers, usually as high as $10^{8}$ , or approximately one hundred million. Notice that the Reynolds number is often quoted in 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 ( $Re = 10^6$ ), viscous effects become increasingly important in the boundary layer, and the lift and drag characteristics undergo profound changes. The aerodynamic characteristics of airfoils at lower chord Reynolds numbers are important because they affect the performance of many smaller-scale flight vehicles, such as UAVs and drones, which often have chord-based Reynolds numbers well below a million. The data in Figure 20 show the profound effects of operating airfoils at lower Reynolds numbers, ranging from 20,000 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 remains relatively low and almost constant over the attached-flow range, 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 significantly more rounded and eventually become 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 lower Reynolds numbers. A laminar boundary layer has relatively low skin-friction drag but also has less near-wall momentum and is more susceptible to separation under an adverse pressure gradient. A turbulent boundary layer has higher skin-friction drag, but its greater near-wall momentum makes it more resistant to separation. At low Reynolds numbers, the boundary layer may separate while still laminar, transition in the separated shear layer, and then reattach as turbulent flow, forming a laminar separation bubble. This process helps explain why low-Reynolds-number airfoil characteristics can become nonlinear and highly sensitive to airfoil shape, surface roughness, and freestream turbulence.

Furthermore, the development of a long laminar separation bubble (LSB) on the airfoil often contributes to these nonlinear characteristics, as shown in Figure 21. 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. 14, for example, is no longer applicable. In general, at lower Reynolds numbers where $Re < 10^5$ , 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 a wind tunnel, where careful ensemble averaging is required.

At low Reynolds numbers, laminar separation bubbles form on the suction side, reducing lift and increasing drag.

The corresponding drag polar for these conditions is shown in Figure 22, providing another helpful way to summarize the effects of Reynolds number on the aerodynamic characteristics. Notice again the profound effects of reducing the Reynolds number below 500,000, which deleteriously affect the lift-to-drag ratio, especially below 50,000. For extremely low Reynolds numbers on the order of $10^4$ or below, conventional airfoil sections typically produce low lift-to-drag ratios, often less than 5, depending on airfoil shape, surface condition, and operating lift coefficient.

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

At this point, a natural question arises: do better airfoil shapes exist for use at low Reynolds numbers? The answer is yes; however, identifying optimal shapes requires a detailed understanding of boundary-layer behavior under these conditions. Michael Selig provides an in-depth discussion of these aerodynamic issues in his lecture notes on low-Reynolds-number designs. More recently, the use of CFD has improved predictions of airfoil performance at low Reynolds numbers, especially when boundary-layer transition and laminar separation bubble behavior are modeled carefully. Because the aerodynamic forces and moments are small, experimental measurements at these scales can be complex and error-prone. CFD therefore plays a valuable role in supplementing wind-tunnel data and revealing systematic trends that may not be readily observed experimentally.

As shown in Figure 23, a notable effect is the sharp increase in drag at low lift coefficients as the Reynolds number decreases, a trend previously discussed. Another observation is the historical performance of legacy airfoils such as the NACA 0012 and the Clark Y. The latter, designed by Virginius E. Clark in 1922, was derived from the Göttingen 398 airfoil. The Clark Y has a maximum thickness-to-chord ratio of 11.7% and is flat on its lower surface aft of 30% chord. While it offers reasonable aerodynamic performance, it is less common in modern optimized designs than purpose-designed low-Reynolds-number airfoils, laminar-flow sections, or proprietary propeller and UAV sections. Contemporary airfoils exhibit much greater sensitivity to Reynolds number than older plate-type shapes. For example, the maximum lift coefficient of a simple 6% cambered plate increases from approximately 1.15 at a Reynolds number of 200,000 to about 1.32 at a Reynolds number of 10 $^6$ .

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.)

However, notice that 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. At the lowest Reynolds numbers, thinner airfoils outperform thicker ones in aerodynamic efficiency, contrary to conventional wisdom. For example, the cambered plate exhibits a maximum lift-to-drag ratio of approximately 23 at a lift coefficient of 0.8. In contrast, the Clark-Y has a lift-to-drag ratio of about half that of the cambered plate at a significantly lower lift coefficient.

Effects of Mach Number

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

A significant amount of research has been devoted to understanding the effects of compressibility on airfoils and wings, as well as to developing suitable airfoil sections and wing shapes that mitigate its adverse effects at higher flight Mach numbers. Figure 24 shows schlieren images of the compressible flow around a symmetric airfoil with a 10% thickness-to-chord ratio at an angle of attack of 2^o in the wind tunnel. The schlieren effect is produced by the refraction of light rays as they pass through regions of varying flow density. (Note: The lower dark vertical line in these images is a model support in the optical path but not in the flow.) Furthermore, it is worth noting that these circular images arise because the schlieren flow-visualization system uses spherical or parabolic concave mirrors.

Schlieren images of the compressible flow around a 10% thick, symmetric airfoil at an angle of attack of 2 °. 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 freestream Mach number $M_{\infty}$ (notice that $M_1$ is used to denote the freestream 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 evolution of shock waves and other compressible flow patterns; the shock waves are the darker zones produced by the schlieren effect.

No strong shock waves are present on the airfoil at $M_{\infty}$ = 0.7. Nevertheless, a close inspection of the image reveals a small, dark line near the 25% chord, indicating a weak shock wave and suggesting that the flow is near, or slightly beyond, the critical Mach number, $M^*$ . At the critical Mach number, the maximum local Mach number first reaches unity; just beyond this condition, a local supersonic pocket and weak shock waves may appear. 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 more distinct weak shock wave has formed. However, no shocks have formed on the lower surface yet.

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 bifurcates at the airfoil surface upon interacting with the boundary layer. The schlieren image suggests that the downstream boundary layer is now relatively thick. At $M_{\infty}$ = 0.90, the boundary layer separates downstream of the shock on the upper surface, a phenomenon known as shock-wave-induced flow separation, commonly referred to as 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. With further increases in freestream Mach number, the flow about the airfoil approaches fully supersonic behavior, with shocks and expansion waves attached to the airfoil surfaces. A summary of the preceding observations is presented as a schematic in Figure 25 for clarity.

As the freestream Mach number increases from subsonic to supersonic, the flow field undergoes profound changes, underscoring the importance of the Mach number in understanding airfoil characteristics.

What is inside a shock wave?

Inside a shock wave, faster upstream molecules penetrate into the downstream gas and collide with slower molecules moving in many directions. These collisions reduce the mean streamwise velocity and increase the random molecular velocity. The reduction in mean motion appears macroscopically as a loss of flow speed, while the increase in random molecular motion appears as higher temperature and pressure. Because this redistribution occurs over only a few mean free paths and is irreversible, entropy increases across the shock.

While the preceding flow visualization images are interesting and valuable, 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 Figure 26. 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 rapidly at higher Mach numbers because of the effects of compressibility. Notice also that the airfoil’s maximum lift decreases as the Mach number increases. As the Mach number approaches the critical Mach number and the onset of transonic effects, the attainable maximum lift coefficient without flow separation or stall is reduced.

Increasing freestream Mach number generally increases the lift-curve slope but decreases the maximum lift coefficient.

The theoretical relationship that represents this latter behavior on the lift-curve slope is known as the Glauert rule, as summarized in Figure 27 and compared with measurements from three NACA symmetric airfoil sections. The Glauert rule states that the lift-curve slope increases according to

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

which is theoretically exact for a flat plate or thin airfoil within linearized subsonic compressible-flow theory. A generalization of this result is to correct the low-Mach-number value of the lift-curve slope using

(22) $\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 higher thickness-to-chord ratios or significant nose camber. The validity of the Glauert rule generally extends only up to near the critical Mach number for the airfoil section, $M^*$ , i.e., the point at which the maximum local Mach number first reaches unity and nonlinear transonic effects begin to appear.

