• Section 8.1 presents fundamental concepts and theories regarding airfoil lift and drag generation. It contains very important definitions. Additionally, it introduces how pressure is distributed along the upper and lower surfaces of the airfoil and how it affects the growth of the boundary layer and, eventually, flow separation.

• Section 8.2 defines important geometric properties of airfoils. It also presents information intended to make the aircraft designer better rounded when comes to identifying various airfoil types, such as NACA airfoils, and understanding of their background. For this purpose, the section introduces a number of airfoils that have gained fame or notoriety in the history of aviation.

• Section 8.3 discusses the generation of forces and moments on the airfoil. It details how various outside agents, such as very high airspeeds, high angle-of-attack, deflection of control surfaces, and even contamination, affects their aerodynamic properties. Finally, it presents a method to help the designer select the proper airfoil for a new aircraft design.

Pressure force:

(8-1)

Force due to fluid flow:

(8-2)

where

ρ = density, m/L³

V = airspeed, L/t

l = characteristic length, L

k = unknown constant of proportionality

a, b, c = exponents to be determined

Inserting the dimensions into Equation (8-2) yields:

(8-3)

Force due to fluid flow:

(8-4)

ρ = density of air, in kg/m³ or slugs/ft³

V = airspeed, in m/s or ft/s

S = reference area, typically wing area, in m² or ft²

C_r = non-dimensional coefficient that relates AOA to the force

The Smeaton Lift Equation (Obsolete)

From a historical standpoint, it is of interest to consider the now-obsolete Smeaton lift equation (which was how the Wright brothers determined the wing area required for their Flyer.) It was attributed to the English civil engineer John Smeaton (1724–1792), who is often referred to as the father of civil engineering, and was in use up until the beginning of the twentieth century. Smeaton’s lift equation is given as follows:

(8-9)

where

Engineers at the time considered the lift coefficient as the ratio of the object’s lift force to its drag force, where the drag was for a flat plate of area A mounted perpendicular to the airstream. Smeaton’s coefficient, k, is the drag of a 1 ft² flat plate at 1 mph. At the turn of the century (∼1900) the accepted value for this coefficient was 0.005 and this had been the value used by Otto Lilienthal in the design of his gliders. In fact, it was Smeaton himself who came up with this particular value; however, other sources claimed it ranged from 0.0027 to 0.005. The Wright brothers concluded the coefficient was wrong and experimentally determined it to be closer to 0.0033. The modern value is 0.00326 [1].

Section Lift Coefficient, C_l

The two-dimensional lift coefficient is commonly called the section lift coefficient. This concept is of great importance to the airplane designer and will be discussed at length later. One of this concept’s most useful properties is that it can indicate both the effective angle-of-attack of the airfoil and how close to stalling it is.

Maximum and Minimum Lift Coefficients, C_lmax and C_lmin

The largest and smallest magnitudes of the lift coefficient are denoted by C_lmax and C_lmin, respectively. The former always has a positive magnitude and the latter a negative one. These values are extremely important because they dictate the stalling speed (at positive and negative loading) of the aircraft (and therefore wing size and airplane weight), as well as impacting other important characteristics such as the maneuvering loads and spin behavior. Generally, the stall is defined as the flow conditions that follow the first lift curve peak, which is where the C_lmax (or C_lmin) occurs [2, p.1].

Lift Curve Slope, C_lα

C_lα indicates how rapidly lift changes with angle-of-attack. The maximum value of the slope is predicted by linear thin airfoil theory for incompressible flow to be 2π (approximately 6.283) and most airfoils indeed achieve a value close to that. For instance, the slope of the red dotted line displaying the linear range in Figure 8-3 is approximately 5.90. The lift curve slope is usually linear at low AOAs; however, it is nonlinear outside this range and can have a negative slope. A nonlinear lift curve slope is indicative of flow separation prevailing on the body.

Angle-of-attack at Zero Lift, α_ZL

This refers to the angle-of-attack at which the airfoil produces no lift. It is important when considering wing washout (see Section 9.3.5, Wing twist – washout and washin, ϕ) and when converting the two-dimensional lift curve to three dimensions, as the change can be approximated by, effectively, rotating the lift curve around this point, toward a shallower slope (see Section 9.5.5, Step-by-step: Transforming the lift curve from 2D to 3D). The α_ZL is shown in Figure 8-4.

Linear Range

The linear range (here shown ranging from AOA = −12° through 8°) is called so because, within it, one can estimate the lift coefficient for any AOA using a simple linear expression, such as the one below:

(8-10)

The extent of this region ultimately depends on the geometry and the operational airspeeds (via Reynolds numbers as discussed in Section 8.3.4, The effect of Reynolds number).

C_l at Zero AOA, C_lo

C_lo is the value of the lift coefficient at zero AOA. This is of great importance in the selection of the airfoil as it will affect the angle-of-incidence at which the wing must be mounted. Generally, this value ranges from 0.0 (for symmetric airfoils) to 0.6 (for highly cambered airfoils). It is negative for under-cambered airfoils (e.g. airfoils used near the root of high subsonic jet aircraft). If the AOA of zero lift, α_ZL, and lift curve slope, C_lα, are known, the value of C_lo can be calculated from:

(8-11)

Minimum Drag Coefficient, C_dmin

C_dmin is the lowest value of the drag coefficient that can be found on the drag polar. Its magnitude is vital to the selection of the airfoil. Ideally, C_dmin should be a low as possible, but it also has to be low where it counts; in the region of intended lift coefficient of cruise.

Lift Coefficient of Minimum Drag, C_lmind

C_lmind is the lift coefficient where the minimum drag coefficient occurs on the drag polar. This location impacts the selection of the airfoil for the same reason as C_dmin.

q = dynamic pressure, in lb_f/ft² or N/m²

p = pressure, in lb_f/ft² or N/m²

p_∞ = far-field pressure, in lb_f/ft² or N/m²

V_∞ = far-field airspeed, in ft/s or m/s

ρ_∞ = far-field density, in slugs/ft³ or kg/m³

M_∞ = far-field Mach number

γ = ratio of specific heats = 1.4 at altitudes where aircraft typically operate

Derivation of Equation (8-13)

From Bernoulli’s equation:

Inserting this into Equation (8-12):

QED

Derivation of Equation (8-14)

Using the equation of state (p = ρRT), the speed of sound can be found from:

Therefore, the dynamic pressure can be written as follows:

Inserting this into Equation (8-12) yields:

QED

The Canonical Pressure Coefficient

The canonical pressure coefficient is regarded by many as a better way to represent airfoil pressure distribution. The concept was introduced by A. M. O. Smith [3] to evaluate the adverse pressure gradient and help determine the onset of flow separation. The approach scales the pressure coefficient, so it varies between 0 and 1. This is done by selecting the peak pressure (at the start of the adverse pressure gradient, i.e. where pressure begins to increase). The canonical pressure coefficient is defined as follows:

(8-15)

“Conventional” Lift Distribution

Figure 8-6 shows a pressure distribution generated by a conventional cambered airfoil (the NACA 4415) at a subsonic condition and an AOA of 2°. The thick solid line shows the pressure along the upper surface and the dashed one along the lower surface. It is evident that, at this AOA, the pressure on the upper surface reaches its lowest value relatively close to the leading edge, or around 20% of the chord length. This means that a favorable pressure gradient extends only as far aft as 20%. In other words, assuming a smooth surface, the laminar boundary layer is promoted only over the first 20% of the chord length. This does not mean a transition will occur immediately thereafter, but this is very likely unless the surface of the airfoil is super-smooth.

Also note that the pressure distribution along the lower surface starts with stagnation condition (C_p = 1 at x = 0), which then develops into a low-pressure dip near the 10% of the chord. This is caused by the airflow accelerating around the curved geometry in the area of the leading edge. It highlights that if curvature is present in a fluid flow, the local pressure will always be reduced – an important fact to keep in mind when trying to reduce hinge moments of control surfaces.

Figure 8-6 also shows the pressure differential (difference between the upper and lower surface pressure). It is negative along the entire chord, indicating all segments of the chord are contributing to the lift, albeit in different capacities. For instance, it is evident that the first 50% of the chord contributes substantially more to the overall lift of the airfoil than does the remaining 50%. This fact begs the question: is it possible to design an airfoil whose chordwise distribution would be uniform? Such an airfoil would have to be the most efficient possible, because all chordwise stations would contribute equally, right? Well, strictly speaking the answer is yes, although this is not achievable in reality due to the tendency of air to separate. However, airfoils that attempt this have been designed and are already in use in many aircraft, in particular, sailplanes and high-performance composite GA aircraft. A discussion of their pressure distribution follows.

Stratford Distribution

In contrast, consider the chordwise distribution of the NLF(1)-0414F airfoil shown in Figure 8-7 at the same AOA of 2°. This distribution forms a distinct flat pressure contour on the upper surface. This contour is commonly referred to as a “rooftop” or Stratford pressure distribution. Such a distribution promotes an extensive laminar boundary layer, provided the surface is sufficiently smooth.

