24 Aerodynamics of Airfoils

Introduction

Understanding the aerodynamic behavior of airfoils and wings is a significant part of aerospace engineering. It is critical to the successful design of all aircraft and many spacecraft too. Any body 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 component of the force 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, as well as the Reynolds number and free-stream Mach number of the flow.

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 examine the aerodynamic characteristics of two-dimensional airfoil sections. 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. Furthermore, this particular approach makes it possible to isolate the other more complicated and interrelated effects associated with the finite span of a wing, including the effects of the wing tip vortices and other aerodynamic effects caused by sweepback, twist, planform (chord) variations, and other things.

Objectives of this Lesson

  • 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.
  • Be aware of some of the aerodynamic characteristics of airfoil sections, both in attached flow and with flow separation, and how these change at different Reynolds numbers and Mach numbers.
  • Understand how to calculate lift and other integrated quantities from the pressure and shear stress distributions about a body.

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 idea being shown in the figure below. These distributions are not uniform and could 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). Boundary layer-induced shear stresses can also be positive (flow moving downstream) or negative (e.g., reversed flow).
The origin of aerodynamic forces on an airfoil section or wing comes from the integrated effects of the distributions of pressure and boundary layer shear stress over its surfaces.

It can be deduced from the figure above that the shear stresses, in aggregate, will act in a direction primarily parallel to the chordline and so will contribute most to the drag force on the wing section. Likewise, the differences in the pressure distribution between the upper and lower surfaces will contribute primarily to the lift force and pitching moment, the shear 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 can be obtained by suitable integration of the effects of the pressures and stresses that act around the surface. 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 near the end of this lesson.

However, it should also be appreciated that the flow over a finite wing will be inherently three-dimensional, 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, thereby affecting the local angles of attack of the wing and so affecting its overall aerodynamic characteristics

The aerodynamics of a finite wing are very three-dimensional, in part because of the effects of the wing tip vortices. Only at sections well away from the wing tip vortices can the flow be assumed nominally two-dimensional.
In the first instance, it seems obvious to understand the wing’s aerodynamic behavior 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 effect, as shown in the figure below. Of course, in a practical sense, a wing of infinite span is impossible. However, it is still possible to mimic two-dimensional wings both theoretically and experimentally, deriving much understanding about the aerodynamics of the airfoil section.
An airfoil section can be considered as a slice of a three-dimensional wing of infinite aspect ratio, the effects of the wing tip vortices sufficient far removed that an airfoil at mid-span behaves effectively as if it were in a two-dimensional flow.

Without the effects of the wing tip vortices, a cross-section of the wing, especially near its mid-span, will behave in a nominally two-dimensional manner, which now forms the basis for further discussion. The word “nominally” must be used in this context because it must be recognized that an actual flow can never be strictly two-dimensional, no matter how large the span or aspect ratio is in practice. Nevertheless, the 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, effectively eliminating the tip vortices. Another common approach 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 approach has seen widespread use, and flow visualization has confirmed the validity of the 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.

An example of a so-called “two-dimensional insert” in a wind tunnel used to measure representative two-dimensional aerodynamic loads on an airfoil section.

Dynamic Pressure

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

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

where \rho is the air density, and V 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 units of N m-2 (Pa) or lb ft-2.

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

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

where \infty means conditions at “infinity” or practically just far away from the wing so it is in the undisturbed “free-stream” flow. In fact, as will be shown, the free-stream dynamic pressure q_{\infty} is often used as the reference pressure in the definition of most of the non-dimensional coefficients used in airfoil and wing aerodynamics.

Airfoil Forces and Moments

The resulting forces acting on an airfoil section from the integrated effects of the pressure and shear can be resolved into a wind-axis system (i.e., in terms of lift L' and drag D') or a chord-axis system (i.e., in terms of normal force N' and a chord force A'), as shown in the figure below.

Resultant forces on an airfoil at angle of attack \alpha. Two statically equivalent axes systems may be used: 1. A wind axis system with lift  per unit span L' and drag per unit spanD'; 2. An airfoil (body) axis system with normal force per unit span N' and axial (chord) force per unit span A'.

As previously discussed, an airfoil can be thought of as a two-dimensional wing. Therefore, for airfoils then, dimensional force and moment quantities per unit span are used, e.g., lift per unit span would be written as L/unit span = L', 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, b = c/2, may be used as a reference length, but this is less common.

By definition, the lift force L or lift force per unit length, L', acts in a direction that is perpendicular to the free-stream velocity, V_{\infty}. The corresponding drag D or drag per unit span, D', is in a direction parallel to V_{\infty}.