The “Glauert rule,” also known as the “Prandtl-Glauert” correction, is grounded in linearized subsonic aerodynamics.

Further examples of the effects of compressibility on the lift and drag characteristics of an airfoil are shown in Figures 28 and 29 below. Notice the increase in the lift coefficient at a given angle of attack as the Mach number increases, but only while the flow remains essentially subcritical and the assumptions of linearized compressible flow remain valid. This outcome means that the lift coefficient is higher for a given angle of attack and dynamic pressure.

However, after the local flow first becomes sonic, transonic effects develop, shock waves, shock-induced separation, and rapid pressure-drag rise can occur. Therefore, for a given angle of attack, the lift coefficient may decrease, and the drag coefficient may increase rapidly once the Mach number enters the transonic range. The attainable stall angle and maximum lift coefficient generally decrease as Mach number increases toward and beyond the critical Mach number.

Summary of the effects of freestream 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 should be used?

Selecting an airfoil for a specific purpose is a deliberate process that requires careful consideration and often considerable time. If conducted correctly, the process will involve both computational methods (for iterative design) and wind tunnel testing (to verify the final airfoil), though this is ideal. Today, many airfoils can be confidently designed solely with computational methods, though this carries some 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, the maximum lift-to-drag ratio, the lift-to-drag ratio at the design point, the maximum lift coefficient, and the 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 performance in off-design conditions. It is generally advisable to design with sufficient robustness to ensure acceptable operation throughout the operational flight envelope. Consideration of maneuvers and gusts may also dictate the airplane’s aerodynamic margins and must be incorporated 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 essential. It is always convenient to place the integrated forces at a point on the airfoil, but the question is: at which 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, as it is the aerodynamic center for a thin airfoil in an incompressible flow. However, even if another reference point was selected, converting from one to another is easy because it just requires applying the rules of statics, as shown in Figure 30.

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

For example, assume the normal force and pitching moment per unit span are known at a point located a distance $x_a$ from the leading edge of the airfoil, and it is desired to find the pitching moment about another point located a distance ${x}$ from the leading edge. Taking moments gives

(23) $\begin{equation*} M'_{x} = M'_{a} + N' \left( x - x_a \right) \end{equation*}$

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

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

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 small angles of attack, $C_n \approx C_l$ , so the corresponding approximation is

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

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

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

Center of Pressure & Aerodynamic Center

The center of pressure and the aerodynamic center are two reference points used to describe how the distributed aerodynamic forces on an airfoil may be represented by an equivalent resultant force and pitching moment. Although both are associated with the chordwise location of the aerodynamic loading, they serve different purposes and behave differently as the angle of attack changes. The center of pressure is the point through which the resultant aerodynamic force acts with zero pitching moment, but its location generally moves with lift coefficient and angle of attack. The aerodynamic center is the point about which the pitching moment remains approximately constant over the linear attached-flow range. Distinguishing between these two concepts is essential when interpreting airfoil data, transferring moments between reference points, and applying sectional aerodynamic characteristics in stability and performance calculations.

Center of Pressure

By definition, the center of pressure is a point about which the pitching moment is zero, i.e., the point where the resultant aerodynamic force can be assumed to act when that resultant force is nonzero. The principle is illustrated in Figure 31, where the center-of-pressure location, $x_{\rm cp}$ , effectively serves as the balance point (or fulcrum) of the aerodynamic forces.

The center of pressure can be determined if the lift and moment coefficients are known about any other reference point, the 1/4-chord often being used, i.e., the values of $C_l$ and ${C_{m_{1/4}}}$ are available. The aerodynamic center, however, requires the variation of moment coefficient with lift coefficient, i.e., the slope $dC_m/dC_l$ , over the attached-flow range. The best way to understand the process is to work through an example using actual airfoil measurements, which are provided 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 Figure 32 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 $C_l$ . Often, the pitching-moment curve has a shallow positive slope, indicating that the effective aerodynamic center is located close to, but slightly forward of, the 1/4-chord. This follows from the moment-transfer relation

(27) $\begin{equation*} \frac{x_{ac}}{c} = \frac{x_{\rm ref}}{c} - \frac{dC_{m_{\rm ref}}}{dC_l} \end{equation*}$

so, when the reference point is the 1/4-chord, a positive slope of $C_{m_{1/4}}$ versus $C_l$ implies $x_{ac} < c/4$ .

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

(28) $\begin{equation*} M'_{\rm LE} = M'_{c/4} - L' \frac{c}{4} = -L' x_{\rm cp} \end{equation*}$

In coefficient form, dividing moments per unit span by $q_{\infty} c^2$ and lift per unit span by $q_{\infty} c$ , this result becomes

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

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

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

For most airfoils with positive camber, the value of ${C_{m_{c/4}}}$ is negative, indicating that the center of pressure is generally located 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 Figure 32. Because the center of pressure is a moving point, it may not be located on the airfoil’s chord and becomes poorly defined when the lift or normal force approaches zero; therefore, the concept of the center of pressure is only sometimes a convenient one 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 the point about which the pitching moment is constant and independent of the angle of attack, at least over the linear attached-flow range. Equivalently, the pitching moment about the aerodynamic center is independent of the lift coefficient over this range. The procedure for finding the aerodynamic center, similar to that for the center of pressure, requires values of the lift and moment coefficients at a point, such as 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

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

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

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

The value of $dC_{m_a}/d C_l$ can be obtained by using

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

Therefore, to calculate the aerodynamic center location, the slopes of the lift curve (in the linear range) and the moment curve (in the same range) are required. Note that the linear range corresponds to conditions in which the flow is fully attached to the airfoil surface. This process involves finding the slopes of the best straight-line fits to the values of $C_l$ versus ${\alpha}$ and then to ${C_{m_{1/4}}}$ 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

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

and

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

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

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

It can be seen that, for the particular airfoil used here, the aerodynamic center is located at 24.25% chord. The location of the aerodynamic center depends on the airfoil section and the operating Mach number. Because the value of the lift-curve slope is always positive, the slope of the moment curve defines the sign of the position of the aerodynamic center relative to the 1/4-chord. Therefore, if this slope is positive, the aerodynamic center is located ahead of the 1/4-chord; if it is negative, it is located 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 the 1/4-chord at low freestream Mach numbers.

It can be concluded that the advantage of using the aerodynamic center to resolve the forces and moments is convenience. Within the linear attached-flow range, it is treated as a fixed point that remains approximately unchanged as the angle of attack varies. However, determining the aerodynamic center location is more challenging because the slopes of the lift and moment curves must be calculated.

What is a least-squares linear fit?

The linear least-squares method is a numerical method for finding the straight line that best fits a dataset. 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.

Representing Nonlinear Airfoil Characteristics

In the attached-flow range, the airfoil aerodynamic coefficients can be represented by relatively simple functions of angle of attack. The lift coefficient increases almost linearly, the pitching moment about the aerodynamic center remains nearly constant, and the drag coefficient changes only gradually. These trends arise because the pressure distribution varies in proportion to the angle of attack, while the boundary layer remains thin and mostly attached.