Consider a hypothetical pressure distribution that is uniform along the chord of the airfoil, from the LE to the TE. This would yield the most efficient generation of lift, as each chordwise segment is contributing equally to the total lift of the airfoil. Such a pressure distribution, on the other hand, is physically impossible because this would require the air pressure to change from a low to ambient pressure instantaneously. In real applications, the air pressure must be allowed to rise over a given distance. If the distance is too short, a separation will occur. If the distance is too long, full advantage is not taken of the benefits of the laminar boundary layer. If the distance is just right, the flow will be on the verge of separation and the flat roof will extend as far aft as possible and skin friction drag will be optimized for that airfoil. Modern airfoils are specifically designed to generate a chordwise pressure distribution along the upper surface that is uniform across much of the chord. However, this can only be achieved for a small range of specific AOAs (say 1°–2°). For this reason, it is imperative to define the mission of the aircraft clearly so the AOA at which it will be operated can be determined and used to design the airfoils.

Center of Pressure

As has already been stated, a body immersed in fluid flow induces a pressure field that, in turn, generates a resultant force and moment. The magnitude of the moment depends on the position on the body about which it is measured. Then, the center of pressure is defined as the point where the magnitude of the moment equals zero.

To better explain the concept, consider the three identical airfoils shown in Figure 8-8, all of which feature hinges in different places; one on the leading edge (left), the next at the center of pressure (center), and the trailing edge (right). Considering the leftmost airfoil, the lift force generates a moment that will rotate it leading edge down (counter-clockwise). By the same token, the rightmost airfoil, which is hinged at the trailing edge, would be rotated in the opposite direction: leading edge up (clockwise). It follows there must be a point between the leading and trailing edge about which no rotation would take place. This point is the center of pressure. Its location depends on the distribution of pressure along the airfoil and will change with AOA.

Aerodynamic Center and Quarter Chord Moment

The aerodynamic center is the point on a body about which the aerodynamic moment is independent of the AOA. Its presence is plotted in all standard NACA wind tunnel test graphs, where its consequence can be seen as the pitching moment curve that is constant over almost the entire α-sweep range. If the slope of the lift curve, , and the pitching moment about the quarter chord, , are known, the aerodynamic center can be computed from:

(8-16)

The pitching moment of airfoils is often reported at the aerodynamic center, although it is needed at the quarter-chord for many stability and control problems. The following expression can be used to transfer the moment from the aerodynamic center to the quarter chord:

(8-17)

Derivation of Equation (8-16)

Based on Figure 8-9, the moment about the aerodynamic center can be obtained by summing the forces (l) and moments (m) as follows:

Dividing through with q·S·c, where and by referencing Equation (8-6), this can be written in a coefficient form as shown below:

This can easily be solved for to give Equation (8-17). Then, differentiating with respect to α and recalling that the definition of the aerodynamic center means that the change in moment with change in α is zero, this above is rewritten as shown:

Using convention in writing the slopes and solving for x_ac results in the following:

QED

Momentum Theorem

The momentum theorem explains lift as the consequence of a wing moving through a mass of air and giving it a downward motion (see Figure 8-10). Since the mass of air is initially at rest, the downward motion means the vertical speed of the air changes from zero to some finite value in a given amount of time. This, in turn, means that a force will be generated in the opposite direction in accordance with Newton’s third law of motion. The magnitude of this force can be estimated using Newton’s second law of motion. The second law of motion states it is the rate of change of momentum of the mass of air that generates the force. The third law states an equal force that acts in the opposite direction of the motion of the mass is also generated. It is this force that we call lift. A common analogy used to describe this phenomenon is the recoil of a firearm. As is well known, the change in the momentum of a bullet generates a force that acts in the opposite direction of its motion. Thus, the sudden down flow imparted on the air effectively generates “continuous recoil” – the lift.

The downward motion of the air is called downwash, here denoted by the letter w, and it represents the vertical speed of air behind the wing. It contrasts with the horizontal speed; a result of the wing moving through air. If we know the downwash and the mass flow of air being deflected, the magnitude of the lift can be estimated, as stated earlier, using Newton’s second law of motion:

(8-18)

where = mass flow rate inside the cylinder = ρV·πb²/4.

The mass flow rate in the stream tube is given by , where A_tube is the cross-sectional area of the stream tube. If it is assumed that the diameter of the stream tube in Figure 8-10 equals that of the wingspan (denoted by b) then the rate of change of momentum (lift force) can be estimated from:

(8-19)

Equating this with the standard expression for lift (Equation (8-6)) allows the magnitude of the downwash to be estimated:

(8-20)

Noting that S = b·C_avg and AR = b/C_avg this can be rewritten as follows:

(8-21)

Since the downwash can approximated by w = εV (see Figure 8-10), we can write:

(8-22)

This equation leads to another result, which as will be shown later, is very helpful in stability and control theory:

(8-23)

Bernoulli Theorem

The Bernoulli theorem has been used for decades to explain the formation of lift to generations of engineers and pilots. It postulates that lift is the consequence of the difference in pressure between the upper and lower surfaces of an airfoil, although this would more properly be explained as the resultant of integrating the pressure over the entire surface of a body. Named after the Swiss mathematician Daniel Bernoulli (1700–1782), the theorem stipulates there is a relationship between the pressure and speed of the fluid at a point and along a streamline that goes through that point. This is expressed as follows:

where

Bernoulli’s theorem is used daily by thousands of people in industry and academia to estimate the aerodynamic forces acting on a body. Computational fluid dynamics (CFD) software uses the theorem to estimate aerodynamic forces and moments acting on a body with great success.

Kutta-Joukowski Circulation Theorem

The Kutta-Joukowski circulation theorem is much more a mathematical method than it is an explanation. Named after the German mathematician Martin Wilhelm Kutta (1867–1944) and the Russian scientist Nikolay Yegorovich Joukowski¹ (1847–1921), the theorem postulates that the lift generated by an airfoil can be considered the product of density (ρ), forward airspeed (V), and a mathematical concept called circulation (Γ). In short, the Kutta-Joukowski theorem states that the airfoil’s lift can be expressed as follows:

(8-26)

where the circulation is calculated using the expression [4]:

(8-27)

where

To better understand the concept of circulation, consider Figure 8-11, which shows the airflow around an airfoil. The top figure shows the far-field airspeed, V_∞, and some representative airspeed V₁ through V₆, positioned at selected locations in the flow-field, above and below the airfoil. Then, it is possible to calculate the difference between each of those airspeeds and the far-field airspeed. This is shown as the airspeeds ΔV₁ through ΔV₆ in the center figure. The important observation is that the airspeed differences along the upper surface are positive and point in the flow direction. Conversely, they are negative along the lower surface and point in a direction opposite that of the general airflow. As can be seen in the bottom figure, these differentials effectively form a path around the airfoil. This path is the circulation.

The Kutta-Joukowski theorem is very useful in a number of computational fluid dynamics (CFD) methods, such as the lifting line method (see Section 9.7.1, Prandtl's lifting line theory), and the vortex-lattice methods, where it is directly used to estimate the lift and induced drag force generated by a lifting surface.

Reynolds Number

The Reynolds number (R_e) is a measure of the ratio of inertial forces to viscous forces in a fluid flow. It is of great importance in the analysis of the boundary layer. It is defined as follows:

A simple expression, valid for the UK system at sea-level conditions only, is (V and L are in ft/s and ft, respectively):

(8-29)

For the SI system at sea-level conditions only, the expression becomes (V and L are in m/s and m, respectively):

(8-30)

The Effect of Flow Separation

Consider two identical aircraft whose only difference is size or scale. Imagine these are immersed in fluid flow at an identical airspeed such that, as far as the aircraft is concerned, it differs in Reynolds numbers only. Now consider them rotated slowly through an alpha-sweep, from 0° to 90°, while we collect their force and moment coefficients. At first, while the flow is mostly attached, both aircraft will effectively generate identical and linear force and moment coefficients. However, after some AOA (perhaps at α = 8°) has been reached, the flow begins to separate on the smaller body, while it remains attached on the larger one. The resulting force and moment coefficients for the small body now turn nonlinear, while remaining linear for the larger one. Eventually, the larger body too begins to experience the same effect (perhaps at α = 12°). The flow separation causes a large increase in drag and reduction in lift. The pitching moment may increase or reduce depending on the overall geometry. This calls for a distinction in the nature of the flow over the two bodies. Generally, three such types are identified, based on the character of the boundary layer that envelopes the body. They are called laminar, turbulent, and separated flow (see Figure 8-12).

Boundary Layer Transition

A body moving in a fluid will generally be exposed to three distinct types of fluid flow; one is characterized by the presence of a laminar boundary layer, another by a turbulent boundary layer, and the third by separated flow which contains molecules that flow in all possible directions, even upstream (see Figure 8-13). A laminar boundary layer occurs when the streamlines inside the boundary layer flow smoothly. In contrast, streamlines inside a turbulent boundary layer are chaotic. An important phenomenon takes place as the initially laminar boundary layer changes to a turbulent boundary layer, due to a process called transition. When the airflow strikes a smooth body a laminar boundary layer will form immediately, but will later degenerate when it transitions into a turbulent boundary layer. This generally occurs when the local Reynolds number equals approximately 5 × 10⁵. This is referred to as the transition Reynolds number.

