Understanding the aerodynamic behavior of airfoils and wings is a significant part of the practice of aerospace engineering, and this understanding is critical to the successful design of all aircraft. Any body shape that is moved through a fluid will create some form of fluid-dynamic force upon it. By definition, the component of this force that acts on the body in a direction perpendicular to the relative free-stream flow or “relative wind direction” is called the lift. The force component on the body in a direction parallel to the relative wind direction is called the drag. The magnitude of the lift and drag forces depends on many factors, including the size and shape of the body and its orientation to the flow, as well as the Reynolds number (based on a characteristic length) and free-stream Mach number.
However, before examining the characteristics of finite wings, i.e., three-dimensional wings with finite span and perhaps with twist and planform taper, it is prudent to investigate the aerodynamic characteristics of two-dimensional airfoil sections. Such two-dimensional airfoils are equivalent to wings of infinite span and aspect ratio. While the concept of a “two-dimensional” wing section may initially sound somewhat artificial, it is possible to mimic a wing of infinite aspect ratio, both experimentally and theoretically, and obtain aerodynamic results that pertain only to the shape of the airfoil section itself. Furthermore, this approach makes it possible to isolate the other more complicated and interrelated effects associated with the finite span of a wing, including the impact of the wing tip vortices and other aerodynamic effects caused by sweepback, twist, planform (chord) variations, and other things.
- Be conversant with the various definitions of aerodynamic forces and moments, as well as lift coefficient, drag coefficient, lift-curve slope, maximum lift coefficient, aerodynamic center, and center of pressure.
- Know some of the aerodynamic characteristics of airfoil sections, both in attached flow and with flow separation, and how these characteristics change at different Reynolds and Mach numbers.
- Appreciate the effects of flaps and other high-lift devices.
- Understand how to calculate lift and other integrated quantities from the pressure and shear stress distributions about a body.
- Know the differences between subsonic, transonic, and supersonic airfoil sections.
Origin of Aerodynamic Forces
The origin of the net aerodynamic forces on an airfoil or wing, such as lift and drag, comes from the integrated effects of the pressure and the boundary layer shear stress distributions acting over its surface, the basic concept being shown in the figure below. These distributions are not uniform and can be positive or negative, e.g., a higher pressure pushing inward toward the surface (as shown in red) or a lower pressure pulling outward away from the surface (shown in green). In addition, boundary layer-induced shear stresses can be positive (flow moving downstream) or negative (e.g., reversed flow).
It can be deduced from the figure above that the shear stresses, in the aggregate, will act in a direction primarily parallel to the chord line, and so will contribute significantly to the drag force on the airfoil section. Likewise, the differences in the pressure distribution between the upper and lower surfaces will contribute primarily to the lift force and pitching moment on the airfoil, the shear stresses having a small net contribution in that direction.
In practice, the net forces and moments can be measured with a balance (e.g., a scale) or obtained by suitable integration of the effects of the pressures and stresses that act around the surfaces. In the meantime, it is possible to proceed under the assumption that either measurement or calculation can be used to obtain these integrated results, a particular approach being detailed at the end of this chapter.
What is Two-Dimensional Flow?
It should be appreciated that the flow over any wing of finite span will be inherently three-dimensional and further complicated by the effects of the vortices that trail behind the wing, as shown in the figure below. The presence of these vortices produces a downwash flow over the wing, affecting the local angles of attack over the span of the wing and, therefore, its lift and overall aerodynamic characteristics. The flow can be assumed nominally two-dimensional only at sections well away from the wing tip vortices.
In the first instance, it seems obvious to understand the aerodynamic behavior of a wing without the influence of the wing tip vortices. To that end, it is possible to think of a case where the wing span and corresponding aspect ratio become infinitely large. Under these conditions, the effects of the tip vortices are moved so far away from the middle part of the wing that they will have a negligible impact, the idea being shown in the figure below. Of course, in a practical sense, a wing of infinite span is impossible. Still, with higher aspect ratios, the effects of the wing tip vortices are sufficiently far removed that an airfoil at mid-span behaves effectively as if it were in a two-dimensional flow. Therefore, it is possible to mimic two-dimensional wings theoretically and experimentally, deriving much understanding about the aerodynamics of the airfoil section by itself.
Without the effects of the wing tip vortices, a single cross-section of the wing, especially near its mid-span, will behave in a nominally two-dimensional manner, which is the basis that now forms the basis for further discussion. The word “nominally” must be used in this context because it must be recognized that an actual flow can never be strictly two-dimensional, no matter how large the span or aspect ratio can reach in theory or practice. Nevertheless, the literal meaning is that all three-dimensional effects, for all practical purposes, will have been reduced to the point that they can be considered negligible.
It is also possible to mimic an infinite span and aspect ratio in the wind tunnel. One approach is to span the wing from wall to wall, thereby eliminating the tip vortices. Another common practice is to test a short-span wing between two “false” walls, as shown in the photograph below. In wind tunnel terminology, this is called a “two-dimensional insert.” This particular approach has seen widespread use for testing two-dimensional airfoils, and flow visualization has confirmed the validity of the nominally two-dimensional flow developments over the wing section at the mid-span, at least up to the onset of the stall. Post-stall, the flow always tends to become inherently more three-dimensional on any lifting surface, regardless of how it is being tested.
The aerodynamic forces acting on a body are directly proportional to dynamic pressure, the pressure associated with the “dynamics” or kinetic energy of fluid movement. Given the symbol , dynamic pressure comes into most (if not all) aerodynamic problems. In general, the dynamic pressure is given by
where is the flow density, and is the flow velocity. Notice that dynamic pressure depends on the squared value of the flow speed. It can be easily confirmed that dynamic pressure has engineering units of pressure, i.e., force per unit area, so in base units of Nm-2 (Pa) in SI or lb ft-2 in USC.