Alternatively, the forces can be decomposed into the sum of two other forces, i.e., the normal force per unit span, N', which acts normal to the airfoil chord, and the leading-edge suction force or axial chord force per unit span, A', which points toward the trailing-edge and acts parallel to the chord. Notice, however, that in some cases, the axial or chord force may be defined as positive when pointing toward the leading edge, so it is essential to know the sign convention being used.

Both of these latter force systems are useful in various forms of 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 can be derived from the other by force resolution using the angle of attack \alpha. For example, to go from a body axis system (N' and A') to a wind axis system (L' and D'), then

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

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 coefficient and drag coefficient.

For example, suppose the lift on a given wing is 90 N, and the corresponding drag is 3 N, these values are undoubtedly helpful to know. However, suppose the wing’s corresponding lift coefficient is 0.6, and its drag coefficient is 0.02. In that case, these latter values reveal much about its aerodynamic operating state, and the coefficients also allow comparisons of the effects of wings of different shapes and sizes. For this reason, airfoil data (either measured or computed) are generally presented in terms of non-dimensional coefficients.

The dimensionless force coefficients for an airfoil section are defined using the free-stream dynamic pressure q_{\infty} and chord c as a reference length and are given by:

Lift coefficient, C_{l} = \displaystyle{\frac{L/ \mbox{unit~span}}{q_{\infty} c}} = \displaystyle{\frac{L'}{q_{\infty} c}}

Drag coefficient, C_{d} = \displaystyle{\frac{D/ \mbox{unit~span}}{q_{\infty} c}} = \displaystyle{\frac{D'}{q_{\infty} c}}

Normal force coefficient, C_{n} = \displaystyle{\frac{N/ \mbox{unit~span}}{q_{\infty} c}} = \displaystyle{\frac{N'}{q_{\infty} c}}

Axial (chord) force coefficient, C_{a} = \displaystyle{\frac{A/ \mbox{unit~span}}{q_{\infty} c}} = \displaystyle{\frac{A'}{q_{\infty} c}}

Pitching moments are defined as positive when the moment tends to increase the angle of attack of the airfoil section, i.e., positive pitching moments are equivalent to nose-up moments. Again, the convention is that a moment M' would represent a moment per unit span. Also, there are several convenient points about which the moments can be conveniently calculated, namely the leading-edge (x=0), 1/4-chord (x = c/4), and the center of pressure x = x_{\rm cp}, as shown in the figure below.

Moments on an airfoil can be taken about any convenient point.

Notice that when the reference point is moved to different locations on the chord then 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, C_{m_{\rm LE}} = \displaystyle{\frac{M_{\rm LE}/ \mbox{unit~span}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_{\rm LE}}{q_{\infty} c^2}}

Moment coefficient at some point a, C_{m_{a}} = \displaystyle{\frac{M_a/ \mbox{unit~span}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_a}{q_{\infty} c^2}}

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

Moment coefficient at center of pressure, C_{m_{cp}} = \displaystyle{\frac{M_{cp}/ \mbox{unit~span}}{q_{\infty} c^2}} = \displaystyle{\frac{M'_{cp}}{q_{\infty} c^2}}

Notice the use of lower-case subscripts on all of the coefficients (e.g., C_l not C_L and C_d not C_D) when applied to an “2-dimensional” airfoil section. When applied to a finite span wing then capital letter subscripts are used as will be discussed in the next lesson; this is the standard notation used to distinguish the finite-span aerodynamic coefficients from the two-dimensional coefficients.

Representative Force & Moment Coefficients

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

Quantities such as maximum lift coefficient, minimum drag coefficient, maximum lift-to-drag ratio, pitching moments, and other metrics can all be important in quantifying the aerodynamic characteristics of airfoils and can also be a basis for airfoil selection. However, interpreting the results on these preceding graphs and finding relevant quantities takes some practice, not least because many of the points and/or curves are almost overlapping. To this end, a clearer presentation is shown in the figure below, in which the lift coefficient C_l, the moment coefficient about the 1/4-chord, C_{m_{1/4}, and the drag coefficient, C_d, are more clearly delineated versus the angle of attack of the airfoil to the free-stream flow.