Unfortunately, the transition is complicated by factors that may change the transition Reynolds number and may cause the transition to happen at a different (lower or higher) Reynolds number [7]. Note that the term early transition means that the transition takes place earlier than indicated by the value of the Reynolds number. By the same token, delayed transition refers to the opposite. Factors that may change the Reynolds numbers:

Flow Separation

In a separated flow the streamlines are separated from the surface. Such flow may actually flow upstream, against the direction of the free stream airflow. Flow separates when the gradient dV/dz equals zero on the surface (z = 0). The external geometry of an airplane should be shaped so the areas of flow separation at the mission condition are minimized, if not eliminated.

All three types usually manifest themselves in fluid flow in the order shown in Figure 8-13. This typically occurs on bodies as shown in Figure 8-14 or Figure 8-15. The former shows flow over an object that features a rounded, small-radius leading edge (LE). The flow forms a laminar boundary layer on the forward part of the object (the bow), followed by a transition to a turbulent boundary layer, and, finally, by the flow separating into a turbulent wake at the stern. Note that the streamlines outside the boundary layer are smooth and can be treated as if they belonged to an inviscid fluid. In fact, for computational efficiency, this is how most Navier-Stokes solvers treat fluid flow: viscid inside the boundary layer, but inviscid outside of it.

Figure 8-15 shows a different scenario, in which an object with a very blunt LE causes flow separation on the LE that results in (highly) turbulent, circulatory flow inside the flow separation region. The laminar streamlines flowing over this region attach aft of this region before separating into the turbulent wake at the stern.

The forward flow separation region is usually referred to as a “separation bubble.” Such bubbles can also form on objects that are considerably less blunt than the one in Figure 8-15. They can even occur on airfoils if their geometry is conducive to such formation. This may be caused by sudden change in curvature. Besides an increase in drag, separation bubbles on airfoils may lead to very detrimental stall characteristics.

Separation bubbles can occur on any aircraft, in particular at the wing/fuselage juncture. However, the phenomenon also occurs on lifting surfaces, in particular on small chords (or low Reynolds numbers). As discussed in Section 8.2.10, Famous airfoils, The NACA 23012 airfoil is used for a number of GA aircraft in spite of its abrupt stall characteristics, which are attributed to the formation of a separation bubble. But they are a prevalent problem for even smaller Reynolds numbers, such as those in which radio-controlled aircraft operate (60,000–500,000). In this region a stable bubble may form in the laminar boundary layer along the leading edge of the wing, increasing the drag of the vehicle. This is referred to by many as “bubble drag” [8]. One way of reducing this drag is to design the airfoil such that its transition ramp² reduces the chance of a bubble formation (refer to Ref. [8] for more details). Another way is to place a transition strip (often called a “trip strip”) along the leading edge to force the laminar boundary layer to transition into a turbulent one without forming the separation bubble. Currently this is a trial-and-error approach and a trip strip that is ideal for one Reynolds number may be detrimental for another one. Research shows their effect is contingent upon the size of the bubble, its intensity, AOA, and geometry of the airfoil.

Factors Affecting Laminar Flow

The laminar boundary layer is very sensitive to a number of factors. Some are out of the control of the designer, whereas others are not and, effectively, depend on his awareness. Among those are (list partially based on Bertin [9]):

Laminar BL:

(8-31)

Turbulent BL:

(8-32)

The dimensions δ_lam and δ_turb are defined in Figure 8-16.

Trailing Edge (TE) Stall

This is undoubtedly the best known among the three types of stalls and typically occurs on thick airfoils, whose thickness-to-chord ratio is around 12% or greater [10]. Such airfoils are characterized by a smooth change in C_l and C_m between the negative and positive lift peaks (see C_lmax and C_lmin in Figure 8-3 and Section 8.1.4, Properties of typical airfoils) and often well beyond those. The shape of the peak of the lift curve is rounded with a mild drop in C_l. Growth in post-stall drag polar is gradual, and the pitching moment curve is without sharp breaks. The flow stays mostly attached to an AOA of about 10°, beyond which it progressively moves forward. At maximum lift the flow is separated to approximately mid-chord the rear half of the airfoil.

Leading Edge (LE) Stall

Leading edge stalls are much less familiar to the General Aviation community than TE stalls. They are associated with sharp leading edge radii of thin airfoils (which are almost never used for GA aircraft – excluding the NACA 23012 airfoil, which is used on a number of GA aircraft). The sharp LE results in a larger pressure peak than airfoils with larger radius LE. This is evident in Figure 8-17, which shows the thinner NACA 4408 airfoil peaks at C_p ≈ −16.5 versus C_p ≈ −6.0 for the thicker NACA 4415 airfoil. As a consequence, the pressure recovery, behind the peak for the thinner airfoil, is much steeper than for the thicker one. This is a problem nature solves by separating the laminar boundary layer. This separation can occur well below the AOA where the airfoil actually stalls. It forces the laminar boundary layer to transition into a turbulent one, forming a bubble of trapped low-energy air between the surface of the airfoil and the boundary layer. An example of such a bubble is shown in Figure 8-18. On a wing the bubble is a spanwise vortex.

Research has shown that the bubbles come in two distinct forms, short or long, and each predicates different scenarios [11]. The difference between the two is determined using Owen’s criterion, which computes the Reynolds number of the displacement thickness of the boundary layer using the following expression:

(8-33)

where V is the velocity at the edge of the boundary layer, ν is the kinematic viscosity and δ₁ is the displacement thickness. In Ref. [12] this value was always found to be greater than 500 for a short bubble and less than 500 for a long one. Note that the size of the bubble, in effect, depends on the free stream Reynolds number, so it is possible the stall behavior of the airfoil changes from one to the other. If the Reynolds number is high enough, chances are no bubble will form. The difference between the two is as follows:

The maximum lift coefficient of thin airfoils that stall because of flow separation at the leading edge can be determined based on the leading edge geometry. A so-called leading-edge parameter, which is defined as the difference between the upper-surface ordinates of the airfoil at the 0.15% and 6% chord stations, has been used for this purpose with good results [13] (for instance see Section 9.5.12, Step-by-step: C_Lmax estimation per USAF DATCOM method 2). Additional correction is required for thicker airfoils. Compressibility effects are important on thick airfoils as they cause reduction in the maximum lift coefficient starting at M ≈ 0.2. Reference [13] also presents a method to estimate the maximum lift coefficient of several airfoils. The three types of airfoil stalls are illustrated in Figure 8-19.

Xfoil and XFLR5

Xfoil is probably the best known of the above codes. It dates back to 1986 and was written by Dr. Mark Drela, an aerodynamics professor at the Massachusetts Institute of Technology. It uses a high-order panel method and a fully-coupled viscous/inviscid interaction method to evaluate drag, boundary layer transition and separation. Xfoil is widely used in the aircraft industry and generally speaking is a reliable tool, although, in the view of this author, it suffers from a poor user interface when compared to many other codes. The user of modern computer operating systems is averse to the unfriendly interfaces of the bygone MS-DOS era. This has been solved in a program called XFLR5 [18], developed by Mr. André Deperrois, which makes Xfoil analyses much easier to perform (see Figure 8-21).

Xfoil allows the user to perform viscous and inviscid analysis of existing airfoils. The user can specify where a laminar boundary layer transitions into a turbulent one, or have the program predict the movement of the transition point with AOA. The viscous analysis can be used to predict C_l, C_d, and C_m to just beyond C_lmax, and uses Karman-Tsien compressibility correction at high subsonic airspeeds (see Section 8.3.6). The program allows the user to simulate control surface deflection by specifying hinge point and deflection angle.

PROFILE (“The Eppler Code”)

The software PROFILE was written by Dr. Richard Eppler of the University of Stuttgart and Dan Somers, a consulting aerodynamicist. The program uses a conformal-mapping method for the design of airfoils for low-speed applications with prescribed velocity-distribution characteristics. The program is claimed to be user-friendly by the distributer of the program, www.airfoils.com/.

AeroFoil

The software AeroFoil was developed by Mr. Donald Reid, a professional nuclear engineer who has a background in aerospace engineering, and “is intended to be the most ‘user-friendly’ of its type,” as stated on its website, aerofoilengineering.com/index.htm. The software uses a vortex-panel method coupled with integral boundary layer equations to calculate the aerodynamic properties of airfoils. It allows up to three airfoils to be compared simultaneously. Validation examples are provided on the website and show the predictions made by the program are in good agreement with experiment.

JavaFoil

JavaFoil is simple and easy-to-use software developed by the German aerodynamicist Dr. Martin Hepperle. The program performs a potential flow analysis using a higher-order panel method, in which the vorticity varies linearly along each panel representing the airfoil. Then, an integral boundary layer method is applied, using a separate boundary layer analysis module. Beginning at the stagnation point, the method solves a set of differential equations to help evaluate the characteristics of the boundary layer. According to information on the developer’s website, the equations and criteria for transition and separation were developed by Dr. Eppler. The program is free and available from www.mh-aerotools.de/airfoils/javafoil.htm. It provides a routine that allows a large number of airfoils to be generated.

Design Process

Once a decision has been made to design an airfoil, the first step in the design is to list the desired characteristics. Such a list can consist of a range of operational lift coefficients and conditions (Mach number, Reynolds number) at which specific properties such as maximum value of the lift should occur, desirable entry into the post-stall region, or where minimum drag should occur, extent of laminar flow, the magnitude of the pitching moment coefficient, as well as desirable geometric characteristics such as thickness and its location along the chord. The next step is to decide on a methodology: direct or inverse method (see below). Some designers use existing airfoils as a baseline and modify them, typically using both methods and a trial-and-error approach until most of the desired characteristics have been achieved.