In particular, the free-stream dynamic pressure is defined as
where means conditions at “infinity” or just far away from the wing where there is an undisturbed “free-stream” flow. In practice, this point will be at least one chord length upstream of the airfoil. As will be shown, the free-stream dynamic pressure is often used as the reference pressure in the definition of most of the non-dimensional coefficients used in airfoil and wing aerodynamics.
Airfoil Forces and Moments
The resulting forces acting on an airfoil section from the integrated effects of the pressure and shear can be resolved into a wind-axis system (i.e., in terms of lift and drag ) or a chord-axis system (i.e., in terms of normal force and a chord force ), as shown in the figure below.
As previously discussed, an airfoil can be considered a “two-dimensional” wing. Therefore, for airfoils, dimensional force and moment quantities per unit span are used, e.g., lift per unit span would be written as /unit span = , i.e., the lift acting over a span of one unit. Furthermore, when force and moment coefficients are defined for airfoils, a reference length is needed, which is usually the chord length. In some cases, the semi-chord, , may be used as a reference length, but this is less common other than in some advanced applications.
By definition, the lift force or lift force per unit length, , acts in a direction that is perpendicular to the free-stream velocity, . The corresponding drag or drag per unit span, , is in a direction parallel to . Alternatively, the forces can be decomposed into the sum of two other forces, i.e., the normal force per unit span, , which acts normal to the airfoil chord, and the leading-edge suction force or axial chord force per unit span, , which points toward the trailing-edge and acts parallel to the chord. Notice, however, that in some cases, the axial or chord force may be defined as positive when pointing toward the leading edge, so it is essential to know the sign convention being used.
Both of these latter force systems are useful in various analyses, although there is usually a preference for using lift and drag. It will be apparent from the preceding that the wind axis and body axis systems are statically equivalent, and one force system can be derived from the other through resolution using the angle of attack . For example, to transform from a body axis system ( and ) to a wind axis system ( and ), then
Force and Moment Coefficients
Aerospace engineers must become familiar with how non-dimensional force and moment quantities are defined and used for airfoils, wings, and other body shapes and what particular values of these coefficients mean. While dimensional values forces (lb or N) and moments (ft-lb or N-m) are helpful in many forms of analysis, it is also convenient to work with non-dimensional aerodynamic quantities, such as lift and drag coefficients.
For example, suppose the lift on a given wing is 900 N, and the corresponding drag is 30 N; these values are undoubtedly helpful to know. However, consider if the wing’s corresponding lift coefficient is 0.6 and its drag coefficient is 0.02. In that case, these values reveal much about the aerodynamic operating state of the airfoil, and the coefficients also allow comparisons of the effects of wings of different shapes and sizes. For this reason, airfoil data (measured or computed) are generally presented in terms of non-dimensional coefficients.
The non-dimensional or dimensionless force coefficients for an airfoil section are defined using the free-stream dynamic pressure and chord as a reference length. The span of a two-dimensional wing is infinite, so the aerodynamic forces and moments are referred to as per unit length or unit span of the wing, i.e., span = 1. A force is pressure times area, so the area of a force per unit span will be . Therefore, the two-dimensional lift coefficient can be defined as
where is the chord of the wing section or the airfoil, which is the distance from its leading edge to its trailing edge. Furthermore, it is assumed that the spanwise lift is uniform from tip to tip, consistent with the two-dimensional assumption.
In summary, the two-dimensional force and moment coefficients can be written as:
Normal force coefficient,
Axial (chord) force coefficient,
Pitching moments are defined as positive when the moment tends to increase the angle of attack of the airfoil section, i.e., positive pitching moments are equivalent to nose-up moments. Again, the convention is that a moment 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 (), 1/4-chord (), and the center of pressure , as shown in the figure below. Notice that when the reference point is moved to different locations on the chord, the values of the moments will change, but the forces acting at that point remain the same. i.e., there is no change in static force equilibrium.
The dimensionless moment coefficients for an airfoil section are defined as:
Moment coefficient at the leading-edge,
Moment coefficient at some point ,
Moment coefficient at 1/4-chord,
Moment coefficient at center of pressure,
Notice the use of lower-case subscripts on all coefficients (e.g., , not and , not ) when applied to a “2-dimensional” airfoil section. When applied to a finite span wing, capital letter subscripts are used, as discussed later; this standard notation distinguishes the finite-span aerodynamic coefficients from the two-dimensional ones.
Representative Force & Moment Coefficients
The data shown in the plots below (indicated by the symbols) show measured values of the lift coefficient, , the drag coefficient, , and the 1/4-chord moment coefficient as functions of the angle of attack for a NACA 23012 airfoil section in a low-speed flow. In this latter regard, a low-speed flow one at low Mach numbers where compressibility effects are relatively small. These results cover an angle of attack range from fully attached flow into the stalled flow condition. They are typical of airfoil characteristics presented in various standard catalogs, e.g., Abbott & Von Doenhoff. The results are often shown with respect to variations of the chord-based Reynolds number and/or the free-stream Mach number.
Quantities such as maximum lift coefficient, minimum drag coefficient, maximum lift-to-drag ratio, pitching moments, and other metrics can all be significant in quantifying the aerodynamic characteristics of airfoils. These parameters can also be a basis for airfoil selection. However, interpreting the results on these preceding graphs and finding relevant quantities takes some practice because many points and curves are almost overlapping, which can be confusing. To this end, a more straightforward presentation is shown in the figure below, in which the lift coefficient , the moment coefficient about the 1/4-chord, , and the drag coefficient, , versus the angle of attack of the airfoil to the free-stream flow, are more clearly delineated.
At low angles of attack, the lifting characteristics of an airfoil are not substantially influenced by viscosity and the presence of the boundary layers. Recall that boundary layers are thin, viscous-dominated regions near the surface. Notice that in this region, the lift coefficient increases almost proportionally with the angle of attack. The parameter is the angle of attack for zero lift or what is usually known as the zero-lift angle of attack. For a symmetric airfoil, ; for positively cambered airfoils, is usually a small negative angle.