For the lift coefficient, recall that the usual linear approximation is

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

where ${C_{l_\alpha}}$ is the lift-curve slope and $\alpha_0$ is the zero-lift angle of attack. The corresponding pitching moment coefficient about a given reference point can be written as

(38) $\begin{equation*} C_{m,{\rm ref}} = C_{m,{\rm ref},0} + C_{m_\alpha}\left(\alpha-\alpha_{\rm ref}\right) \end{equation*}$

where $C_{m_\alpha}=dC_{m,{\rm ref}}/d\alpha$ over the attached-flow range. If the reference point is the aerodynamic center, then this slope is approximately zero, so that

(39) $\begin{equation*} C_{m,{\rm ac}} \simeq \text{constant} \end{equation*}$

The drag coefficient does not vary linearly because, even in attached flow, the drag usually changes with the lift coefficient. Over a narrow range of angle of attack, it may be represented locally as

(40) $\begin{equation*} C_d \simeq C_{d,{\rm ref}} + C_{d_\alpha}\left(\alpha-\alpha_{\rm ref}\right) \end{equation*}$

but over a wider attached-flow range, a quadratic form is usually more useful, i.e.,

(41) $\begin{equation*} C_d = C_{d_0} + d_1 (\alpha -\alpha_0) + d_2 (\alpha-\alpha_0)^2 \end{equation*}$

where $C_{d_0}$ is the zero-lift drag coefficient and $d_1$ and $d_2$ are empirically derived constant coefficients. An alternative form is

(42) $\begin{equation*} C_d = C_{d_0} + d^{\prime}_1 \, C_l + d^{\prime}_2 \, C_l^2 \end{equation*}$

where again, $d^{\prime}_1$ and $d^{\prime}_2$ are coefficients.

These attached-flow relationships are among the most useful approximations for representing airfoil aerodynamics, but they apply only while the flow remains mostly attached. As the angle of attack increases further, the pressure recovery over the upper surface becomes more severe, and the resulting adverse pressure gradient thickens the boundary layer. Local flow separation may then begin, marking the onset of static nonlinear aerodynamic behavior when measured or modeled under steady conditions at a fixed angle of attack.

The first signs of this behavior are usually a rounding of the lift curve, a more rapid increase in drag, and a nose-up change in pitching moment. The lift coefficient may continue to increase with angle of attack to a point, but at a decreasing rate, until the maximum lift coefficient, $C_{l_{\rm max}}$ , is reached. Beyond this point, further increases in angle of attack produce more extensive separation, a loss of lift, a significant increase in pressure drag, and a substantial nose-down pitching moment as the center of pressure shifts aft. The details of what occurs depend strongly on airfoil geometry, Reynolds number, Mach number, surface roughness, and freestream turbulence.

The issue now is how to represent this nonlinear behavior in a form suitable for practical calculations. Several methods can be used, including table look-up from measured or computed data, empirical curve fits, and semi-empirical stall models. Each method provides a way to estimate $C_l$ , $C_d$ , and $C_m$ beyond the range where the linear relationships are adequate.

Table Look-Up

One common engineering method for representing nonlinear static airfoil characteristics is a table look-up process. In this approach, the aerodynamic coefficients are stored in a numerical table rather than being represented by analytical formulas. For example, the table may contain measured or computed values of $C_l$ , $C_d$ , and $C_m$ at a sequence of angles of attack for a specified Reynolds number and Mach number. A simple table might have the form

(43) $\begin{equation*} \begin{array}{|c|c|c|c|} \hline \alpha & C_l & C_d & C_m \\ \hline -4^\circ & C_l(-4^\circ) & C_d(-4^\circ) & C_m(-4^\circ) \\ -2^\circ & C_l(-2^\circ) & C_d(-2^\circ) & C_m(-2^\circ) \\ 0^\circ & C_l(0^\circ) & C_d(0^\circ) & C_m(0^\circ) \\ 2^\circ & C_l(2^\circ) & C_d(2^\circ) & C_m(2^\circ) \\ 4^\circ & C_l(4^\circ) & C_d(4^\circ) & C_m(4^\circ) \\ 6^\circ & C_l(6^\circ) & C_d(6^\circ) & C_m(6^\circ) \\ 8^\circ & C_l(8^\circ) & C_d(8^\circ) & C_m(8^\circ) \\ \hline \end{array} \end{equation*}$

This type of table allows a computer program to extract the airfoil coefficients at an operating condition that usually does not fall exactly on one of the tabulated entries.

Suppose, for example, that the required angle of attack is $\alpha = 3^\circ$ , but the table contains data only at $\alpha = 2^\circ$ and $\alpha = 4^\circ$ . The program first locates the two tabulated angles that bracket the required value. It then interpolates between the corresponding coefficient values. For linear interpolation, the lift coefficient can be estimated as

(44) $\begin{equation*} C_l(3^\circ) = C_l(2^\circ) + \frac{3^\circ-2^\circ}{4^\circ-2^\circ} \bigg[ C_l(4^\circ)-C_l(2^\circ) \bigg] \end{equation*}$

and the same procedure would be applied separately for $C_d$ and $C_m$ . In this way, the table is effectively converted into a continuous numerical approximation to the discrete airfoil data.

In a more general implementation, the coefficients may also depend on Reynolds number and Mach number, i.e., $C_l = C_l(\alpha, Re, M)$ , $C_d = C_d(\alpha, Re, M)$ , and $C_m = C_m(\alpha, Re, M)$ . The program must then locate the surrounding tabulated values for all relevant variables. For example, if the required condition lies between two angles of attack, two Reynolds numbers, and two Mach numbers, then the program interpolates within the small block of data surrounding that condition. This process is called multidimensional interpolation. The basic idea is the same as in the one-dimensional example: intermediate data are obtained by linearly interpolating neighboring data values.

This type of table lookup method is widely used for aircraft performance calculations, flight simulation, rotorcraft analysis, and aerodynamic prediction because it allows nonlinear airfoil behavior to be included without requiring a simple closed-form expression for the coefficients. In the attached-flow region, where $C_l$ , $C_d$ , and $C_m$ vary smoothly with angle of attack, interpolation between tabulated values is usually well behaved. Near stall, however, the coefficients can change rapidly over a small angle-of-attack interval. A coarse data table may then miss the stall angle, underestimate or overestimate the maximum lift coefficient, or smooth out abrupt changes in drag and pitching moment. For this reason, airfoil tables usually require finer angle-of-attack spacing near stall and careful limits on extrapolation beyond the range of available data.

Direct Curve Fitting

A second approach is direct curve fitting. Below stall, a low-order polynomial may represent the lift, drag, or pitching moment with acceptable accuracy. For example, the lift coefficient may be approximated as

(45) $\begin{equation*} C_l = \sum_{n=0}^{N} a_n \alpha^n \end{equation*}$

where the coefficients $a_n$ are obtained from measured data, usually by a least-squares fit. In practice, the lowest polynomial order that gives acceptable accuracy should be used. Higher-order polynomials may fit the data more closely over a limited range, but they often perform poorly outside that range and may even distort the linear behavior. Polynomial fits are especially troublesome when the aerodynamic coefficients change rapidly with angle of attack. Nevertheless, as shown in Figure 33, reasonable results can be obtained with low-order polynomial fits.

Direct curve-fitting on discrete airfoil data can often give reasonable polynomial representations for use in analysis.

An alternative curve-fitting method is to use a ratio of polynomials, such as

(46) $\begin{equation*} C_l = \frac{ \displaystyle \sum_{n=1}^{N} a_n \left(\alpha-\alpha_0\right)^{2n-1} } { \displaystyle 1 + \sum_{m=1}^{M} b_m \left(\alpha-\alpha_0\right)^{2m} } \end{equation*}$

where the coefficients $a_n$ and $b_m$ are again obtained from measured data. This form can give a smoother representation than a simple polynomial over a wider range of angle of attack. However, it remains an empirical curve fit and must be used cautiously outside the range of data from which it was obtained. Another practical approach is to use piecewise curve fits. For example, one expression may be used in the attached-flow range, another through the stall region, and another in the post-stall range. The curves must be matched carefully at the break points to avoid artificial discontinuities in the aerodynamic coefficients or their slopes. This approach can be useful, but it becomes more complicated when the breakpoints and curve-fit coefficients also vary with the Reynolds and Mach numbers.