Direct Analysis Method

A direct analysis is the evaluation of the pressure field around an airfoil that has already been defined. In direct analysis the ordinates of the airfoil are entered into software such as Xfoil, PROFILE, or AeroFoil and then the software will predict lift, drag, and pitching moment of the airfoil at the angles-of-attack specified. The most important feature of such software is an accurate prediction of flow separation growth with AOA, subsequent stall, and width and depth of the drag bucket at lower AOA. The software packages cited above are all capable of such predictions, although the accuracy of the results is not being commented on. The software cited sometimes yields different predictions for an identical airfoil at identical conditions. It is always the responsibility of the user to learn to properly use the software and this requires the predicted results to be compared with reliable wind tunnel tests.

The direct analysis method is not an ideal way to design an airfoil for a desired pressure distribution, as this would require substantial trial and error to accomplish. The inverse airfoil design method discussed below is a far better approach.

Inverse Airfoil Design Method

The inverse method allows the airfoil designer to specify a specific velocity distribution along the surface, which is then used to calculate the geometry that will generate such a distribution. The knowledgeable designer will know the consequences of a specific speed distribution, where it will promote laminar flow or cause early separation, and so on. For this reason, airfoil design is a field of specialization and calls for the evaluation of large numbers of airfoils to build a database of behavior based on experience. Inverse methods were responsible for significant advances in airfoil design in the 1950s, when enough computational power was realized to allow integral boundary layer methods to be coupled with potential-flow solutions. The computational prowess of the modern computer is now used to implement such methods using the full Navier-Stokes equations.

Thickness, Mean-line and Camber

NACA defines airfoils based on a specific thickness distribution and mean-line. The camber is defined as the maximum distance between the mean-line and the chordline. Camber strongly affects the downwash behind the airfoil and, thus, how much lift is generated. The rule-of-thumb is that the larger the camber, the greater the maximum lift of the airfoil and greater the thickness the greater the stall angle-of-attack and drag. Of course there are exceptions. Generally, the greater the camber the greater is the drag as well.

LE Radius

The leading edge radius impacts the maximum lift of the airfoil, as well as its drag in cruise. Generally, the larger the radius the more lift will be generated at high AOA, as this delays flow separation near the LE. This often manifests itself as a less abrupt reduction in lift at stall (“stall break”). Large radius can also increase the airfoil’s drag, although this is also dependent on the geometry of the airfoil downstream.

Square TE

A square trailing edge is sometimes employed to decrease adverse pressure gradients⁴ on airfoils. This is important for NLF airfoils (which feature the maximum camber way back along the chord) to help stabilize the boundary layer on the aft part of the airfoil. This way, the formation of a separation bubble is prevented and, consequently, both lift and drag characteristics are improved. It is of importance how the TE is squared. A sharp trailing edge cannot just be made blunt, as this will not increase the thickness of the airfoil upstream. Rather, the TE must be deliberately thickened to improve adverse pressure gradients. Blunt TE airfoils are sometimes called “flatbacks.”

Geometric Description

The geometric description of airfoils is generally given in so-called ordinate tables. Such tables often list the airfoils upper and lower surfaces separately in terms of x and z coordinates. The x-axis is called the chord line and the x-ordinate represents the chordwise station and the z-ordinate the height above or below the x-axis. Then we define the mean-line and thickness as follows:

(8-34)

(8-35)

If the mean-line and thickness are tabulated rather than the ordinates, then these can be obtained from:

(8-36)

(8-37)

Applications

The airfoils are widely used in GA aircraft, with the best-known aircraft being a family of Cessna airplanes. Cambered versions are used for wings, while symmetric ones are used for HT and VT. Symmetric airfoils are also used for helicopter rotors, antennas, and even for some supersonic aircraft and missile fins.

Numbering System

As shown in Figure 8-24, the numbering system is based on the geometry of the airfoil. The first digit indicates the camber in percentage of chord. The second digit indicates the distance from the leading edge to the maximum value of the camber in tenths of the chord. The last two digits indicate the thickness of the airfoil in percent of the chord. Thus, the NACA 4415 airfoil has a 4% camber located at 40% of the chord and is 15% thick. Also, NACA 0009 is a symmetrical airfoil as indicated by the first two digits 00. The airfoil is 9% thick.

Computation of Airfoil Ordinates

The geometry of the airfoil can be determined by computing the x- and y-values in accordance with the Step-by-step that follows. The algorithm is based on the report NACA TN-460 and is set up for analysis using a spreadsheet. Note how the ordinates of the upper and lower surfaces are calculated. These are rotated with respect to the slope of the mean-line. Thus, the x-value of the point on the upper surface is not the same as that of the point on the lower surface. This necessitates the evaluation of the slope of the mean-line and the rotation about the point (x, y_c) as shown in Figure 8-25.

The definition of airfoil is always based on the assumption that the leading edge is located at x = 0 and the trailing edge at x = 1. The resulting chord is of a unit length. The advantage of defining the airfoil in terms of a unit chord is that it can be easily scaled up by a direct multiplication of the ordinates by the desired chord. The geometric definition is in the form of a table containing the x- and y-coordinates of the airfoil. As stated in the previous section, such a table is called an ordinate table, where the word ordinate refers to the y-value of a specific co-ordinate.

The ordinate table lists the x-values for the airfoil by distributing them along the x-axis. While the x-values are sometimes distributed uniformly, a more desirable distribution is based on a cosine scheme as shown in Figure 8-26. Doing so will guarantee a better definition of the leading edge, where curvature is greater. This scheme consists of a unit circle, which is uniformly sectored at an angle Δϕ. The value of Δϕ is determined from the number of points as follows:

(8-38)

Now consider the thick dark line extending from x = −1 to 0 in Figure 8-26 (QII). The x-values of the intersection of the sector lines and the circle are projected vertically on to this line, revealing a tight separation of points close to x = −1. The spacing of the vertical line gradually gets more sparse as we approach x = 0. If we shift this pattern by adding 1 to each x so obtained, the resulting set of x-values results in a tight separation close to x = 0 (the leading edge of our airfoil) and more sparse as we approach the trailing edge at x = 1.

To prepare this scheme we number each x using indexes ranging from 1 through N, where N is the number of points. Mathematically, we can write:

(8-39)

These definitions allow us to prepare the following Step-by-step to calculate the geometry of any 4-digit airfoil.

Step 1: Preliminary Values

Decide the thickness ratio, camber, and location of the camber using the variables t, C, and x_camber, respectively. If the thickness ratio is 15%, then t = 0.15. If the camber is 4%, then C = 0.04. If the location of the camber is 40%, then x_camber = 0.4. (These numbers represent the NACA 4415 airfoil.)

Step 2: Airfoil Resolution

Decide how many x-values (and thus y-values) to include in the analysis. Here, let’s call that value N (e.g. N = 100 for 100 points).

Step 3: Prepare Ordinate Table

Tabulate the x-ordinates along the unit chord using the cosine scheme where Δϕ is calculated using Equation (8-38). These values should range from 0 to 1, where x = 0 represents the leading edge and x = 1 the trailing edge.

Step 4: Calculate Thickness

Calculate the thickness of the upper and lower surfaces of the airfoil for each x-value, from:

(8-40)

Step 5: Compute the y-value for the Mean-line

The next step involves computing the y-value of the mean-line for each x, which we call y_c. This depends on whether x is larger or smaller than the location of the camber and is given by:

Step 6: Calculate the Slope of the Mean-line

Calculate the slope of the mean-line at the point:

Step 7: Calculate the Ordinate Rotation Angle

Calculate the angle θ as follows:

(8-45)

Step 8: Calculate the Upper and Lower Ordinates

Calculate the upper and lower surface ordinates as follows:

(8-46)

Generation of the NACA 4415 – an Example Implementation

This procedure has been implemented as shown in Figure 8-27, for a NACA 4415 airfoil, using N = 30. More points should be used for a serious design project. Let’s perform a sample calculation for the row with ID number 10 or i = 10 (see the highlighted row in the figure). Beginning with column 2, we calculate the cumulative sector angle ϕ based on Equation (8-38) as follows:

where Δϕ = 3.1034° and i is the index in the first column (titled “ID”).

Column 3 is the x-value using the cosine scheme, calculated using Equation (8-39):

Column 4 is the thickness, calculated using Equation (8-40):

Column 5 is the y-value of the mean-line, calculated using Equation (8-41) since x₁₀ < x_camber:

Column 6 is the slope of the mean-line, calculated using Equation (8-43) since x₁₀ < x_camber:

Column 7 is the slope of the mean-line in degrees, calculated using Equation (8-45):

The remaining columns 8–12 are calculated using Equation (8-46). The spreadsheet can be used to determine the geometry of any NACA four-digit airfoil by simply changing the thickness fraction, camber, and the location of the camber.

Applications

The airfoils are widely used in GA aircraft, commuters, and business jets, where they are used for wings. Among aircraft using five-digit airfoils are a number of models manufactured by Beechcraft.