As the angle of attack is increased further, then there comes the point where the boundary layer thickens, and the aerodynamic characteristics start to become non-linear with respect to the angle of attack. The airfoil will soon reach its maximum lift coefficient, . A further increase in the angle of attack is followed by the onset of flow separation on the upper surface, a sudden loss of lift, and a significant increase in drag; the process is called a stall. The essential flow physics of stall has been previously discussed and is illustrated schematically in the figure below. At low subsonic Mach numbers, the onset of stall usually occurs at an angle of attack between 12 and 15 depending on the airfoil section and the Reynolds number. Higher Reynolds numbers inevitably delay the onset of flow separation and stall, increasing the value of .
Pitching moments about the 1/4-chord usually stay relatively low on most airfoils, but for a significantly cambered airfoil, as in the case shown, the moments are generally non-zero. Usually, the moment curve has a shallow positive slope because the aerodynamic center (defined later) is close to but just forward of the 1/4-chord.
- The legend on the graphs shows that for a Reynolds number of 5.7 x 106, the results are marked by square symbols. Referring to the left side plot, this zero-lift angle for the smooth airfoil is about -2.1o.
- The maximum lift coefficient is about 1.68 for the smooth airfoil.
- For the smooth airfoil, the corresponding stall angle of attack is about 17.6o.
- These values can be found from the right side plot, the minimum drag coefficient for the smooth airfoil being about 0.06 at a lift coefficient of 0.4.
- The legend shows that triangular symbols mark the results in this case, so referring to the left side plot, this zero-lift angle for the rough airfoil is unchanged at about -2.1o.
- The maximum lift coefficient is less and is about 1.21 for the rough airfoil.
- The corresponding stall angle of attack is about 15.1o for the rough airfoil.
- These values can be found from the right side plot, the minimum drag coefficient being higher than for the smooth airfoil and is about 0.098 at a lift coefficient of 0.18.
Notice from the drag curves that below the onset of the stall the drag coefficient of an airfoil remains relatively low and is reasonably constant. However, it will be seen that the effects are different with surface roughness, in which the airfoil’s leading edge is roughened up to represent the in-service degradation of a wing’s surface. Roughness always causes a boundary layer to become turbulent almost immediately after it is formed, thereby creating a higher average drag and a more rapid increase in drag with an increasing angle of attack.
At lower angles of attack below stall, the lift on the airfoil section is almost proportional to its angle of attack and the local dynamic pressure. This linear relationship is exact within the framework of what is known as linearized inviscid aerodynamic theory. The lift per unit span can be written as
where is called the lift-curve slope (i.e., of the slope of the lift curve in the linear part of the graph) and is measured per degree or radian angle of attack. It is a common mistake to assume that the value of is dimensionless, but remember, it has units per angle of attack. At low Mach numbers per radian or 0.11 per degree.
Remember that the preceding relationship in Eq. 6 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. 6 up to an angle of about 10 to 12 degrees, depending on the Mach number and Reynolds number, i.e., up to the point of stall.
In each case, the relevant equation is
The lift-curve slope is 6.1 per radian angle of attack. Notice that 360 radians so 1 radians, i.e., = 6.1 per radian is equal to 0.1065 per degree.
- . For 5 then = 0.532, for 10 then = 1.065, and for 15 then = 1.597.
- . For 5 then = 0.660, for 10, then = 1.193, and for 15 then = 1.725.
The simplicity of the relationship in Eq. 6 is advantageous in various engineering analyses, although its limitations to airfoil operation below stall must be recognized. However, it should be appreciated that no simple equations can be used to represent the variation of the aerodynamic coefficients in the post-stall regime because they vary non-linearly with the angle of attack.
Effect of Flaps
The figure below shows the effects of a flap on the lift characteristics of a wing. First, notice that the flap deflection introduces a form of trailing-edge camber, so the lift curve far shifts to the left, giving a much lower zero lift angle of attack, changing from about -2o to nearly -16o. Second, notice the significant increase in the maximum lift coefficient from about 1.6 to 2.6 before the airfoil stalls. Such significant increases in the lift coefficient before the onset of stall are beneficial because they will allow a wing to fly at a lower airspeed. Therefore, flaps will significantly decrease an aircraft’s takeoff and landing distances. Notice, however, that there is also a decrease in the stall angle of attack.
Further increases in maximum lift coefficient may be possible using different types of flaps, such as slotted flaps, fowler flaps, and double-slotted or triple-slotted flaps. In conjunction with leading edge slats, significant stall speed reductions can be achieved with larger airliner types of airplanes. The penalties, however, with more complicated flap systems are an increase in weight, cost, and maintenance.
The variation of the drag with the angle of attack and/or with the lift coefficient is of great interest because of its essential effects on aircraft performance, i.e., drag generally always acts to diminish the aircraft’s performance. Some representative results are shown in the figure below in terms of drag coefficient versus lift coefficient for both a conventional airfoil (in this case, the NACA 23012) and a so-called “laminar flow” airfoil (a NACA 63-series airfoil). The variation of the drag coefficient for the NACA 23012 airfoil is typical. The drag stays low until the angle of attack and the corresponding lift coefficient have increased to the point that significant boundary layer thickening occurs. Then, the drag grows rapidly when a stall occurs.
Notice again that the effects of surface roughness substantially increase the values of drag. Although these particular results are from wind tunnel tests, the effects of roughness, in this case, can simulate the typical “in-service” abrasion of the leading edge of wings compared to when they were new from the factory (and so were initially smoother), the effects of roughness are very significant. In airplane design, it is necessary to consider surface roughness effects so that the aircraft’s potential performance is not overestimated by assuming a perfectly smooth airfoil shape that is unlikely to be unrealizable in practical flight operations.