Kirchhoff-Helmholtz Model

A more physically motivated semi-empirical approach is based on the Kirchhoff-Helmholtz separated-flow model.^[2] The idea is to represent the effect of trailing-edge separation through an effective trailing-edge separation point, denoted by $\scriptstyle f$ , measured nondimensionally from the leading edge, as shown in Figure 34. For a fully attached flow, $\scriptstyle f$ is close to unity, meaning that the flow remains attached almost to the trailing edge. As the separation point moves forward on the chord, $\scriptstyle f$ decreases. The separated flow region then reduces the lift relative to the value predicted by attached-flow theory.

In this type of model, the lift coefficient can be approximated by

(47) $\begin{equation*} C_l = C_{l_{\alpha}}(Re,M_\infty) \left( \frac{1+\sqrt{f}}{2} \right)^2 (\alpha-\alpha_0) \end{equation*}$

where $C_{l_{\alpha}}(Re,M_\infty)$ is the lift-curve slope at the appropriate Reynolds number and Mach number, $\alpha_0$ is the zero-lift angle of attack, and $\scriptstyle f$ is the effective trailing-edge separation point. When $f=1$ , the flow is fully attached, and the usual linear result is recovered. As $\scriptstyle f$ decreases, the separated region grows, and the lift coefficient decreases.

The intermediate part of this method is determining the effective separation point $\scriptstyle f$ . If static airfoil data are available, then $\scriptstyle f$ can be inferred from measured values of $C_l$ by rearranging Eq. 47, giving

(48) $\begin{equation*} f = \left[ 2 \sqrt{ \frac{C_l} {C_{l_{\alpha}}(Re,M_\infty)(\alpha-\alpha_0)} } -1 \right]^2 \end{equation*}$

which requires good estimates of $C_{l_{\alpha}}(Re,M_\infty)$ and $\alpha_0$ from the measured lift curve, as previously described. The resulting value of $\scriptstyle f$ should be interpreted as an effective separation point rather than as a direct measurement of the physical separation location on the airfoil; it is a modeling parameter that allows the nonlinear reduction in the lift coefficient to be represented compactly.

For practical calculations, the variation of $\scriptstyle f$ with angle of attack can be represented using empirical curve fits. One commonly used form is

(49) $\begin{equation*} f = \begin{cases} 1 - 0.3 \exp\left( \dfrac{(\alpha-\alpha_0)-\alpha_1}{S_1} \right), & \alpha \leq \alpha_1 \\[8pt] 0.04 + 0.66 \exp\left( \dfrac{\alpha_1-(\alpha-\alpha_0)}{S_2} \right), & \alpha > \alpha_1 \end{cases} \end{equation*}$

where $\alpha_1$ is a break point associated with the onset of strong nonlinear behavior, and $S_1$ and $S_2$ control how gradually or abruptly the separation develops.

The idea behind this type of model is illustrated in Figure 35. The left plot shows measured and reconstructed lift coefficient curves for a NACA 0012 airfoil at several Mach numbers. The right plot shows the corresponding inferred separation point, $\scriptstyle f$ , as given by Eq. 48 and fitted with Eq. 49, where $f \approx 1$ indicates mostly attached flow, and decreasing $\scriptstyle f$ indicates progressive forward movement of the separated-flow region. The nonlinear lift curve is reconstructed by allowing the effective separation point $\scriptstyle f$ to move forward on the chord as the angle of attack increases. At low angles of attack, $\scriptstyle f$ remains close to unity, and the flow behaves almost as in attached flow. Near the stall, $\scriptstyle f$ decreases rapidly, which reduces the lift coefficient relative to the value predicted by the linear attached-flow model.

Reconstruction of nonlinear static lift characteristics using an effective trailing-edge separation point model.

The pitching moment can also be modeled semi-empirically by relating the movement of the center of pressure to the growth of the separated region. As separation grows, the pressure distribution changes, and the center of pressure may shift significantly. One possible representation is

(50) $\begin{equation*} \frac{C_{m_{1/4}}}{C_l} = k_0 + k_1(1-f) + k_2 \sin(2\pi f^m) \end{equation*}$

where $k_0$ , $k_1$ , $k_2$ , and $m$ are empirical constants obtained by fitting measured moment data. The coefficient $k_0$ represents the attached-flow moment behavior, while the remaining terms model the moment changes caused by the growth of the separated-flow region. This type of expression should not be interpreted as a fundamental law; rather, it is a compact semi-empirical representation of the observed moment break associated with separation and stall.

Post Stall Behavior

At still higher angles of attack, beyond the normal static-stall range, the airfoil enters a fully separated-flow regime. In this regime, the detailed airfoil shape is less important than in attached flow, and the aerodynamic behavior approaches that of a stalled flat plate. Representative high-angle-of-attack data for a NACA 0012 airfoil are shown in Figure 36. The lift coefficient changes sign as the angle of attack passes through $90^\circ$ and $-90^\circ$ , while the drag coefficient reaches values of order 2 when the airfoil is broadside to the flow.

High-angle-of-attack lift and drag characteristics for a NACA 0012 airfoil over the full range from $-180^\circ$ to $180^\circ$ .

For approximate engineering estimates in the fully separated high-angle-of-attack range, the quasi-steady lift coefficient may be represented by

(51) $\begin{equation*} C_l = A \sin\big( 2(\alpha-\alpha_0)\big) \end{equation*}$

where $A$ is an empirical constant. This form captures the approximate antisymmetric lift behavior expected for a body with fully separated flow. The corresponding drag coefficient may be represented approximately by

(52) $\begin{equation*} C_d = D + E \cos\big( 2(\alpha-\alpha_0)\big) \end{equation*}$

where $D$ and $E$ are empirical constants selected to match the measured drag level. For this form to give a maximum drag near $\alpha-\alpha_0 = 90^\circ$ , the fitted value of $E$ would normally be negative. The constants must be obtained from data for the airfoil and Reynolds number of interest, although values giving $C_d$ of order 2 near $\alpha-\alpha_0 = 90^\circ$ are typical of a fully separated flat-plate-like flow.

The foregoing expressions used to represent “nonlinear” static aerodynamics are useful only as rough post-stall approximations. They do not capture the detailed effects of airfoil geometry, Reynolds number, Mach number, surface condition, three-dimensionality, or unsteady vortex shedding, but they can provide reasonable trends when detailed airfoil data are unavailable. More generally, nonlinear static models provide compact representations of airfoil behavior beyond the linear attached-flow regime, but their reliability depends on the quality and range of the data used for calibration. Near stall and in the post-stall regime, small changes in Reynolds number, Mach number, surface roughness, freestream turbulence, and airfoil geometry can produce substantial changes in $C_{l_{\rm max}}$ , stall angle, drag rise, and pitching moment. Therefore, nonlinear airfoil characteristics are usually obtained from wind-tunnel measurements, validated CFD, or semi-empirical models calibrated against reliable data.

Pressure & Shear Stress Coefficients

Pressure coefficients (i.e., non-dimensional pressures) are commonly used to present pressure distributions around body shapes and airfoil sections. The pressure coefficient, ${C_p}$ , is defined as

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

where $p$ is the local static pressure and $p_{\infty}$ is the freestream static pressure. Again, note the use of the freestream dynamic pressure as the reference pressure.

Suppose the local pressure equals the freestream static pressure $p_{\infty}$ , then $C_p = 0$ . If the local pressure is less than the freestream static pressure $p_{\infty}$ , then $C_p < 0$ . At a stagnation point, where the local flow velocity is brought to zero, $C_p = 1$ for incompressible flow when the Bernoulli equation is applied. In compressible flow, however, the stagnation-point pressure coefficient depends on Mach number. For isentropic subsonic compressible flow,

(54) $\begin{equation*} C_{p,0} = \frac{2}{\gamma M_\infty^2} \left[ \left(1+\frac{\gamma-1}{2}M_\infty^2\right)^{\gamma/(\gamma-1)} -1 \right] \end{equation*}$

which approaches unity only in the low-Mach-number limit.