Numbering System

Per Ref. [21], “the first digit is used to designate the relative magnitude of the camber.” NACA R-824 [23] adds that “the first digit indicates the amount of camber in terms of the relative magnitude of the design lift coefficient; the design lift coefficient in tenths is thus three-halves of the first integer.” Thus, considering the airfoil NACA 23012, the 2 means that the maximum camber height is 2%/100 = 0.02 and the design lift coefficient is (2/10)·(3/2) = 0.3. The second digit, when divided by 20, would give the chordwise location of the maximum camber (0.15). The third digit is ‘0’ for normal camber and ‘1’ for reflexed airfoils like those used for flying wings. The last two digits represent the thickness of the airfoil (0.12 or 12%). Using this nomenclature, the various members of the family of five-digit airfoils would be represented as shown in Table 8-2.

Computation of Airfoil Ordinates

This information can now be used to mathematically compute the shape of any 5-digit airfoil as shown in the Step-by-step below:

Step 1: Preliminary Values

Decide the thickness ratio, camber, and location of the maximum camber using the variables t, C, and x_camber, respectively. If the thickness ratio is 12%, then t = 0.12. If the camber is 2%, then C = 0.02. If the location of the maximum camber is 15%, then x_camber = 0.15. (These numbers represent the NACA 23012 airfoil.)

Step 2: Airfoil Resolution

Decide how many x-values (and thus y-values) to include in the analysis. Here, let’s call that value N (e.g. N = 100 for 100 points).

Step 3: Prepare Ordinate Table

Tabulate the x-ordinates along the unit chord using the cosine scheme where Δϕ is calculated using Equation (8-38). These values should range from 0 to 1, where x = 0 represents the leading edge and x = 1 the trailing edge.

Step 4: Calculate Thickness

Calculate the thickness of the upper and lower surfaces of the airfoil for each x-value, using Equation (8-40), repeated here for convenience:

(8-40)

Step 5: Compute the y-value for the Mean-line

The next step involves computing the y-value of the mean-line for each x, which we call y_c. This depends on whether x is larger or smaller than the location of the camber and is given by:

Step 6: Calculate the Slope of the Mean-line

Calculate the slope of the mean-line at the point using Equations (8-43) and (8-44), also repeated for convenience:

Step 7: Calculate the Ordinate Rotation Angle

Calculate the angle θ using Equation (8-45):

(8-45)

Step 8: Calculate the Upper and Lower Ordinates

Calculate the upper and lower surface ordinates using Equations (8-46):

(8-46)

It is of interest to note that it is possible to calculate the constant m and k₁ for any position of the x_camber using the following expressions, which are obtained through simple curvefit to the data in the table in Step 5:

(8-49)

Note that the equation for k₁ did not have a perfect correlation like the one for m. For this reason, values of k₁ calculated using Equation (8-49) will differ slightly from those in the table, although the accuracy gives acceptable results. This allows the evaluation of alternative versions of the five-digit series airfoils. Corresponding values for m and k₁ for reflexed airfoils (211, 221, etc.) are given in Ref. [21].

This procedure has been implemented as shown in Figure 8-29, for a NACA 23012 airfoil, using N = 30. More points should be used for a serious design project. In fact, the primary difference between this scheme and the one for the four-digit airfoil is the determination of m, k₁, and then the calculation of the mean-line. Here m = 0.2025, k₁ = 15.957, although it is calculated using Equation (8-49) A sample calculation for the row with ID number 10 or i = 10 (see the highlighted row in the figure) is provided. First, columns 2 and 3 are identical to the previous approach. The thickness calculated in column 4 also uses Equation (8-40), but this time it yields 0.0494.

Column 5 is the y-value of the mean-line, calculated using Equation (8-47) since x₁₀ < x_camber:

Column 6 is the slope of the mean-line, calculated using Equation (8-43) since x₁₀ < x_camber:

Column 7 is the slope of the mean-line in degrees, calculated using Equation (8-45):

The remaining columns 8–12 are calculated using Equation (8-46). The spreadsheet can be used to determine the geometry of any NACA 5-digit airfoil by simply changing the appropriate parameters.

Applications

Widely used for aircraft and ship propellers.

Numbering System

Typical 1-series airfoils are designated by a five-digit number such as NACA 16-212. The first integer ‘1’ represents the series. The second digit ‘6’ denotes the distance in tenths to the chordwise location of the minimum pressure when the symmetrical airfoil is at zero lift (60%). The first number following the dash ‘2’ is the amount of camber in terms of the design lift coefficient in tenths (0.2). The final two digits ‘12’ represent the thickness of the airfoil (0.12 or 12%).

Applications

The airfoils are widely used for aircraft ranging from WWII era fighters to high-performance GA aircraft, business jets, and military trainers.

Numbering System

The NACA 6-series airfoils (Figure 8-31) feature a six-digit designation with an indicator of the mean-line used. For instance, the designation for the airfoil NACA 65₃-415, a = 0.5, the ‘6’ refers to the 6-series airfoils, ‘5’ denotes the chordwise location of the maximum camber in tenths of the chord (50%). The subscript ‘3’ gives the range of lift coefficients below and above the design lift coefficient in which favorable pressure gradients exists on both surfaces (± 0.3). The ‘4’ following the dash indicates the design lift coefficient in tenths (0.4). In this way, the particular airfoil is expected to sustain laminar flow for lift coefficients ranging from 0.1 to 0.7 (i.e. 0.4 ± 0.3). The last two digits indicate the airfoil thickness as percent of the chord (0.15 or 15%). The designation “a = 0.5” refers to the mean-line used, but this is used in the derivation of the ordinates. When a mean-line designation is omitted the default value of a = 1.0 is used.

The airfoil designation has a number of variations. For instance, the above airfoil (NACA 65₃-415) is sometimes represented as NACA 65(3)-415 or 65,3-415.

The camber line can be a combination of more than one such line and then the design lift coefficient is the sum of all the camber lines used; for instance, the “2” in 218 in the following airfoil designation:

If the thickness distribution of an airfoil is obtained by linearly increasing or decreasing the ordinates of one of the original thickness distributions, then the airfoil is denoted as shown in the example below:

Where the number ‘3’ in the parentheses denotes the low-drag range and the ‘18’ denotes the thickness ratio of the original thickness distribution. NACA 6-series airfoils whose thickness ratio is less than 0.12 usually do not have a subscript as their low-drag range is less than 0.1 (meaning the subscript should be a fraction). In this case, the subscript is omitted, as shown below:

There are other deviations, but an interested reader is directed to Ref. [23] for more details.

Applications

Not widely used. The University of Illinois at Urbana-Champaign (UIUC) airfoil database indicates five airplane types that use such airfoils; the Brändli BX-2 Cherry, DRS Sentry HP (UAV), Jameson RJJ-1 Gypsy Hawk, MacDonald S-20, and the RFB RW.3-A3 Multoplane.

Numbering System

The NACA 7-series have their own numbering system best explained by considering a typical type: NACA 747A315. The ‘7’ indicates the series number. The ‘4’ indicates the extent of favorable pressure gradient over the upper surface of the airfoil in tenths of the chord length (40%). The next, ‘7’, similarly, indicates the extent of favorable pressure region over the lower surface in tenths of the chord length (70%). This means that as long as the surface qualities are smooth, the laminar boundary layer should extend at least that far aft. The series of three numbers ‘315’ following the letter ‘A’ is identical to that of the 6-series, i.e. ‘3’ is the design lift coefficient (0.3) and ‘15’ is the thickness ratio in hundredths (0.15 or 15%). The intent of the ‘A’ is to distinguish between airfoils that have properties that would lead to an identical digit, but differ in camber or thickness distribution. As an example, another airfoil of the series with an equal coverage of favorable pressure gradients, but with a different camber-line or thickness distribution, would be distinguished from the first one using the serial letter ‘B’. As with the 6-series airfoils, the 7-series also feature camber lines that are the combination of two or more lines.

Applications

No known application.

Numbering System

The numbering system for typical NACA 8-series airfoils is best explained by considering a representative type, e.g. NACA 835A216. The first digit, ‘8’, identifies the series. The second one, ‘3’, denotes the position of the minimum pressure on the upper surface in tenths (0.30). Similarly, the third digit, ‘5’, is the position of the minimum pressure on the lower surface. The letter ‘A’ has an identical function as in the 7-series airfoils, as do the remaining three digits, where the first ‘2’ is the design lift coefficient in tenths (0.2) and the two remaining characters are the airfoil thickness in hundredths (0.16 or 16%).

CLARK Y

The Clark Y airfoil (see Figure 8-35) is famous for being one of the most widely used airfoils in the history of aviation, although its use is mostly in airplanes designed before World War II. It is named after Colonel Virginius E. Clark (1886–1948), who designed many airfoils around the time of World War I. The Clark Y airfoil, in particular, was designed in 1922. One of its distinguishing features is the flat lower surface, which extends from 30% chord to the trailing edge. The aerodynamic properties of the Clark Y airfoil can be found in NACA R-502 [29]. Among famous aircraft types that feature this airfoil are the Ryan NYP Spirit of St. Louis, flown by Charles Lindbergh across the Atlantic in 1927; the Lockheed Vega, flown by Amelia Earhart across the Atlantic; and Wiley Post (which he named “Winnie Mae”) twice around the globe. The University of Illinois at Urbana-Champaign (UIUC) airfoil database cites this airfoil 493 times for the 7420 aircraft type instances.