In the lower angle of attack regime, the drag coefficient on an airfoil can be represented by the equation
where , , and 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 , then it is also possible to write that
Again, it is essential to recognize that this preceding equation can only represent airfoil characteristics below stall; they are invalid with any significant amounts of flow separation or in the stalled flow regime. In the case of a symmetric airfoil (where then and will be zero.
Also shown in the figure above are results for a NACA 63-series laminar flow airfoil, which is a design that maximizes the extent of the laminar boundary layer over the leading edge of the airfoil, hence reducing skin friction drag. These airfoils (with very smooth surfaces) tend to produce low drag “buckets” where the drag is relatively lower, but typically only when the airfoil operates at low angles of attack and low lift coefficients. The geometric shapes of these laminar flow airfoils tend to have a point of maximum thickness much closer to the 1/2-chord than more conventional airfoils, which will have a maximum thickness nearer to the 1/4-chord, as shown in the figure below. This geometric feature produces a favorable pressure gradient over more of the leading edge, thereby encouraging the boundary layer to be laminar for longer. The downside is that such airfoils typically produce lower values of maximum lift coefficient, i.e., a stall occurs at lower angles of attack.
It can also be inferred from the previous discussion that surface roughness encourages turbulent mixing and destroys the laminar boundary layer in short order. The resulting increase in skin friction drag makes the airfoil perform comparably (or sometimes worse) than a conventional airfoil. This reason is why laminar flow airfoil sections are challenging to use successfully in practice because some form of surface roughness on the wing is inevitable. However, such airfoils have seen better success for us on the smooth, almost glass-like wings of sailplanes, although the wings must be polished and always be kept completely clean and free of bugs, etc., for laminar flow to prevail. Nevertheless, obtaining an extensive laminar flow region on an airfoil can profoundly reduce its drag, at least over some smaller range of angles of attack and lift coefficients. Unfortunately, a practical and robust means for achieving extensive laminar flow regions on an airplane wing remains a research challenge, although certain surface coatings have been found to help.
Types of graphical presentations of versus or versus are often called “drag polars” or just “polars,” an example being shown in the figure below. Polars are a helpful way of presenting airfoil section characteristics because results for different operating conditions (e.g., different Reynolds numbers of Mach numbers) or different airfoils can be readily compared and contrasted on a single plot.
One advantage of the versus form of the presentation shown above is that the slope of a straight line running from the origin (0, 0) of the graph to any point on the polar is a measure of the lift-to-drag ratio and hence is a quantitative measure of the airfoil’s aerodynamic efficiency. Notice that the tangent to the polar gives the highest slope. So this point will be the best lift-to-drag ratio, i.e., the operating conditions when the airfoil reaches its highest aerodynamic efficiency. Notice also that there is only one point and operating lift coefficient for best efficiency, which depends on the particular airfoil shape and its operating Reynolds number and Mach number.
Effects of Reynolds Number
The previous results have also indicated some effect of Reynolds number, at least over a limited range, but all the results are higher than one million. Remember that for an airfoil, the Reynolds number will be based on the chord, so
The “c” subscript is often dropped, so just is used.
Flight vehicles typically experience higher chord Reynolds numbers, often as high as or “one-hundred million.” Notice that the Reynolds number is frequently quoted in terms of millions. For example, for , 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 () and the effects of viscosity begin to manifest more strongly, the lift and drag characteristics change more profoundly. The aerodynamic characteristics of airfoils at lower chord Reynolds numbers are particularly important as they affect the performance of many smaller-scale flight vehicles, such as UAVs and drones, which may have chord-based Reynolds numbers well below a million. The data in the figure below shows the profound effects of operating airfoils at lower Reynolds numbers, which in this case range from as low as 20,000 to 3,000,000 (three million).
At the highest Reynolds number, the lift coefficient varies almost linearly with the angle of attack, and the drag is nominally constant, typical of airfoil behavior at higher Reynolds numbers, as previously discussed. However, as the Reynolds number decreases below a million, the lift and drag curves become much more “rounded” and eventually become significantly nonlinear for variations in the angle of attack. This latter behavior directly results from the relatively thicker boundary layers on the airfoil surfaces at these lower Reynolds numbers. In these cases, the lift’s nonlinear behavior becomes much harder to generalize as a function of the angle of attack, and the relationship in Eq. 6, for example, is no longer applicable.
The corresponding drag polar is shown in the plot below, which is another helpful way of summarizing the effects of the Reynolds number on the aerodynamic characteristics. Notice again the profound effects of reducing Reynolds number, which deleteriously affects the lift-to-drag ratio, especially below values of 50,000.
Effects of Mach Number
During the 1950s, aeronautics continued to advance quickly, and with the advent of the turbojet engine, airplanes began to fly much faster. Soon, the effects of compressibility on their performance became increasingly apparent, including the production of higher drag and phenomena such as wave drag, shock-induced flow separation, and buffeting. The significant increase in drag and buffeting on an aircraft as it approached the speed of sound (Mach 1) was initially referred to as the sound barrier. Still, it is now known there is no intrinsic aerodynamic barrier to attaining supersonic flight if sufficient thrust is available to overcome the drag.
Much research has been devoted to understanding compressibility effects on wings and developing airfoil sections and wing shapes to reduce the adverse impact of compressibility at higher flight Mach numbers. The figure below shows schlieren images of the compressible flow about a 10% thickness-to-chord ratio symmetric airfoil set at an angle of attack of 2 degrees in the wind tunnel. The refraction of light rays produces the schlieren effect as they pass through the regions of different flow densities. The lower dark vertical line in these images is a model support in the optical path but not in the flow. Furthermore, it should be noted that these circular images arise because the schlieren flow visualization system uses spherical or parabolic concave mirrors.