The shear stress or local skin friction coefficient is defined as

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

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

(56) $\begin{equation*} { C_f = \frac{1}{L} \int_{s=0}^{s=L} c_f \, ds } \end{equation*}$

If two surfaces are exposed to the flow, such as the upper (u) and lower (l) surfaces of an airfoil, then the section skin-friction drag coefficient can be written approximately as

(57) $\begin{equation*} C_{d_f} \approx \frac{1}{c}\left( \int_u c_{f_u} \, ds_u + \int_l c_{f_l} \, ds_l \right) \end{equation*}$

If a small slope assumption is made, then the integral can be simplified to

(58) $\begin{equation*} C_{d_f} \approx \frac{1}{c}\left( \int_u c_{f_u} \, dx + \int_l c_{f_l} \, dx \right) \end{equation*}$

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 integration process around the body contour can then be used to determine coefficients for lift, drag, and pitching moment. 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 drag, shear stress must be accounted for, as it is the dominant contributor 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 are shown in Figure 37. 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 knowledge 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 underlying the integration process used to determine lift is illustrated in Figure 38. Let $x$ be measured along the chord from the leading edge to the trailing edge, and let $y_u(x)$ and $y_l(x)$ denote the upper and lower surface ordinates. The normal force $N'$ is taken as positive upward, normal to the chord, and the axial or chord force $A'$ is taken as positive toward the trailing edge. For an elemental chordwise interval $dx$ , the pressure and shear contributions may be written as

(59) $\begin{eqnarray*} dN'_u & = & -p_u \, dx + \tau_u \, dy_u \\[10pt] dA'_u & = & p_u \, dy_u + \tau_u \, dx \end{eqnarray*}$

on the upper surface, and

(60) $\begin{eqnarray*} dN'_l & = & p_l \, dx + \tau_l \, dy_l \\[10pt] dA'_l & = & -p_l \, dy_l + \tau_l \, dx \end{eqnarray*}$

on the lower surface. The opposite signs of the pressure terms in $dN'_u$ and $dN'_l$ are important because pressure acts inward normal to each surface. Therefore, the pressure contribution to the normal force depends primarily on the pressure difference between the lower and upper surfaces.

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

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

(61) $\begin{eqnarray*} N' & = & \int_0^c \left[ p_l - p_u + \tau_u \frac{dy_u}{dx} + \tau_l \frac{dy_l}{dx} \right] dx \\[10pt] A' & = & \int_0^c \left[ p_u \frac{dy_u}{dx} - p_l \frac{dy_l}{dx} + \tau_u + \tau_l \right] dx \end{eqnarray*}$

where the shear stresses are signed quantities in the local downstream direction. After $N'$ and $A'$ have been determined, then $L'$ and ${D'}$ can be found using

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

For thin airfoils at small angles of attack, the pressure contribution to the normal force reduces to the familiar pressure-difference result,

(63) $\begin{equation*} C_l \simeq C_n \simeq \int_0^1 \left(C_{p_l}-C_{p_u}\right) d\left(\frac{x}{c}\right) \end{equation*}$

with shear making only a small direct contribution to lift but an important contribution to drag.

The pitching moment about the leading edge is obtained by weighting the normal and axial force components by their moment arms, remembering that positive pitching moment is nose-up, i.e.,

(64) $\begin{equation*} M'_{\rm LE} = -\int_0^c x \, dN' + \int_0^c y \, dA' \end{equation*}$

or, when the upper and lower surfaces are retained separately,

(65) $\begin{equation*} M'_{\rm LE} = -\int_{\rm LE}^{\rm TE} x \, dN'_u -\int_{\rm LE}^{\rm TE} x \, dN'_l +\int_{\rm LE}^{\rm TE} y_u \, dA'_u +\int_{\rm LE}^{\rm TE} y_l \, dA'_l . \end{equation*}$

From the geometry, $dy=(dy/dx)dx$ , so these integrals can be evaluated in terms of $x$ using the upper and lower surface shapes, $y_u(x)$ and $y_l(x)$ , together with their slopes.

Typical Pressure Distributions

The standard practice is to calculate (or measure) the pressure distributions around airfoils and plot the results in terms of the pressure coefficient, ${C_p}$ , as a function of a non-dimensional distance or chord, $x/c$ . As previously discussed, the freestream dynamic pressure is required to determine the pressure coefficient, which in turn requires calculating air density from measurements of ambient pressure and temperature.

An example of a measured (discrete) pressure distribution around an airfoil at subsonic conditions is shown in Figure 39. 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; therefore, at zero angle of attack (or almost at $-$ 0.1^o in this case), the upper surface pressure distribution is identical to that of the lower surface. Increasing the angle of attack produces significant negative (suction) pressure on the upper surface, with the lowest value 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 Figure 40. 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 subsonic flow, where the lowest pressure is much closer to the leading edge, resulting in an adverse pressure gradient over most of the chord. The steep adverse pressure gradient near the shock wave renders the boundary layer highly susceptible to local thickening and often leads to shock-induced flow 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 and the center of pressure on a two-dimensional body where the pressure coefficient ${C_p}$ distribution at low angles of attack is described by

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

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

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

where $A$ = 300, $B$ = 2.2777 and $C$ = 0.95.

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

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

The center of pressure location is obtained from the first moment of the pressure difference about the leading edge, i.e.,

$\frac{x_{cp}}{c} = \frac{\displaystyle{\int_0^1 (C_{p_{l}} - C_{p_{u}})\, \overline{x}\, d\overline{x}}}{C_l}$

For the first interval, then

$\int_0^{0.1} (C_{p_{l}} - C_{p_{u}})\, \overline{x}\, d\overline{x} = \int_0^{0.1} \left( 300 \overline{x}^3 - 0.95 \overline{x}^2 \right) d\overline{x} = \Bigg[ 75 \overline{x}^4 - \dfrac{0.95}{3}\overline{x}^3 \Bigg]_0^{0.1} = 0.00718$

For the second interval, then

$\int_{0.1}^{1} (C_{p_{l}} - C_{p_{u}})\, \overline{x}\, d\overline{x} = \int_{0.1}^{1} 3.2277(\overline{x} - \overline{x}^2) d\overline{x} = 3.2277 \Bigg[ \dfrac{1}{2}\overline{x}^2 - \dfrac{1}{3}\overline{x}^3 \Bigg]_{0.1}^{1} = 0.52289$

Therefore, the total moment is

$\int_0^1 (C_{p_{l}} - C_{p_{u}})\, \overline{x}\, d\overline{x} = 0.00718 + 0.52289 = 0.53007$

and so

$\frac{x_{cp}}{c} = \frac{0.53007}{1.4025} \approx 0.378$

Numerical Integration of Pressure

Rarely are analytic (continuous) pressure distributions available about a body, as in the preceding example. Pressure values are typically known at discrete points, such as those obtained from measurements on a wing in a wind tunnel or from computational fluid dynamics (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. Pressure integration can determine the pressure-drag contribution, but it cannot determine the skin-friction contribution unless the surface shear stresses are also known. A momentum-deficiency approach is often used to determine the total drag.

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

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

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.,

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

where $N$ is the number of discrete points for which ${C_p}$ or $\Delta C_p$ is known. The process is shown in Figure 41. 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 usually available at different values of $x/c$ on the upper and lower surfaces. This means that the integral must be found in two parts, i.e., a numerical summation of the form

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

where $N_u$ is the number of points on the upper surface and $N_l$ is the number of points on the lower surface, as shown in Figure 42.