USA-35B

The USA series of airfoils was designed by engineers of the United States Army (USA) in the era before 1920. The USA35B (see Figure 8-36) is the best known. It is a flat-bottomed airfoil used in many well-known aircraft, especially in a line of aircraft made by Piper Aircraft, such as the Piper J-3 Cub, PA-25 Pawnee, PA-23 Apache and Aztec twin-engine aircraft. Like the Clark Y airfoil, it features a substantial camber and generates a high maximum lift coefficient and gentle entry into the post-stall region, but also high drag.

NACA 23012

The NACA four-digit airfoils were developed in the 1930s. This work led to the development of the 23-thousand series airfoils. One of the best known of those airfoils is the NACA 23012 (see Figure 8-37), which is used in a number of aircraft. The UIUC airfoil database cites the airfoil 397 times for the 7420 aircraft type instances, including very common aircraft such as Beech’s Bonanza and B1900 Airliner, just to name a few. It is also featured on the twin-engine Britten-Norman BN-2 Islander, Cessna 208 Caravan and even the Citation jet.

Research on the lift and drag properties of the symmetrical four-digit NACA airfoils showed that while the drag was quite small, the same could be said of the maximum lift. It was proposed that the maximum lift could be improved by introducing a small camber to the airfoil. This was accomplished by simply deflecting the forward part of the leading edge. As can be seen in Figure 8-38, the 23012 airfoil is effectively a NACA 0012 airfoil whose forward 15% has been deflected leading edge down. This new airfoil showed great promise in wind tunnel testing, offering a low C_dmin, very high C_lmax, and low C_m, all are great properties that make the airfoil very tempting for the aircraft designer. However, what was less touted in the NACA literature was the abruptness of its stall. And this is a serious problem. In fact, it is not just the NACA 23012 that poses such problems, but a host of airplanes that feature a root/tip combination of 23015/23009 and similar.

As stated in Section 8.1.11, Airfoil stall characteristics, a leading-edge separation bubble may form on airfoils whose thickness-to-chord ratio is 12% or less. Compounding this issue on the NACA 230XX series airfoil is the nature of its geometry: the leading edge deflection at 15% chord. This introduces a sharp discontinuity in the curvature of the airfoils that may also contribute to the formation of the separation bubble [30]. As the AOA of the airfoil is increased, the airflow will reattach behind the bubble (see schematic to the left in Figure 8-39) and separate again at the trailing edge. With further increase of AOA the second separation point will continue to move forward, eventually and suddenly combining with the leading-edge separation bubble. When this happens, there will be an abrupt and large drop in the lift coefficient. This is clearly visible as the sharp drop in Figure 8-39, from the wind tunnel test data presented in NACA TR-537 [31].

Once featured on a wing, fabrication differences between the two wing halves on either side of the plane of symmetry will result in a high probability that one wing will stall (abruptly) before the other. In other words, an uncontrollable roll-off at stall is a very plausible scenario the designer should be strongly aware of. And while it is certainly possible to “beat” this detrimental stall behavior into submission with adequate wing washout and stall strips, this is likely to take extra development time and cost, making the series of airfoils something the aircraft designer should stay away from. There are multiple other options at the disposal to the designer which offer similar favorable properties but with far more benign stall characteristics.

GA(W)-1 or LS(1)-0417

The GA(W)-1 (see Figure 8-40) was designed in 1972 specifically for GA aircraft using computational fluid dynamics by Robert T. Whitcomb and associates at the NASA Langley Research Center [32]. Details of its characteristics can be found in the report NASA TN D-7428 [33]. It is also known as NASA/Langley/Whitcomb LS(1)-0417. The report tested the airfoils at Reynolds numbers ranging from 2 × 10⁶ to 20 × 10⁶ and Mach numbers ranging from 0.15 to 0.28 and demonstrated C_Lmax ranging from 1.6 to 2.0. A selection of lift-to-drag ratios between 65–85 were obtained at a climb lift coefficient of 1.0. These characteristics appeared very promising and the airfoil was featured on at least two aircraft designs from the 1970s: the Piper PA-38 Tomahawk and Beechcraft 77 Skipper. Other aircraft using the airfoil are the cancelled American Aviation Hustler, the Edgley Optica, and the cancelled military trainer Fairchild T-46 Eaglet.

Among other favorable characteristics is the structural depth offered by its thickness-to-chord ratio of 17%. This provides the aircraft with ample volume for fuel and should result in a lighter wing structure. The cusp (reflexed curvature) near the trailing edge of the airfoil usually presents an issue when using NLF airfoils, as it results in a thin trailing thickness that makes it practically impossible to fit ribs that extend to the trailing edge. The last 30% of the chord is usually allotted to control surfaces and usually these are constructed using conventional stressed skin aluminum construction. The cusp will then require ribs whose flanges must be curved, which requires costly tooling to fabricate. If the control surfaces are built using composite materials the cusp can be accommodated, although such surfaces usually end up being heavier than if they were made from aluminum. Overall, the cusp presents a hard detail in real production environment and sometimes forces the manufacturer to modify the geometry by making the control surfaces flat-bottomed.

Davis Wing Airfoil

The Davis wing is the topic of discussion in online Appendix C1.5.1, The Davis wing, but it stood for a wing design philosophy used for many military aircraft, of which the Consolidated B-24 Liberator is probably the best known. The airfoil, which was inspired by what, at the time, was thought to be the shape of a teardrop (see Figure 8-41), had a number of interesting characteristics, most prominent of which was low drag and high lift. The airfoil was evaluated in NACA WR-L-677 [34]. An Xfoil analysis of the airfoil at a Reynolds number of 9 × 10⁶ matches the experiment reasonably well and indicates it has many interesting properties. Among those are gentle stall characteristics, LD_max in excess of 160, and a C_lmind of 0.65 (suitable for a lumbering heavy transport aircraft). Xfoil under-predicts the C_lmax of about 1.35 versus 1.4 from experiment, but the C_lmind between prediction and experiment match well.

“Peaky” Airfoils

A peaky airfoil is a name used for airfoils that were designed in the 1960s for use in aircraft operating in the transonic range (see Figure 8-42). When conventional airfoils accelerate to high subsonic airspeeds a shockwave begins to form well below Mach 1. This takes place as the speed of air flowing around the airfoil locally reaches and exceeds the speed of sound on the airfoil. As soon as shocks begin to form in the flow there will be a sharp rise in the drag of the airfoils, as shown in Figure 8-50. The airspeed at which this drag rise begins is the critical Mach number (see Section 8.3.7, The critical Mach number, M_crit). M_crit for typical NACA four-digit or five-digit airfoils may be as low as a hair over Mach 0.6, depending on thickness. This renders them impractical for use in high-speed aircraft. A strong shock wave will form on such airfoils, greatly increasing their drag and reducing their lift, and may even introduce aeroelastic problems to the vehicle [35]. It is possible to delay these effects by increasing the sweep of the wing, but eventually they are inescapable.

As soon as the shock forms, the center of lift of the airfoil moves from approximately the quarter-chord to the mid-chord, further compounding the issue by substantially increasing stability. Additionally, the shock will induce separation, which thickens the wake aft of the shock (see Figure 8-43), increasing the drag further [36]. What follows is a complex interaction of the movement of the upper and lower shockwaves that ultimately results in a drastic drop in the lift coefficient being produced, causing a shock-stall.

Research on such high-speed effects began early, or in the 1940s, but serious work on modifying airfoils began in 1955 with the pioneering work of H. H. Pearcey, who tried to experimentally obtain “an essentially shock-free flow” [37]. Pearcey showed that by reshaping the leading edge of an airfoil it was possible to weaken the shock. The process was to allow the flow to expand rapidly from the stagnation point and become supersonic in the leading edge area. This would form a series of compression shock waves that slowed the local Mach number, so the final shock wave was substantially weakened [38]. Less drag was one important benefit from this change. It was thought the modified airfoils would be capable of weakening the shock for maximum local Mach numbers as high as 1.4. The resulting pressure distribution has a prominent pressure peak near the leading edge and was described as being peaky. The airfoil has a relatively flat upper surface, which necessitates a cusp in the trailing edge to improve its lift generation.

Experience in using peaky airfoils on combat aircraft has revealed that high-speed, high-g maneuvers (which call for high AOA) result in the formation of significant shock strength that may cause sudden loss of lift. When introduced to the Hawker-Siddeley Kestrel FGA.1, the prototype of the production Hawker-Siddeley Harrier, this shock would occur above Mach 0.8, causing a serious wing-rocking at an AOA where a gentle buffet would be expected [38].

Supercritical Airfoils

Supercritical airfoils are often referred to as one of the three major contributions to aviation made by the famous aerodynamicist Richard T. Whitcomb (1921–2009), whom many historians of aviation call the most distinguished alumnus of the NASA Langley Research Center. His other two contributions being the area-rule and the winglet.