This sequence of images shows what happens to the flow about the airfoil as the free-stream Mach number (or is used in the caption to these images) is gradually increased from a subsonic flow just below the critical Mach number through transonic conditions and until the flow becomes supersonic. The differences delineate the developments of the shock waves and other compressible flow patterns; the shock waves are the darker zones produced by the schlieren effects.
No strong shock waves appear on the airfoil at . However, a close inspection of the image shows a small dark line has formed near the 25% chord, suggesting a high-speed flow and that the critical Mach number, , is being reached, i.e., the flow about the airfoil becomes locally supersonic. By , a series of small shocklets (which are baby shockwaves) can be seen between 10% and 30% of the chord, confirming the existence of a supersonic pocket, and by , a weak shock wave has formed. However, no shocks have yet formed on the lower surface.
By , a more substantial shock wave has formed on the upper surface and moved aft on the chord, and by , 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 foot of the upper shock wave, resulting from the adverse pressure gradient produced there (see also the pressure distributions discussed later in this lesson). 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 , both the upper and lower surface shock waves continue to grow in strength, the upper surface shock becoming bifurcated at the airfoil surface, and the flow visualization here suggests that the downstream boundary layer is now relatively thick. By , the boundary layer has separated downstream of the shock on the upper surface, a phenomenon known as shock-induced flow separation. By , both shocks have reached the trailing edge and become significantly bifurcated from their interaction with the relatively thick boundary layer. After that, the flow becomes entirely supersonic. A summary of these preceding observations is shown below as a schematic for clarity.
While the preceding flow visualization images are interesting and of much value, the resulting forces and moments on the airfoil are also important. The effects of compressibility on the lift curve are shown in the figure below. Typically, flow with a Mach number lower than 0.3 is considered incompressible, and an airfoil will have a lift-curve slope of about per radian or 0.11 per degree. Nevertheless, it can be seen that the lift-curve slope increases quickly at the higher Mach numbers because of the effects of compressibility. Notice also that the maximum lift of the airfoil decreases with increasing Mach number. As the Mach number approaches the transonic region (about 0.7), the attainable maximum lift coefficient without producing flow separation and stall is relatively low.
The theoretical relationship representing this latter behavior on the lift-curve slope is called the Glauert rule, summarized in the figure below and compared to measurements made on three NACA symmetric airfoil sections. The Glauert rule states that the lift-curve slope increases according to
which is theoretically exact for a thin airfoil in subsonic flow. A generalization of this result is to correct the low Mach number value of the lift coefficient using
These latter relationships hold for many airfoils of practical interest. Still, there are exceptions, especially for airfoils with large thickness-to-chord ratios or large amounts of nose camber.
Further examples of the effects of compressibility on the lift and drag characteristics of an airfoil are shown in the two figures below. Notice the increase in the lift coefficient for a given angle of attack with increasing Mach number but to a point. This outcome means that the lift is higher for a given dynamic pressure. However, also notice that for a given angle of attack, reductions in lift coefficient and increases in the drag coefficient will occur when the Mach number increases beyond a critical value, i.e., the stall angle of attack of the airfoil decreases with increasing Mach number.
The behavior of the pitching moment on an airfoil is also important. It is always convenient to place the integrated forces at some convenient point on the airfoil, but the question is: At what point? The answer is that the forces can be located at any point if the corresponding moment about that same point is also defined.
In many aerodynamic applications, the 1/4-chord point is used as a reference point, i.e., . The 1/4-chord has theoretical significance, this being the aerodynamic center for a thin airfoil in an incompressible flow. However, even if another reference point were selected, converting from one reference point to another is easy because it is just the application of the rules of statics, as shown in the figure below.
For example, assume the lift force (or normal force) and pitching moment per unit span are known at a point at a distance from the leading edge of the airfoil, and it is desired to find the pitching moment about another point, say at a distance behind the leading-edge. Taking moments about the leading edge in each case gives
Converting to coefficient form by dividing by gives
where the overbar means that the length scale has now been non-dimensionalized with respect to the chord, i.e., . As an example, if the known pitching moment is about the leading edge, , then
and the above equation becomes
Center of Pressure & Aerodynamic center
The concepts of center of pressure and aerodynamic center are used routinely in aerodynamic analysis, and it is essential to understand their differences. They are frequently confused in practice, though they are quite different.
Center of Pressure
By definition, the center of pressure is a point about which the moments are zero, i.e., a point where the resultant forces can be assumed to act. The principle is illustrated in the figure below, the center of pressure location effectively being the balance point (or fulcrum) of aerodynamic forces.
The center and pressure (as well as the aerodynamic center) can be determined if the lift and moment coefficients are known about any other point, the 1/4-chord often being used as a reference point, i.e., the values of and are available. The best way for the student to understand the process is to work through an example using actual airfoil measurements, which are given in the table below for a NACA 0012 (symmetric) airfoil.
|Angle of attack||Lift coefficient||Drag coefficient||Moment coefficient|
These data are shown in the figure below in the form of lift coefficient, , versus angle of attack, , the pitching moment coefficient about the 1/4-chord versus , and the center of pressure location versus . The slope of the lift curve (i.e., the lift-curve slope, , and the slope of the moment curve, i.e., .
To find the position of the center of pressure, say 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
and in coefficient form (divide by ) this becomes
so that the center of pressure (as a fraction of the chord) is given by
For most airfoils with positive camber, the value of is negative, so the center of pressure is generally behind the 1/4-chord. In particular, notice that the center of pressure will be a function of the lift coefficient (and hence also the angle of attack), so it is not a fixed point, as shown in the above figure. Because the center of pressure is a moving point and may not even be on the airfoil’s chord, the center of pressure is only sometimes a convenient concept to use in aerodynamics. Hence, the center of pressure to resolve the forces and moments is used sparingly in practice, even though the pitching moment is zero at this point.
By definition, the aerodynamic center is a point where the moment is constant and independent of the angle of attack. The procedure for finding the aerodynamic center, like that for the center of pressure, requires values of the lift and moment coefficient about any other point, say a distance from the leading-edge, are needed.