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

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

An extra ${\overline{x}}$ term appears in the equation, representing the 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 midpoint of each trapezoid, i.e., giving an equation of the form

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

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

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

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

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

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

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

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 many points are also 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 Figure 43. 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 provided below; you can copy and paste it into your MATLAB environment. Running the code yields $C_l$ = 1.4025, which agrees with the value obtained from the analytic integration. Of interest is the effect of varying the number of points used by the linspace function in numerical integration.

clc
clear
close all

figure
hold on

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;
cl = trapz(x,dcp) % section lift coefficient using the trapezoidal rule

plot(x,cpu)
plot(x,cpl)

set(gca,’YDir’,’reverse’)
axis([0.0 1.0 -2.5 1.2])

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, which can use relatively rounded leading edges and substantial camber, supersonic airfoils are usually thin and have sharp leading edges. This geometry helps the flow turn through the attached oblique shocks and expansion waves when the airfoil operates within its intended Mach number and angle-of-attack range. These waves produce pressure changes over the airfoil surfaces and strongly affect the lift, drag, and pitching moment. Common supersonic airfoil sections include angled-plane double-wedge airfoils and opposed circular-arc biconvex airfoils, as shown in Figure 44.

The sharp leading edges of supersonic airfoils help prevent the formation of a detached bow shock ahead of the airfoil, thereby reducing wave drag. For example, in the schlieren image in Figure 45, the supersonic flow passes smoothly over the airfoil’s sharp nose, resulting in relatively low drag. For a supersonic airfoil, the thickness and camber distributions are designed to minimize energy losses associated with flow compression and expansion. However, this goal is often achieved only over a narrow range of angles of attack.

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

Consider the flow around an airfoil in the form of a double wedge (or diamond) in a supersonic flow, as shown in Figure 46. Unlike a subsonic airfoil with smooth surface curvature, a double-wedge airfoil is ideal for supersonic flow. Notice that disturbances cannot propagate upstream in a supersonic flow, so the flow ahead of the airfoil remains essentially undisturbed. The streamlines turn abruptly through the shock and expansion waves generated by the airfoil rather than curving smoothly ahead of it as they do in subsonic flow.

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

Compression and expansion waves form wherever the airfoil surfaces turn the supersonic flow. A turn into the flow produces an oblique compression wave (shock), decreasing the Mach number and increasing the static pressure, whereas a turn away from the flow produces an expansion fan, increasing the Mach number and decreasing the static pressure. For a double-wedge airfoil at a positive angle of attack, the pressure increases and decreases over the four surface panels in a manner determined by the local surface inclination relative to the freestream. The resulting pressure differences between the lower and upper surfaces produce the lift, while the wave system also produces wave drag. Downstream, additional compression or expansion waves occur as the flow is turned again at the points of maximum thickness and at the trailing edge.

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

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

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

The results in Eq. 21 (subsonic) and Eq. 74 (supersonic) are classical solutions in 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 2-D airfoils, as shown in Figure 47. Linearized supersonic theory also provides the basis for finite-wing supersonic theory, although the lift-curve slope of a finite wing depends on aspect ratio, sweep, Mach number, and whether the leading edges are subsonic or supersonic.

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

For thin symmetric airfoils at small angles of attack, linearized supersonic theory shows that the lift coefficient is independent of the thickness distribution to a first order. For example, the lift coefficient produced at a given angle of attack is the same for a flat plate, a symmetric diamond-wedge airfoil, or a symmetric biconvex airfoil. Camber can change the zero-lift angle and must be treated separately. The drag, however, differs because the dominant source in supersonic flight is wave drag, which depends strongly on the airfoil shape.

For example, for a symmetric diamond-wedge airfoil with maximum thickness at mid-chord, the surface inclination caused by thickness is approximately $t/c$ . Therefore, the wave-drag coefficient is

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

which shows that wave drag increases with the square of the airfoil’s maximum thickness-to-chord ratio, $t/c$ . In general, the wave drag of a supersonic airfoil can be expressed as

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

where $t/c$ is its maximum thickness-to-chord ratio and $f/c$ is its maximum camber ratio. 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.

Check Your Understanding #5 – Calculating the lift & drag coefficients on a 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^oat a Mach number of 3.5. Using linearized supersonic theory and considering only wave drag, calculate the corresponding lift-to-wave-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

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

The lift coefficient is

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

For a symmetric double-wedge airfoil with its maximum thickness at mid-chord, the surface inclination associated with thickness is approximately $t/c$ . Therefore, the wave drag coefficient is

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

Therefore, the lift-to-wave-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) speeds, they tend to exhibit 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 wing’s entire span. Another method is to droop the leading edge using camber, which maintains supersonic performance while markedly improving low-speed characteristics. Nevertheless, supersonic aircraft have relatively high takeoff and landing speeds and may require drogue parachutes to reduce landing distance.

Supercritical Airfoils

Because commercial jet airliners must achieve increasingly high cruise speeds approaching the speed of sound (i.e., at transonic Mach numbers), this requirement has led to the design of unique wing shapes known as supercritical wings. A supercritical wing also uses a supercritical airfoil shape to reduce the strength of shock waves, thereby reducing wave drag and increasing the drag divergence Mach number. The principle of this airfoil is illustrated in Figure 48, where the upper surface is relatively flat at the leading edge compared to a conventional airfoil.

A supercritical airfoil has a relatively flat upper surface and aft camber designed to weaken shock waves at a given Mach number, thereby reducing wave drag and delaying boundary-layer separation.

The classic approach to increasing the drag-divergence Mach number is to reduce wing thickness, i.e., the maximum thickness-to-chord ratio. However, this has several other disadvantages, including a reduced fuel tank capacity and increased 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 NASA. Today, supercritical airfoils are used on nearly all commercial jet aircraft, enabling them to cruise at higher flight Mach numbers, typically between 0.8 and 0.85.

A supercritical airfoil shape is notably distinctive. It has a point of maximum thickness fairly aft along the chord, with a relatively flat upper surface and slight camber. The aerodynamic principle used in transonic wing and airfoil design is to control the flow’s expansion to supersonic speed and its subsequent recompression, thereby 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 lift loss at a given angle of attack. However, such transonic airfoils have significant camber at their trailing edges, which helps recover the lost lift.

The net effect is a significant increase in the critical and drag-divergence Mach numbers, as shown in Figure 49. 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. However, the drag-divergence Mach number can vary with design factors such as the thickness-to-chord ratio, $t/c$ . A supercritical airfoil generally limits the onset of drag divergence to a higher freestream 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.

Observing a Shockwave on an Airliner’s Wing

When cruising at Mach 0.8 in an airliner, a faint shadow or shimmering band on the wing may be visible, as shown in Figure 50. Here, the optical effect of the upper-surface shock wave is observed. For a transonic supercritical wing, this shockwave is located quite far back from the leading edge, often near the mid-chord. The physics behind this observed effect can be explained by the changes in air density and the refraction of light rays as they travel through regions of varying flow density.

Science from your airplane window! The shadow or shadowgram of a shockwave on an airliner’s wing can often be seen if the sunlight strikes the wing from the correct angle.