Like the peaky airfoils, supercritical airfoils are intended for high-speed aircraft. They feature a large radius (blunt) leading edge, considerably flatter upper surface than the peaky airfoils, and a significant cusp near the trailing edge (see Figure 8-44). The blunt leading edge softens the suction peak of the smaller radius peaky airfoil. The flatter top surface helps keep down the local Mach number and keeps down adverse pressure gradients. The cusp was introduced to help generate lift without lowering M_crit by forming a high-pressure region under the airfoil.

The characteristics of supercritical airfoils can be summarized as follows: (1) the drag rise Mach number, M_crit, is substantially higher than for more conventional airfoils. As an example, at a section lift coefficient of 0.65, the M_crit of one such airfoil is 0.79, which compares very favorably to the 0.67 of a NACA 64A-series airfoil of an equal thickness ratio [39]. (2) The section pitching moment coefficient for supercritical airfoils is substantially more negative than for conventional airfoils. This is caused by the high-pressure region that forms at the lower surface due to the cusp. (3) The supercritical airfoil also increases M_crit at off-design section lift coefficients. (4) The airfoil also increases the maximum lift coefficient at high subsonic Mach numbers.

Wing design that uses supercritical airfoils offers many advantages to a high-speed aircraft. First, and as has already been stated, is the increase in M_crit for a given thickness and wing sweep. Second, their thickness offers structural depth and volume for fuel. Third, they allow the designer to reduce the wing sweep, which offers a host of benefits. Among those are improvements in low-speed characteristics and performance capabilities.

Joukowski Airfoils

Joukowski airfoils are a much more clever mathematical tool used to demonstrate aerodynamic theory than a practical solution to aircraft design. The airfoils are named after the Russian scientist Nikolay Yegorovich Joukowsky (sometimes spelt Zhukovsky, 1847–1921), who is also the originator of the circulation theorem (see Section 8.1.8, The generation of lift). The airfoils are generated in conformance with the transformation of a circle on to the complex plane, using the complex number ζ:

(8-50)

Consider the complex number ζ to be given by . Inserting this into Equation (8-50) and manipulating yields the two spatial variables x and y that are given by the following relation:

(8-51)

By varying the values of a and b between −1 and 1 (e.g. try a = −0.1 and b = 0.1) the shape of the airfoil can be modified (see Figure 8-45). Then, using elementary flow concepts, such as uniform flow, sources, sinks, and vortex flow, airflow around the resulting airfoil can be simulated. Joukowski airfoils are a subject in most texts on theoretical aerodynamics, although they are rarely used on actual aircraft. In fact, the UIUC airfoil database indicates five instances, all of them sailplanes (the Schempp-Hirth Gö-4 III Goevier, Schleicher ASK 14, ASK 16, ASK 18, and Schleicher Ka-6 Rhönsegler).

Liebeck Airfoils

The unconventionally shaped airfoil shown in Figure 8-46 is named after Dr. Robert Liebeck, who was one of the first to try and find out what sort of “airfoil-shaped” geometry would yield the highest possible C_lmax. The answer is the shape devised by Dr. Liebeck, which is remarkable for achieving the highest lift-to-drag ratio of any airfoil. The airfoils are designed to utilize Stratford distribution (see Section 8.1.6, Chordwise pressure distribution) and achieve an L/D ratio in excess of 600. The airfoils are intended for low Reynolds number applications. There is a large amount of scientific literature available that investigates the properties of these airfoils.

Maximum Theoretical Lift Coefficient

Questions such as “what is the maximum value of the lift coefficient?” and “what kind of shape yields that value?” are often asked. Answers to these questions are addressed in a classic paper by Smith [3]. Knowing the theoretical limits is important for comparison reasons. It allows efficiency and potential for improvements to be quantified. Figure 8-48 shows a cylinder of diameter d, rotating in a uniform inviscid flow of velocity V, exposed to three values of the circulation, Γ. Since the flow is inviscid it will not separate, but rather flow around the cylinder as shown. While such a flow exists only in the imagination, it is still of considerable importance in aerodynamics. Not only does it show what could be were it not for viscosity, it also allows the aerodynamicist to determine theoretical limits.

Using potential flow theory (PFT), it is possible to simulate airflow around a rotating cylinder by adding the elementary flow functions of uniform and vortex flow. For instance, consider the Figure 8-48(a), which shows two stagnation points, A and B. In Figure 8-48(b), the circulation is strong enough to bring the two stagnation points together, and in (c) it is strong enough to move the stagnation point off the surface. It can be shown using PFT that the circulation required to create the flow in Figure 8-48(b) amounts to 2πVd. For instance see Anderson [40, pp 249–251]. Using Equation (8-26), we can write:

But it is also possible to write lift using Equation (8-52). Therefore, the lift coefficient, C_l, can be found to equal:

The corresponding lift coefficients for values of the circulation on either side of 2πVd result in lift coefficients that are less than 4π. A geometry that yields a higher value of the lift coefficients is not known by this author. The value C_lmax = 4π can therefore be considered the maximum theoretical lift coefficient.

Compressibility Effect on Lift

Figure 8-50 shows the effect compressibility has on the airfoil’s lift curve and drag. Compressibility increases the slope of the lift curve. In practice, this means that a smaller AOA is required to generate a given lift coefficient, as is evident from the left graph in Figure 8-50. The designer of GA aircraft is generally only interested in the stall portion of the lift curve at low Mach numbers because such airplanes always stall at low subsonic airspeeds. However, the stall portion is important in fighter aircraft design as such aircraft pull high gs at high airspeeds at AOAs near or exceeding stall.

Compressibility Effect on Drag

Compressibility also has important effect on the drag polar (also discussed in Section 15.4.2, The effect of Mach number). The drag coefficient at a given AOA (or C_l) remains relatively constant until the airfoil approaches Mach numbers in the range 0.6 to 0.8. These are called the critical Mach number (see Section 8.3.7, The critical Mach number, M_crit). Increasing the airspeed further will result in a sudden rise in drag. This is a consequence of a shock forming on the airfoil, as the local velocities exceed the speed of sound. The effect is shown as the sharp rise in the compressible drag coefficient, C_DC, in the right graph of Figure 8-50. These are but a part of the intriguing (if not surprising) behavior of fluids near this critical airspeed (M = 1). The so-called drag divergence Mach number, denoted by M_DD, is defined after the coefficient has clearly begun to diverge from its incompressible value. Generally, there are two common definitions by two prestigious manufacturers; Boeing and Douglas (which is now a part of Boeing). The Boeing approach defines M_DD as the Mach number where the drag coefficient exceeds its incompressible value (C_{Dincompressible} in Figure 8-50) by 0.0020. The Douglas approach defined it as the Mach number where the slope dC_Dc/dM first reaches 0.10 [41]. Section 15.5.12, Drag due to compressibility effects, presents a method to account for this drag.

Compressibility Effect on Lift and Drag Exemplified

Compressibility affects both the section lift coefficient and the section drag coefficient detrimentally at high Mach numbers, as shown in Figure 8-51. These show that at a fixed AOA, there is an abrupt reduction in lift and a sharp rise in drag when the airspeed approaches sonic conditions. This effect happens at the critical Mach number (see Section 8.3.7, The critical Mach number, M_crit). It is indicative of local airspeeds over the airfoil having exceeded Mach one, forming a normal shockwave that alters the pressure distribution, causing the large deviations seen.

Figure 8-52 shows the effect the M_crit has on the lift curve slope and the zero AOA lift coefficient. The first observation is that the lift curves at M = 0.4 and 0.6 do not show a large change. However, there is a sharp reduction in the lift curve slope at M = 0.8 (note that the values of C_lα and C_lo shown are based on the four points at 1° through 4°). This is followed by a slight recovery at the minimum value of the lift coefficients, which occurs at a Mach number of 0.87; however, the zero AOA lift is now negative and almost doubled in magnitude. Any airplane featuring this airfoil (NACA 2412) operating at these Mach numbers would suffer serious stability issues.

Compressibility Effect on Pitching Moment

Compressibility has a major impact on the behavior of airfoils and wings at high Mach numbers. It is appropriate to discuss both at the same time. This is shown schematically in Figure 8-53 for a hypothetical aircraft trimmed at M = 0.5. It is of interest to consider the effect in terms of the pitch stability, denoted by C_mα, and stick force, which is the force the pilot must exert in order to control the airplane longitudinally. Of course the graph would be different for other aircraft, CG locations, control system, airfoil types, trim points, and so on. However, considering these at the moment is not necessary to make the following point. As will be discussed in Chapter 11, The anatomy of the tail, static stability of aircraft requires C_mα to be negative (here, for the example aircraft, it is near −1.0 at low Mach numbers. This means that at low airspeeds, the pilot must pull on the yoke (or stick or column) in order to slow down further. Alternatively, he or she must push in order to fly at Mach numbers higher than 0.5.

With respect to the airfoil, as the far-field airspeed increases, a shock-wave begins to form and change the pressure distribution over the wing. The airplane has entered the transonic region. The effect of the shock-wave depends on where on the wing it forms first (inboard, midspan, outboard) and grows with further increase in the airspeed. It is possible to experience a number of different scenarios, besides the one shown in Figure 8-53. The bottom line is that once a given airspeed is exceeded, the pitch stability and stick force will rise rapidly. This is caused by the center of pressure moving from approximately the quarter-chord to the mid-chord. The effect is akin to the CG being moved forward by that amount at subsonic speeds. If the airplane has not been designed for this, it is possible that the magnitude of the stick force will render the airplane uncontrollable – it will enter a dive (recall the moment tends to point the nose down), which will increase the airspeed further, and from which it may not recover.