If the aerodynamic center is at a distance behind the leading edge, then the application of statics (as explained previously) gives
The objective is to find the location of such that the value of the pitching moment at that point is constant. Differentiating the above equation with respect to gives
The value of can be obtained by using
Therefore, to calculate the location of the aerodynamic center, the slope of the lift curve (in the linear range) and the slope of the moment curve (also in the linear range) are needed. Remember that the linear range corresponds to the conditions where the flow would be fully attached to the airfoil surface. This process is performed by finding the slopes of the best straight line fit to the values of versus and then versus in the linear range (attached flow).
Following on with the previous example, a least-squares fit to the measurements below the stall gives
Therefore, because the 1/4-chord is being used as a moment reference, the aerodynamic center, in this case, is
It can be seen that for the particular airfoil data, the aerodynamic center location is at 24.25% chord. The location of the aerodynamic center depends on the airfoil section and the Mach number at which it operates. Because the value of the lift-curve slope is always positive, then the slope of the moment curve defines the sign of position of the aerodynamic center relative to 1/4-chord. So, if this slope is positive, the aerodynamic center is in front of the 1/4-chord; if it is negative, the aerodynamic center is behind the 1/4-chord. For thin airfoils, the value of is almost constant, so the aerodynamic center is generally always close to 1/4-chord at low free-stream Mach numbers.
It can be concluded that the advantage of using the aerodynamic center about which to resolve the forces and moments is convenient because it is a fixed point and does not change with the angle of attack. However, finding the aerodynamic center location takes more work because the slope of the lift curve and the moment curve must be determined.
What is a Least-Squares Linear Fit?
The linear least-squares method is a numerical process of finding a straight line to represent best the trends shown in a particular data set. The main objective is to reduce as much as possible the sum of the squares of the differences or residuals between the fitted straight line and all of the data points, as shown in the figure below.
It can be assumed that the given data points are , , , …., , where is the number of data points. A straight line is , where is the slope, and is the intercept on the axis. The formulas to calculate the slope and intercept of the best straight-line fit are given by
The process can also be performed in MatLab by using the polyfit function.
Pressure & Shear Stress Coefficients
Pressure coefficients (i.e., non-dimensional pressures) are usually employed when presenting pressure distributions around body shapes and airfoil sections. The pressure coefficient, , is defined as
where is the local static pressure and is the free-stream static pressure. Again, notice the use of free-stream dynamic pressure as a reference pressure. If the local pressure equals the free-stream static pressure , then . If the local pressure is less than the free-stream static pressure , then . At a stagnation point (where the flow velocity is brought to zero), then , at least if the flow is assumed incompressible and the Bernoulli equation is being applied.
The shear stress or local skin friction coefficient is defined as
where is the boundary layer shear stress, which is proportional to the velocity gradient at the surface.
The total skin friction coefficient, denoted by , is obtained by integrating the local skin friction coefficient over the surface. For example, if the surface exposed to the flow has a length , then the total shear stress coefficient is
If two surfaces are exposed to the flow, such as the upper (u) and lower (l) surfaces of an airfoil, then the total skin friction coefficient would be equivalent to the skin friction drag coefficient, i.e.,
Integration of Pressures & Shear Stresses
It is possible to calculate or measure pressure distributions over body shapes, such as on airfoil sections. The process of integration around the body contour can then be used to find quantities such as lift, drag, and pitching moment coefficients. The pressure values are typically two orders of magnitude greater than the shear stress, so adequate results for the lift can often (but not always) be obtained by considering only the pressures. However, for the drag, it is necessary to account for the shear stress because this is the dominant contribution to the drag at low angles of attack.
The pressure distributions around airfoil sections are non-uniform, with higher and lower pressure regions relative to ambient pressure, with typical variations being shown in the figure below. While pressure is a scalar, in this presentation, the local pressure, , is plotted as a vector directed perpendicular to the local surface slope of the airfoil, i.e., in terms of where is the unit normal vector. The green zone represents lower than ambient pressure, and the red zone is higher than ambient pressure. Airfoil sections also produce substantial pressure gradients, especially near the leading edge at higher angles of attack. Therefore, the challenge is to determine the integrated quantities, which requires a knowledge of the values of the pressures and their detailed distributions.
For example, the principle behind the integration process to find the lift can be explained with reference to the figure below. On the upper surface, the per unit span force components acting on an elemental area of width are
and on the lower surface, they are
Integration from the leading edge to the trailing edge produces the total per unit span forces, i.e.,
After and have been determined then and can be found using
The pitching moment about the leading edge is the integral of these forces weighted by their moment arms and with appropriate signs remembering the moments are positive nose-up, i.e.,
From the geometry, then , , which allows all the above integrals to be performed in terms of by using the upper and lower shapes of the airfoil, i.e., and as well as the slopes and , respectively.
Typical Pressure Distributions
The standard practice is to calculate (or measure) these pressure distributions around airfoils and plot the results in terms of pressure coefficient, , as a function of a non-dimensional distance or chord, . Remember that to determine the pressure coefficient, the value of free-stream dynamic pressure is needed, as discussed previously, which requires that the density of the air be obtained from measurements of ambient pressure and temperature.
An example of a measured (discrete) pressure distribution around an airfoil at subsonic conditions is shown below. Notice that the 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 values are on the top half of the plot.
The NACA 0012 is a symmetric airfoil, so at zero angle of attack (or almost at -0.1 degrees in the case), the upper surface pressure distribution is identical to the lower surface. Increasing the angle of attack produces significant suction (negative) pressures on the upper surface, the lowest at the leading edge. At an angle of attack of 8.1, the negative pressures reach a peak value of about .