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

At low angles of attack, the lifting characteristics of an airfoil are not substantially influenced by viscosity and the presence of the boundary layers. Recall that boundary layers are thin, viscous-dominated regions near the surface. Notice that in this region, the lift coefficient increases almost proportionally with the angle of attack. The parameter \alpha_0 is the angle of attack for zero lift or what is usually known as the zero-lift angle of attack. For a symmetric airfoil, then \alpha_0 = 0; for positively cambered airfoils, then \alpha_0 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 now soon reach its maximum lift coefficient, C_{l_{\rm max}}. A further increase in the angle of attack is then followed by the onset of flow separation on the upper surface, a sudden loss of lift, and a significant increase in drag, the process being called stall. The basic flow physics of the stall is illustrated schematically in the figure below. At low subsonic Mach numbers, the onset of stall usually occurs at an angle of attack between 12^{\circ} and 15^{\circ} depending on the airfoil section and the Reynolds number. Higher Reynolds numbers inevitably delay the onset of flow separation and stall, increasing the value of C_{l_{\rm max}}.

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

Notice also from the drag curves below the onset of the stall that the drag coefficient of an airfoil remains relatively low and is reasonably constant. However, it will be seen that the effects are different with roughness, which always causes the 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 increasing angle of attack.

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.

Example #1 – Obtaining Airfoil Characteristics from Graphs

The graphs below shows the aerodynamic characteristics of a NACA 2412 airfoil section directly from Abbott & Von Doenhoff. i.e., the lift coefficient C_l, the drag coefficient C_d, and the pitching moment coefficient about the 1/4-chord axis C_{m_{1/4}}. Use these graphs to find for a Reynolds number of 5.7 x 106 and for both the smooth and rough surface cases:  1. The zero-lift angle of attack, \alpha_0. 2. The maximum lift coefficient, C_{l_{\rm max}}, 3. The stall angle of attack, \alpha_s, 4. The minimum drag coefficient C_{d_{0}} and the lift coefficient at which it occurs.
Measured values of lift and drag coefficient shown plotted on a graph.
For the smooth airfoil at Reynolds number of 5.7 x 106:
  1. The legend on the graphs shows that for a Reynolds number of 5.7 x 106 the results are marked by square symbols. Referring to the left side plot this zero-lift angle \alpha_0 for the smooth airfoil is about -2.1o.
  2. For the smooth airfoil then the maximum lift coefficient C_{l_{\rm max}} is about 1.68.
  3.  For the smooth airfoil then the corresponding stall angle of attack \alpha_s is about 17.6o.
  4.  These values can be found from the right side plot, the minimum drag coefficient C_{d_{0}} for the smooth airfoil being about 0.06 at a lift coefficient of 0.4.
For the rough airfoil at Reynolds number of 5.7 x 106:
  1. The legend shows that the results in this case are marked by triangular symbols, so referring to the left side plot this zero-lift angle \alpha_0 for the rough airfoil is unchanged at about -2.1o.
  2. For the rough airfoil then the maximum lift coefficient C_{l_{\rm max}} is less and is about 1.21.
  3. For the rough airfoil then the corresponding stall angle of attack \alpha_s is about 15.1o.
  4. These values can be found from the right side plot, the minimum drag coefficient C_{d_{0}} being higher than for the smooth airfoil and is about 0.098 at a lift coefficient of 0.18.

Lift Characteristics

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. In fact, this linear relationship is exact within the framework of what is known as linearlized inviscid aerodynamic theory. The lift per unit span can be written as

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

where C_{l_{\alpha}} is called the lift-curve-slope (i.e., of the slope of the lift curve in the linear part of the graph) and is measured in per degree or per radian angle of attack. Remember that the value of C_{l_{\alpha}} has units of per angle of attack. At low Mach numbers C_{l_{\alpha}} \approx 2\pi per radian or 0.11 per degree.

In coefficient form then

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

Remember that the foregoing relationship in Eq. \{linearCL} 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. 5 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.

Example #2 – Calculating Lift Coefficients

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

In each case, the relevant equation is

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

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

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

The simplicity of the relationship in Eq. 5 is advantageous in various forms of engineering analysis, 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 angle of attack.

Drag Characteristics

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

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

Notice again, however, 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 the process of airplane design, it is necessary to take surface roughness effects into account 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

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

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

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

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

Also shown in the figure above are results for a NACA 63-series laminar flow airfoil, which is a particular design that maximizes the extent of the laminar boundary layer over the leading edge of the airfoil, hence reducing skin friction drag. These airfoils (with very smooth surfaces) tend to produce low drag “buckets” where the drag is relatively lower, but typically only over a small range of angle of attack when the airfoil operates at low angles of attack and/or at 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 compared to more conventional airfoils that will have a maximum thickness nearer to the 1/4-chord, as shown in the figure below. This geometric feature produces a favorable pressure gradient over more of the leading edge, thereby encouraging the boundary layer to be laminar for longer. The downside is that such airfoils typically produce lower values of maximum lift coefficient, i.e., stall occurs at lower angles of attack.