Prandtl-Glauert [42]:

(8-56)

Kármán-Tsien:

(8-57)

Laitone:

(8-58)

Frankl-Voishel correction for skin friction drag:

(8-59)

while compressibility effects can be included with the lift and moment coefficients as follows (here using the Prandtl-Glauert model):

(8-60)

(8-61)

where the subscripts ‘0’ refer to the incompressible values. The same correction does not hold for the drag coefficient, as it incorporates skin friction as well as pressure drag and will give erroneous results.

Step-by-step: Determining M_crit for a Body

The M_crit can be determined for a body, such as an airfoil or a fuselage, by the application of the following method, based on Anderson [40, pp. 674–679] and Perkins and Hage [43].

Step 1: Establish the Minimum Pressure Coefficient

Using either a theoretical or experimental method determine the low-speed, incompressible minimum pressure coefficient, C_po, on the body. For airfoils this can be done by any of the freely (and commercially) available computer codes cited in Section 8.1.14, Airfoil design. References [23,44] also list airspeed ratios for a number of NACA airfoils.

Step 2: Select Compressibility Correction Method

Using any of the methods presented with Equations (8-56), (8-57), or (8-58), select the appropriate compressibility correction. Note that Equations (8-57) and (8-58) are generally considered more precise than the Prandtl-Glauert method.

Step 3: Solve to Determine M_crit

Depending on the compressibility correction method selected, the predicted M_crit can be found by setting it equal to the following expression, which determines the pressure coefficient required for sonic conditions to prevail at the point of minimum pressure:

(8-62)

An example of this procedure is shown in Figure 8-55, as it is applied to the NASA NLF(1)-0414F airfoil in Figure 8-7.

Derivation of Equation (8-62)

As the airspeed (Mach number) increases, the local velocities over an airfoil increase as well. Consider point A on the airfoil in Figure 8-54. Denote the static pressure in the far field with p_∞ and at A using p_A. Assuming the flow to be isentropic (adiabatic and irreversible) the pressure in the far-field and at A can be related to the total pressure as follows:

Therefore, the pressure at point A can be related to that in the far-field by dividing Equation (i) by (ii) as follows:

(iii)

Of special interest is the case when the airspeed at point A becomes sonic, i.e. M_A = 1. Then Equation (iii) can be rewritten as shown below:

(iv)

Inserting Equation (iii) into Equation (8-14) and renaming M_∞ as M_crit yields Equation (8-62).

QED

If R_e < 20 × 10⁶: ΔC_d = 0.000453

If R_e ≥ 20 × 10⁶: ΔC_d = 0.00308

Impact on Drag

The geometry of the airfoil will affect where the laminar boundary layer transitions into a turbulent one. This can have a major effect on the magnitude of the skin friction drag and, therefore, the performance of the airplane. The geometry of the airfoil dictates whether or not an airfoil can develop extensive laminar flow. In order to take advantage of NLF airfoils the designer must ensure the mission C_L resides inside the drag bucket. If the width of the drag bucket envelopes the C_L for best rate of climb, this will also improve climb performance. This can be done by the selection of the appropriate wing area.

Impact on Flow Separation

The geometry also affects at what AOA flow separation begins and how rapidly it grows with additional increase in AOA. Modern NLF airfoils are quite sensitive to imperfections and this may pose a problem if it causes asymmetry between the left and right wings at stall. This may lead to undesirable stall handling, such as an uncontrollable roll-off to the left or right. Additionally, leading edge radius and smoothness for the first 15–20% of the chord can have a significant effect on the stall characteristics.

Impact on Maximum Lift and Stall Handling

The geometry of the airfoil dictates what its maximum lift coefficient will be. The more prominent the camber the higher is the maximum lift coefficient. The smaller the leading edge radius, the lower will be the maximum lift coefficient. A small leading edge radius may also promote a sudden drop in lift at stall, with the associated detrimental effect on stall handling. The shape of the airfoil may also encourage the formation of a separation bubble that may have a profound detrimental effect on stall handling (e.g. see discussion of the NACA 23012 series airfoils in Section 8.2.10, Famous airfoils).

Impact on Longitudinal Trim

The geometry of the airfoil dictates its pitching moment coefficient, C_m. This may have important implications for the sizing of the horizontal tail and trim drag in cruise. Generally, the larger the camber the greater is the magnitude of the pitching moment coefficient. A symmetrical airfoil will have a zero C_m about its aerodynamic center. Cambered airfoils will have negative C_m. Deploying flaps will make it even more negative (stable), but requires a larger horizontal tail area or greater elevator deflection to balance.

Critical Mach Number

The geometry of the airfoil is instrumental in how early the shockwaves begin to form at high subsonic Mach numbers. A thick airfoil will cause a greater local acceleration of air, which will form a shockwave at a lower M than will a less thick airfoil. An airplane designed to cruise at, say, M = 0.5 should not feature airfoils much thicker than 15% without modifying the thickness distribution to prevent shocks at dive speed. This is particularly important in the juncture between the fuselage and wing, where the local airspeeds are even greater.

Impact on Wing-fuselage Juncture

The wing-fuselage juncture can present particularly challenging effects on airflow, where it accelerates to even greater airspeeds than elsewhere on the wing. The effect is generally twofold: acceleration to a higher airspeed means a lower critical Mach number; and acceleration to a higher airspeed means that air must decelerate that much more. If this deceleration (called pressure recovery) takes place over a short distance a flow separation will occur. Such a separation may even take place at a low AOA. The consequence is higher drag and impaired performance.

Impact on Structural Depth

The geometry of the airfoil dictates entirely the structural depth of the wing. A thick airfoil provides a large structural depth that accommodates a taller spar. A taller spar, in turn, brings down bending stresses in the spar caps and results in a lighter wing structure that offers greater fuel volume. For this reason, selecting a thick rather than thin airfoil is essential, as long as it does not increase section drag too much. Modern NLF airfoils are thicker and generate less drag than previous airfoils, which is one of the reasons why established manufacturers design mission-based airfoils.

Evaluation of a Target Zero-lift AOA

It is of importance to identify what zero lift AOA, α_ZL, to aim for in the airfoil selection process. Using the data of Ref. 23 (Equation [16]), Ref. 13 presents a helpful expression that can be used to estimate target α_ZL for the selection process. Assuming the designer knows the design AOA, denoted by α_i (in degrees), and the design lift coefficient, denoted by C_li, the zero-lift AOA is given by:

Consider an aircraft intended to cruise so its NACA 6-series airfoil is at an AOA near 1° and its section lift coefficient is 0.40. Using Equation (8-63) we readily find the target α_ZL should be near −2°. Generally, this method is accurate, although it is recommended that test data is used when possible.

Selection of an Airfoil

Assuming the designer is aware of the above limitations and issues, the following guidelines can be kept in mind when selecting an airfoil.

NACA Recommended Criteria

The conclusion section of Ref. [23] lists a number of things to keep in mind when selecting airfoils. These are paraphrased below.

Airfoils permitting extensive laminar flow, such as the NACA 6- and 7- series, have substantially less drag at typical cruise lift coefficients than other kinds of airfoils. However, these characteristics are realized only if the quality of the lifting surface is smooth. Wind tunnel tests have shown that extensive laminar flow is possible on smooth three-dimensional wings if the surface quality is smooth and like that provided by sanding in the chordwise direction with No. 320 carborundum sand paper.

Wings of moderate thickness ratios with such surface qualities can achieve a minimum drag coefficient of the order of 0.0080. In fact, the C_Dmin depends more on the surface quality than on the chosen airfoil. Thus, at high Reynolds numbers where laminar flow is no longer achievable, drag can be kept low by ensuring smooth surface qualities.

The maximum lift coefficient for moderately cambered 6-series airfoils are as high as those achieved using NACA 24- and 44-series airfoils. The NACA 230-series airfoils with thickness ratio less than 20% generally achieve the highest maximum lift coefficients. The maximum lift coefficient with flaps is about the same for moderately thick 6-series airfoils as it is for the NACA 23012 with flaps. However, the thinner 6-series airfoils have substantially lower C_lmax with flaps. The lift curve slope for smooth 6-series airfoils is slightly steeper than that of the 24-, 44-, and 230-series airfoils. It exceeds the theoretical value (2π) for thin airfoils.

Leading edge contamination (roughness) causes large reductions in C_lmax for plain and flapped airfoils. The magnitude of the reduction is similar for both. The leading edge contamination also reduces the lift curve slope, especially for thicker airfoils that have the location of minimum pressure farther aft than do thinner ones. This lends support to the importance of understanding the impact of poor surface quality and leading edge contamination on the overall characteristics of the airfoils and, thus, the performance of the aircraft when designing wings.

The NACA 6-series airfoils have higher critical Mach numbers than the earlier airfoil types at typical cruise lift coefficients. However, their critical Mach numbers are lower at higher lift coefficients than are the same types of airfoils. The 6-series airfoils also offer better lift coefficients at higher Mach numbers than the earlier airfoils.