A representative pressure coefficient distribution around an airfoil in transonic flow is shown in the figure below. Notice in this case that the region of supersonic flow over the leading edge region produces a more uniform low pressure terminated by an abrupt pressure recovery, indicative of the presence of a shock wave. This outcome contrasts with the subsonic flow, where the lowest pressure value is much closer to the leading edge, giving an adverse pressure gradient over most of the chord. The steep adverse pressure gradient near the shock wave makes the boundary layer more prone to thickening and often produces flow separation.
The lift coefficient is given by
which for this problem can be split into two integrals, i.e.,
Taking the first integral gives
Taking the second integral gives
Therefore, adding the two parts of the integral gives
Numerical Integration of Pressure
Rarely are analytic (continuous) distributions of pressure available about a body as in the preceding example. The pressure values are usually known at discrete points, e.g., from measurements made on a wing in the wind tunnel or from CFD calculations. Therefore, if the lift and moment coefficients are to be obtained from the pressure distributions, then a numerical integration approach must be adopted. It is generally impossible to find the drag this way, however, at least using measurements, and usually, a momentum deficiency approach is used.
The lift force coefficient, , can be determined by evaluating the integral
which will be apparent as just the area under the versus the 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.,
where is the number of discrete points for which or is known. The idea is shown in the figure below. Notice that is an index, i.e., and that for points, there would be incremental area contributions to the integral. The more points available with pressure values, the more accurate the numerical integration will be.
The preceding is a satisfactory approach if the values of are known along the chord at all discrete points (both on the upper and lower surfaces). The problem is that pressure points are available at different values of on the upper and lower surfaces. So, this means that the integral (area under the curve) must be found in two parts, i.e., a numerical summation like
where would be the number of points on the upper surface and would be the number of points on the lower surface, the idea being shown in the figure below.
The corresponding pitching moment about the leading edge of the airfoil can be determined from
where now an extra term appears in the equation, which is a moment arm. Notice the sign on the pitching moment, which can be recalled as being positive in a nose-up sense.
Again, this latter integral would be converted to a numerical summation where the moment arms will be included in the summation, which is calculated at the mid-point of each trapezoid, i.e., giving an equation of the form
The pitching moment about 1/4-chord can be determined from
After that, the pitching moment about any other point can be found by using simple statics from the values of and , i.e., for the moment about a point that is a non-dimensional distance from the leading-edge then
Finally, the axial force (or chord force) coefficient can be determined by evaluating
where . 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., values and the slope of the surface) but a lot of points is needed around the nose of the airfoil (where there are steep pressure gradients and relatively high values of surface curvature) to get good accuracy, as shown in the figure below. Caution should also be exercised to ensure the correct signs (positive or negative) are used.
Some MATLAB code is given below, which can be copied and pasted by the student. Running the code gives = 1.4025, which agrees with the value from the analytic integration given in Worked Example #3. Of interest is to vary the number of points used for the numerical integration in the linspace function.
axis([0.0 1.0 1.0 5.0])
x1 = linspace(0,0.1,500);
x2 = linspace(0.1,1,500);
cpu_1 = 1.-300*x1.^2;
cpu_2 = -2.2277 + 2.2777.*x2;
x = [x1 x2];
cpu = [cpu_1 cpu_2];
cpl = (1.0-0.95*x);
dcp = cpl-cpu;
trapz(x,dcp) % to find the section cl using the trapezoidal rule
legend(‘Upper surface’, ‘Lower surface’)
A supersonic airfoil shape is designed to operate at Mach numbers greater than unity. Unlike subsonic airfoils, supersonic airfoils experience shock waves from the compression of air. These shock waves create significant pressure changes, affecting the airflow and aerodynamic performance of the airfoil. The well-rounded, cambered airfoil sections well-suited to subsonic flight speed are generally inappropriate for high-speed and supersonic flight. Supersonic airfoils are distinctive in their geometric shapes in that they are thin (i.e., have a low thickness-to-chord ratio) with sharp leading edges. Supersonic airfoils generally have thinner sections constructed of either angled planes called double-wedge airfoils or opposed circular arcs called biconvex airfoils, as shown in the figure below.
The sharp leading edges on supersonic airfoils prevent the formation of a detached bow shock in front of the airfoil, which is a high source of drag called wave drag. For example, in the schlieren image shown below, the supersonic flow passes smoothly over the sharp nose of the airfoil profile, thereby creating minimum drag. For a supersonic airfoil, the thickness and camber shapes are designed to minimize energy losses associated with the compression and expansion of the flow. However, often only this goal can be achieved over a small range of operating angles of attack.
Consider the flow about an airfoil in the form of a double-wedge or diamond shape experiencing a supersonic flow, as shown in the schematic below. Unlike a subsonic airfoil with smooth surface curvature, a double-wedge airfoil is ideal for supersonic flow. Notice that the upstream flow gets no warning of the approaching airfoil in a supersonic flow, so the streamlines have no curvature.
Oblique compression shock waves occur at the leading edge of the airfoil. The Mach number across the shocks decreases (but remains supersonic), and the static pressure increases over the free-stream value. Notice the more significant pressure increase on the lower surface, contributing significantly to the lift. At the points of maximum thickness, expansion waves appear, which causes the Mach number to increase and the pressure to decrease after the expansion is complete. The upper surface of the airfoil now contributes to the lift. Rarefaction shock waves form at the trailing edge, increasing the Mach number and returning the pressure to the free-stream value.
This latter result shows that the lift-curve slope is lower in supersonic than subsonic flight and also decreases with increasing Mach number.
The results in Eq. 10 (subsonic) Eq. 41 (supersonics) are classical solutions in the airfoil theory and allow the determination of the lift coefficients on airfoils at small angles of attack. These thin airfoil approximations agree well with measurements of the lift-curve slope made on thin two-dimensional airfoils, as shown in the figure below. They also have good validity when extended to three dimensions, such as predicting the lift-curve slope of a finite wing with a higher aspect ratio.