At cruise, the flow over the upper surface of the wing accelerates to supersonic speeds and then returns to subsonic speed as it passes through a shock wave. Across this thin shock, the air density increases abruptly, say from $\varrho_1$ to $\varrho_2$ . Because the refractive index of air depends on its density according to the Gladstone-Dale relation, i.e.,

(77) $\begin{equation*} n = 1 + K_{\!GD}\,\varrho \end{equation*}$

then there is an associated jump in the refractive index of the air, i.e.,

(78) $\begin{equation*} \Delta n = K_{\!GD}(\varrho_2-\varrho_1) \end{equation*}$

Whenever light passes through a region where the refractive index changes according to Snell’s law, i.e.,

(79) $\begin{equation*} n_1 \sin\theta_1 = n_2 \sin\theta_2 \end{equation*}$

where ${n_1}$ and ${n_2}$ are the refractive indices on either side of the shock wave, and $\theta_1$ and $\theta_2$ are the angles of the ray before and after refraction. For small angles, the deflection angle can be approximated by

(80) $\begin{equation*} \Delta\theta \approx K_{\!GD}(\varrho_2 - \varrho_1)\,\tan\theta_1 \end{equation*}$

Sunlight streaming past the airplane is, therefore, bent by small angles where it crosses the shock wave, and it also incurs a small change in optical path. Along each ray, the optical path length is

(81) $\begin{equation*} \mathrm{OPL} = \int n\,ds \end{equation*}$

where $n$ is the local refractive index and ${ds}$ is an element of distance along the ray, so the shock wave introduces an optical-path difference given by

(82) $\begin{equation*} \Delta \mathrm{OPL} = \int \Delta n\,ds \;\approx\; K_{\!GD}\,(\varrho_2-\varrho_1)\,\delta \end{equation*}$

where $\delta$ is the effective thickness of the shock wave along the line of sight. The transverse gradient of refractive index bends the rays so that, before reaching the wing surface and the observer’s eye, neighboring rays diverge or converge slightly. The recorded brightness variation $\Delta I$ scales with the second derivative of the refractive index, i.e.,

(83) $\begin{equation*} \Delta I \ \propto\ \int \frac{\partial^2 n}{\partial y^2}\,dz =K_{\!GD}\int \frac{\partial^2 \varrho}{\partial y^2}\,dz \end{equation*}$

where $y$ is the transverse coordinate normal to the shock and $z$ is the line-of-sight coordinate. This quantity is relatively large even in the presence of a weak shock. Where the bundle of rays diverges, the surface appears darker, and where it converges, it appears brighter. The result is a visible spanwise stripe across the wing that shifts as the shock position changes with changes in airspeed or wing angle of attack.

This shadow should not be mistaken for a surface feature of the wing, such as a structural seam or joint, but is the direct manifestation of the refraction of light across the shock wave. It constitutes a natural shadowgraph, in which the Sun serves as the light source, the shock acts as the optical interface that bends the rays, and the human eye functions as the detector along the optical path. In this way, the physics of a compressible flow can be observed directly, using nothing more than sunlight and access to a passenger window.

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] airfoil shapes can be tailored 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 for lifting surface(s) may involve multiple aerodynamic design goals and can lead to more efficient aircraft or other aerodynamic components.

General Approach

The aerodynamic shape optimization process requires defining an objective function, mathematically parameterizing the airfoil’s shape, applying formal optimization algorithms, and iterating (carefully!) until an optimal airfoil shape is achieved. The process combines aerodynamic solutions, optimization techniques, and sensitivity analysis to refine the airfoil shape, ensuring it meets 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 account for off-design aerodynamic performance. In this regard, off-design operation is often referred to as “robustness.” Some degree of robustness is desirable because an aircraft cannot operate precisely at the design point.

The shape optimization 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 freestream velocity, angle of attack, Mach number, and Reynolds number. These parameters significantly affect aerodynamic performance and must be considered in the design process.
Use a panel method to solve aerodynamic problems, in which the airfoil shape is discretized into panels and integral equations are solved to determine the resulting pressure distribution. Alternatively, consider using computational fluid dynamics (CFD), although this approach will be significantly more time-consuming.
Implement optimization algorithms to adjust the airfoil shape to meet performance objectives. 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 ${J}$ quantifies the difference between the desired and actual performance. For a pressure distribution, say $p(x)$ , this can be expressed as

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

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 that the shape must satisfy, such as geometric and aerodynamic constraints, can now be introduced. For example, constraints may be imposed on the maximum or minimum thickness-to-chord ratio, on the amount of camber, or on both. The feasibility of manufacturing the shape may also be considered a design constraint, especially when different airfoils are used along the span of a wing or other lifting surface, such as a propeller, helicopter rotor, or wind turbine blade.

Typically, the airfoil’s shape must be parameterized mathematically using a set of shape variables. If control points or coefficients describe the shape, it can be expressed in the form

(85) $\begin{equation*} (x, y) = (x_0, y_0) + \sum_{i=1}^N \delta_i \, \vec{\phi}_i \end{equation*}$

where $(x, y)$ is the modified shape of the airfoil, $(x_0, y_0)$ is the initial or baseline shape, $\delta_i$ are shape parameters, and $\vec{\phi}_i$ are vector-valued basis functions that describe the corresponding changes in the $x$ and $y$ coordinates. For example, in the case of using a standard Fourier series

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

where ${\cos(n \pi \overline{x})}$ and ${\sin(n \pi \overline{x})}$ represents the nondimensional 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 ${\cos(n \pi \overline{x})}$ and ${\sin(n \pi \overline{x})}$ are referred to as the basis functions used to represent the shape.

Well-established gradient-based optimization methods can then be used to compute the gradient of the objective function with respect to the shape parameters that describe the airfoil. The shape parameters can then be updated using

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

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. Typically, the relaxation parameter should be kept relatively small (e.g., less than 0.05) to prevent significant shape changes in the airfoil during any design iteration, which could cause the method to fail. As part of the process, changes in shape parameters can be explored to assess their effects on the objective function, with sensitivity quantified by $\dfrac{\partial J}{\partial\delta_i}$ .

Finally, the shape parameters are updated via the optimization algorithm, thereby recalculating the objective function. This process can be continued until the convergence criteria are met, i.e., $\| \nabla J \| < \epsilon$ , where $\epsilon$ is an acceptable tolerance level based on the design goals. If the process runs smoothly, the numerical approach and shape changes must be carefully monitored to obtain the new airfoil shape.

Figure 51 shows representative outcomes of this process for a transonic airfoil section. The initial “conventional” airfoil shape exhibits a strong shock wave and a relatively low drag-divergence Mach number. The optimization objective is to maximize the drag-divergence Mach number while maintaining a thickness-to-chord ratio of at least 13% for structural integrity. The optimization process results in an airfoil with a flatter upper surface. This design modification results in a weaker shock wave and a higher drag divergence Mach number, thereby enhancing the airfoil’s performance at transonic speeds. Additionally, the trailing edge of the optimized airfoil exhibits increased camber, improving aerodynamic efficiency while satisfying the structural constraint.

The process of inverse airfoil design determines the shape of an airfoil to achieve 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, when available, can verify aerodynamic performance by placing a scaled or full-sized wing-section model in a controlled airflow environment. Pressure measurements over the surface of the airfoil can be compared with the predictions made, and any aerodynamic improvements can be quantified against the original baseline airfoil. While the method for optimizing the airfoil shape is well-established and relatively quick, the wind-tunnel verification process may be much longer.

Rooftop & Stratford Pressure Distributions

The upper surface or suction-side pressure distribution governs the state of the boundary layer and, therefore, the attainable lift of an airfoil. A design principle useful for both laminar and turbulent boundary layers is to allow the boundary layer to develop until it remains just below the threshold for flow separation. In this way, the suction-side pressure can be kept low over a larger portion of the surface, thereby increasing the lift generated before boundary-layer separation occurs.

Two canonical design conditions are of interest. The first condition is the laminar “rooftop” constant-pressure condition, which maintains the laminar boundary layer for as long as possible to minimize skin friction. The second condition is the Stratford turbulent recovery, which prescribes the steepest adverse pressure gradient that a turbulent boundary layer can withstand without separating. Inverse design methods for airfoil shapes typically allow the two conditions to be combined sequentially. Hence, the suction side maintains a nearly constant low pressure upstream of the transition point and then recovers according to Stratford’s recovery criterion downstream of the transition point.