 

Shapes of two NACA low-drag airfoil sections compared with the NACA 23012 airfoil section.

It can also be inferred from the previous discussion that any surface roughness would completely destroy the laminar boundary layer. 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 always be kept clean and free of bugs, etc., for the laminar flow to prevail. Nevertheless, obtaining an extensive region of laminar flow on an airfoil can profoundly reduce its drag, at least over some smaller range of angles of attack and lift coefficients. Unfortunately, a practical and robust means for achieving extensive laminar flow regions on an airplane wing remains a research challenge, although certain surface coatings have been found to help.

Drag Polars

Types of graphical presentations of C_l versus C_d or C_d versus C_l are often called “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 for different airfoils can be readily compared and contrasted on a single plot.
An example of a “drag polar” for a two-dimensional airfoil.

One advantage of the C_l versus C_d form of the presentation shown above is that the slope of a straight line running from the origin (0, 0) of the graph to any point on the polar is a measure of the lift-to-drag ratio and hence is a quantitative measure of the airfoil’s aerodynamic efficiency. Notice that the tangent to the polar gives the highest slope, and so this point will be the point of 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 will also depend on the particular airfoil shape as well as its operating Reynolds number and Mach number.

Effects of Reynolds Number

The results shown previously have also indicated that there is 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 chord so

(8)   \begin{equation*} Re = \frac{\rho_{\infty} \, V_{\infty} \, c}{\mu_{\infty}} \end{equation*}

Many flight vehicles will typically experience chord Reynolds numbers, often as high as 10^{8} or “one-hundred million.” Notice that the Reynolds number is often quoted in terms of millions. For example, for Re = 3 \times 10^6, the value of the Reynolds number is usually referred to as “A Reynolds number of three million.”

As the Reynolds number decreases below one million (Re = 10^6) and the effects of viscosity begin the 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).

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

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, etc., becomes much harder to generalize as a function of the angle of attack, and the relationship in Eq. 5, 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 has a deleterious effect on the lift-to-drag ratio, especially below values of 50,000.

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

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 barrierbut 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 effects 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 mirrors.

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

This sequence of images shows what happens to the flow about the airfoil as the free-stream Mach number M_{\infty} (or M_1 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.

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

By M_{\infty} = 0.82, a stronger shock wave has formed on the upper surface and moved aft on the chord, and by M_{\infty} = 0.84, a shock develops on the lower surface almost directly below the one on the upper surface. In addition, there is clear evidence of boundary layer thickening at the 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 M_{\infty} = 0.88, both the upper and lower surface shock waves continue to grow in strength, the upper surface shock becoming bifurcated at the airfoil surface, and the flow visualization here suggests that the downstream boundary layer is now relatively thick. By M_{\infty} = 0.90, the boundary layer has separated downstream of the shock on the upper surface, a phenomenon known as shock-induced flow separation. By M_{\infty} = 0.95, both shocks have reached the trailing edge and become significantly bifurcated from their interaction with the relatively thick boundary layer. After that, the flow becomes entirely supersonic.

A summary of these preceding observations is shown below as a schematic for clarity. Notice again that as the free-stream Mach number increases from subsonic through transonic to supersonic, there are profound changes to the flow fields, emphasizing the importance of Mach number in understanding (and predicting) airfoil characteristics.

While the preceding flow visualization images are interesting and of much value, the resulting forces and moments on the airfoil are also important. The effects of 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 2\pi per radian or 0.11 per degree. Nevertheless, it can be seen that the lift curve slope increases quickly at the higher Mach numbers because of the effects of compressibility. Notice also that the maximum lift of the airfoil decreases with increasing Mach number. As the Mach number approaches the transonic region (about 0.7), the attainable maximum lift coefficient without producing flow separation and stall is relatively low.

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

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

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

The Glauert rule states that the lift curve slope increases according to

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

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

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

These latter relationships hold true for many airfoils of practical interest but there are also exceptions, especially for airfoils that have 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 to a point. This outcome means that the lift is higher for a given dynamic pressure. However, also notice that for a given angle of attack, reductions in lift coefficient and increases in the drag coefficient will occur when the Mach number increases beyond a critical value, i.e., the stall angle of attack of the airfoil decreases with increasing Mach number.