The linearized supersonic airfoil theory shows that the lift coefficient is independent of the airfoil shape, i.e., the lift coefficient produced by a supersonic airfoil at a given angle of attack is the same for a flat plate, a diamond-wedge airfoil, or a biconvex airfoil. The drag, however, is different because the dominant source of drag in supersonic flight is wave drag, which depends strongly on the airfoil shape. For example, for a diamond-wedge airfoil, the drag coefficient is
which shows that wave drag increases with the square of the airfoil thickness-to-chord ratio, . In general, the wave drag of a supersonic airfoil can be expressed as
where is its maximum thickness-to-chord ratio and is its maximum camber. The values of the constants and depend on the exact shape of the airfoil. Therefore, it becomes clear why the airfoils used on the wings of supersonic airplanes must be very thin and mildly cambered compared to those used on subsonic airplanes. You can read more about supersonic airfoils and wings in a later chapter of this book.
In this case, the values given lead to
The lift coefficient is
The wave drag coefficient is
Therefore, the lift-to-drag ratio is
A significant problem with thin supersonic airfoils is at low (subsonic) flow flight speeds when they tend to produce leading-edge flow separation and stall even at moderate angles of attack. Therefore, many aircraft that employ supersonic airfoils must use high-lift devices for takeoff and landing, such as large leading-edge slats along the entire wing. Nevertheless, supersonic aircraft have high takeoff and landing speeds and may need drogue parachutes to reduce their landing distances.
Because commercial jet airliners need to reach higher and higher cruise speeds approaching the speed of sound, i.e., for flight at transonic Mach numbers, this requirement has led to the design of unique wing shapes called supercritical wings. A supercritical wing also uses a supercritical airfoil to reduce the strength of shock waves, thereby reducing wave drag and increasing the drag divergence Mach number, the principle of which is shown in the figure below. The classic approach to increasing the drag divergence Mach number is reducing wing thickness, i.e., thickness-to-chord ratio. However, this has several other disadvantages, including less fuel tank capacity and higher structural weight for the same structural strength and stiffness. Supercritical airfoils were initially studied in the 1950s by Herbert Pearcey at NPL in England and refined during the 1960s by Richard Whitcomb at NACA. Today, supercritical airfoils are used on nearly all commercial jet aircraft, allowing them to cruise at higher flight Mach numbers between 08 and 0.85.
A supercritical airfoil shape is notably distinctive. It has a point of maximum thickness fairly aft on the chord, with a relatively flat upper surface with a slight camber. The aerodynamic principle used in transonic wing and airfoil design is to control the expansion of the flow to supersonic speed and its subsequent recompression, making the leading edge pressure distribution less “peaky” and more uniform. The small amount of leading-edge camber helps limit the strength of the shock waves at the expense of some loss of lift at a given angle of attack. However, such transonic airfoils have significant camber at their trailing edges, which helps to recover the lost lift on the airfoil.
The net effect is a significant delay in the drag divergence Mach number, as shown in the figure below. Increasing the critical Mach number allows the aircraft to operate at higher speeds without encountering excessive drag or other aerodynamic issues associated with transonic flight. Notice, however, that drag divergence Mach number can vary depending on various design factors, including its thickness-to-chord ratio, . However, in general, using a supercritical airfoil limits the onset of drag divergence to a higher free-stream Mach number and improves the aerodynamic performance of aircraft that operate near the speed of sound.
Summary & Closure
Understanding the aerodynamic characteristics of two-dimensional airfoil sections is a prerequisite to understanding the characteristics of finite wings. While airfoil characteristics can be predicted, the most reliable results still come from measurements made in the wind tunnel, especially near the maximum lift or into the stall. Naturally, the importance of testing techniques here must be noticed, and airfoil measurements must be made carefully using established methods. Nevertheless, most of the understanding of airfoil section characteristics, including the effects of geometric shape, has come from measurements.
It is essential for engineers how to interpret and use two-dimensional airfoil characteristics. To this end, the ability to access, interpret, and use graphs from catalogs of airfoil characteristics is critical. In a design problem, for example, it may be necessary to select an airfoil shape to meet a set of aerodynamic requirements. Quantities such as maximum lift coefficient, minimum drag coefficient, maximum lift-to-drag ratio, etc., can all be necessary for quantifying the aerodynamic characteristics of airfoils and can also be a basis for airfoil selection. In some cases, the characteristics of candidate airfoils may need to be compared, which should always be done on a non-dimensional basis in terms of force coefficients, such as in the form of a polar.
- Research how lift and drag measurements of a “two-dimensional” airfoil section could be best performed in a wind tunnel environment.
- Explain why the aerodynamic center for an airfoil moves aft on the airfoil with increasing free-stream Mach number. Hint: Consider the nature of the pressure distributions in subsonic, transonic, and supersonic flow.
- What is the role of the angle of attack in determining the characteristics of an airfoil section?
- Can you explain the terms “stall” and “critical angle of attack” in relation to airfoil sections?
- How do airfoil sections contribute to the efficiency and performance of aircraft wings?
- What are the factors that affect the lift-to-drag ratio of an airfoil section?
- Describe the phenomenon of boundary layer separation and explain its impact on airfoil performance.
- Research shadowgraph and schlieren flow visualization systems for use in the wind tunnel. What are their relative advantages, and which one may be preferred?
- Integration of the pressures to find the lift is usually a numerically accurate and successful process, but what is the most reliable method of finding the drag?
Check out some of these additional online resources about the aerodynamics of airfoil sections:
- Smoke flow visualization video of wing stall.
- Aerodynamics – pressure profile around airfoil – a video by the NACA.
- Video: Wind tunnel pressure data for a NACA 0012 symmetric airfoil
- Video on understanding aerodynamic lift.
- With this simulation, you can investigate how a wing produces lift and drag.
- Prandtl-Glauert rule video from GaTech.
- Video: Wind tunnel pressure data for a NACA 0012 symmetric airfoil.