Stratford’s turbulent separation criterion^[4] was derived using the boundary layer equations in conjunction with turbulent boundary layer measurements in adverse pressure gradients, and it remains one of the most useful results in practical aerodynamics. Starting from the suction peak at $\overline{x}_m = x_m/c$ , where the velocity has reached its maximum and the pressure its minimum, the criterion states that the subsequent recovery, written in terms of the nondimensional chordwise coordinate $\overline{x}=x/c$ , must satisfy

(88) $\begin{equation*} C_p^2 (\overline{x}-\overline{x}_m) \frac{dC_p}{d\overline{x}} = S \end{equation*}$

where $S$ is an empirical constant, for which $S\approx 0.35$ is used for most applications. Equation 88 expresses the combination of pressure and pressure gradient that will lead to the onset of flow separation. Once $\overline{x}_m$ and the overall recovery endpoint are specified, the criterion states that there exists only one pressure distribution that keeps the turbulent boundary layer attached. If the pressure recovery is more adverse, the flow separation will occur, resulting in a loss of lift and an increase in drag.

A reparameterization of the Stratford criterion is useful for inverse airfoil design methods, which is obtained by defining

(89) $\begin{equation*} C_p^\ast = 1 - \left(\frac{U_e}{U_{\max}}\right)^2 \end{equation*}$

where $U_e$ is the local flow velocity at the edge of the boundary layer, together with an effective length $x^\ast$ measured from the leading-edge suction peak. The Stratford criterion then takes the form

(90) $\begin{equation*} C_p^\ast \,\sqrt{x^\ast\frac{dC_p^\ast}{dx^\ast}} = K\left(\frac{Re^\ast}{10^6}\right)^{0.1} \end{equation*}$

where $Re^\ast = U_{\max} x^\ast/\nu$ is a Reynolds number based on the peak velocity, and $K$ is an empirical constant of order unity that depends on the surface curvature. Another formulation, attributed to Cebeci and Smith,^[5] incorporates the effective development length of the turbulent boundary layer, which accounts for the accumulated history of the adverse environment, i.e.,

(91) $\begin{equation*} \frac{C_p^{10/7}}{(\overline{x} - \overline{x}_m)^{1/2}(U_e^{5/2})^{2/7}} \left(\frac{dC_p}{d\overline{x}}\right) = \text{const} \end{equation*}$

This refined criterion is relevant when the transition occurs just upstream of the recovery, as it adjusts the separation limit to account for the relatively short development of the turbulent layer. In practice, Stratford’s criterion has been found to be conservative, i.e., predicting separation slightly earlier than it occurs. By shaping the suction-side pressure recovery to follow this criterion, inverse design methods^[6] produce airfoil shapes that can generate high lift coefficients before flow separation occurs.

Stratford’s pressure recovery criterion provides a basis for designing airfoils with high lift coefficients by delaying turbulent boundary-layer separation.

A laminar boundary layer cannot withstand a strong pressure recovery. Therefore, to encourage the development of a laminar boundary layer, the suction side is shaped to produce an approximately constant-pressure, or “rooftop,” distribution, as shown in Figure 52. After the leading-edge acceleration has produced a suction peak at $\overline{x}_m$ , the external velocity is held nearly constant downstream, i.e., $U_e(\overline{x})\approx U_{\max}$ and $C_p(\overline{x})\approx$ constant, so that the laminar boundary layer avoids a strong adverse pressure gradient before transition. Any attempt to recover pressure too early can lead to laminar separation; therefore, an approximately constant-pressure plateau is a useful idealization for maintaining laminar flow over part of the suction surface.

The extent of the rooftop region, defined by $C_{p,\min}$ , and its length $L_{\rm roof}$ are the key design variables. At typical Reynolds numbers, the suction peak occurs at $x/c \approx$ 0.07–0.15, with a stable rooftop region extending to $x/c \approx$ 0.15–0.60 to the transition point. Lower suction pressures result in higher lift coefficients but induce earlier transition and may also increase the airfoil’s sensitivity to surface roughness. Shallower rooftops extend the laminar flow and reduce drag, but limit the attainable maximum lift coefficient. The design of a laminar-flow airfoil is therefore a composite exercise: the suction side is given a rooftop plateau extending from $\overline{x}_m$ to the transition point $\overline{x}_{\rm tr}$ , beyond which the recovery follows Stratford’s law, prescribing the steepest pressure rise a turbulent boundary layer can withstand before separating.

Compressibility effects impose further limits on airfoil design. At transonic Mach numbers ( $M_\infty \sim$ 0.75–0.85), a deep rooftop produces a supersonic pocket that terminates in a shock, and shock-boundary-layer interaction often triggers flow separation before Stratford’s criterion is even reached. The suction-side sequence then consists of rapid acceleration to the rooftop pocket, a terminating shock, and a turbulent recovery downstream. Upstream, the rooftop must be moderated to keep the shock weak and attached, while downstream, the Stratford recovery must be relaxed to account for compressibility and the thickened boundary layer. The ultimate limit on lift in this regime is therefore set by both the onset of turbulent separation and shock-induced flow effects.

Summary & Closure

Understanding the aerodynamic characteristics of two-dimensional (2-D) airfoil sections is a prerequisite for understanding those of finite wings. While airfoil characteristics can be predicted, the most reliable results still come from wind-tunnel measurements, especially near the maximum lift coefficient or during stall. The importance of testing techniques in this context must be acknowledged, and airfoil measurements must be conducted carefully using established methods. Nevertheless, most of the understanding of airfoil section characteristics, including the effects of geometric shape, has come from measurements.

Engineers need to know how to interpret and use 2-D 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 the maximum lift coefficient, minimum drag coefficient, and maximum lift-to-drag ratio are necessary for quantifying the aerodynamic characteristics of airfoils. They can also serve as 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 using force and moment coefficients and drag polars.

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 freestream 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 the context of 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 wind tunnels. What are their relative advantages, and which one may be preferred?
The integration of pressure to find lift is usually numerically accurate and successful, but what is the most reliable method for finding 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.

Where ${\infty}$ means conditions at "infinity" or just far away from the wing where there is an undisturbed "freestream" flow. ↵
This approach is based on the classical Helmholtz-Kirchhoff free-streamline model for separated flow, later adapted in semi-empirical airfoil models by Beddoes. See Thwaites, B., Incompressible Aerodynamics, Oxford University Press, 1960; Beddoes, T. S., "Representation of Airfoil Behaviour," Vertica, Vol. 7, No. 2, 1983, pp. 183–197. ↵
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, Vol. 87, pp. 48–61, 2019, doi:10.1016/j.ast.2019.01.051 ↵
Stratford, B. S., "The Prediction of Separation of the Turbulent Boundary Layer," Journal of Fluid Mechanics, Vol. 5, 1959, pp. 1–16. ↵
Cebeci, T., & Smith, A. M. O., Analysis of Turbulent Boundary Layers, Academic Press, New York, NY, 1974, pp. 341–349. ↵
See: Eppler, R., Airfoil Design and Data. Springer-Verlag, Berlin, 1990. (Provides comprehensive use of rooftop distributions and Stratford recoveries in inverse airfoil design). ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 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

Introduction

Origin of Aerodynamic Forces

Two-Dimensional Flow

Dynamic Pressure

Pressure Coefficient

Airfoil Forces and Pitching Moments

Aerodynamic Coefficients

Representative Aerodynamic Coefficients

Lift Characteristics

Maximum Lift & Stall Types

Effect of Flaps

Drag Characteristics

Drag Polars

Effects of Reynolds Number

Effects of Mach Number

Pitching Moments

Center of Pressure & Aerodynamic Center

Center of Pressure

Aerodynamic Center

Representing Nonlinear Airfoil Characteristics

Table Look-Up

Direct Curve Fitting

Kirchhoff-Helmholtz Model

Post Stall Behavior

Pressure & Shear Stress Coefficients

Integration of Pressures & Shear Stresses

Typical Pressure Distributions

Numerical Integration of Pressure

Supersonic Airfoils

Supercritical Airfoils

Observing a Shockwave on an Airliner’s Wing

Inverse Airfoil Design and Shape Optimization

General Approach

Rooftop & Stratford Pressure Distributions

Summary & Closure

License

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