Summary of the effects of free-stream Mach number as it increases from subsonic through transonic to supersonic conditions.
Drag coefficient characteristics of the NACA 23015 airfoil section as function of Mach number for several angles of attack.

Pitching Moments

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

In many aerodynamic applications, the 1/4-chord point is used as a reference point, i.e., a = c/4. In fact, 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 was to be selected, converting from one reference point to another is easy because it is just the application of the rules of statics, as shown in the figure below.

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

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

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

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

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

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

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

Center of Pressure & Aerodynamic center

The concepts of center of pressure as well as aerodynamic center are used routinely in aerodynamic analysis, and it is important to understand the differences between them. They are frequently confused in practice, though they are quite different.

Center of Pressure

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

The center of pressure location is the effective balance point on the airfoil where there is no pitching moment.
The center and pressure (as well as the aerodynamic center) can be determined if the lift and moment coefficients are known about any other point, the 1/4-chord often being used as a reference point, i.e., the values of C_l and C_{m_{1/4}} are available. The best way to understand the process is to perform an example using actual measurements, which are shown in the figure below.
Representative measurements of airfoil characteristics that can be used to find the center of pressure and the aerodynamic center.

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

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

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

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

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

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

For most airfoils, the value of C_{m_{c/4}} is negative, so the center of pressure is generally behind the 1/4-chord. In particular, notice that the center of pressure will be a function of the lift coefficient (and hence also the angle of attack), so it is not a fixed point, as shown in the above figure. Because the center of pressure is a moving point and may not even be on the chord of the airfoil, 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 not used much in practice even though the pitching moment is zero about this point.

Aerodynamic Center

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

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

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

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

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

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

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

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

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

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

and

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

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

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

It can be seen that for the particular airfoil data then, 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, and if it is negative, the aerodynamic center is behind the 1/4-chord. For thin airfoils, the value of C_{m_{1/4}} is almost constant, so the aerodynamic center is generally always close to 1/4-chord at low free-stream Mach numbers.

It can be concluded that the advantage of using the aerodynamic center 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 some more work because the slope of the lift curve and the moment curve must both be determined.

Pressure & Shear Stress Coefficients

Pressure coefficients (i.e., non-dimensional pressures) are usually employed when it comes to the presentation of pressure distributions around body shapes and airfoil sections. The pressure coefficient, C_p, is defined as

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

where p is the local static pressure and p_{\infty} is the free-stream static pressure. Again, notice the use of free-stream dynamic pressure as a reference pressure. If the local pressure is equal to the free-stream static pressure p_{\infty}, then it is clear that C_p = 0. At a stagnation point (where the flow velocity is brought to zero) then C_p = 1, at least if the flow is assumed incompressible and so the Bernoulli equation applies.

The shear stress or local skin friction coefficient is defined as

(24)   \begin{equation*} c_f = \frac{\tau_w}{\frac{1}{2} \rho_{\infty} V_{\infty}^2} = \frac{\tau_w}{q_{\infty}} \end{equation*}

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

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

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

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

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

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 magnitude of 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. In this case, 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.

Representative static pressure distributions around symmetric and cambered airfoils. In this case, the magnitude of the local pressure is plotted perpendicular to the local surface slope of the airfoil.

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

(27)   \begin{eqnarray*} dN'_u & = & (p_u \cos \theta_u - \tau_u \sin \theta_u ) ds_u \\ dA'_u & = & (p_u \sin \theta_u + \tau_u \cos \theta_u ) ds_u \end{eqnarray*}

and on the lower surface they are

(28)   \begin{eqnarray*} dN'_l & = & (p_l \cos \theta_l - \sin \theta_l ) ds_l \\ dA'_l & = & (p_l \sin \theta_l + \tau_l \cos \theta_l ) ds_l \end{eqnarray*}

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

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

(29)   \begin{eqnarray*} N' & = & \int_{\rm LE}^{\rm TE} dN'_u + \int_{\rm LE}^{\rm TE} dN'_l \\ A' & = & \int_{\rm LE}^{\rm TE} dA'_u + \int_{\rm LE}^{\rm TE} dA'_l \end{eqnarray*}

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

(30)   \begin{eqnarray*} L' & = & N' \cos \alpha - A' \sin \alpha \\ D' & = & N' \sin \alpha + A' \cos \alpha \end{eqnarray*}

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

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

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

Typical Pressure Distributions

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

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

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

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

A representative pressure coefficient distribution around an airfoil in transonic flow is shown in the figure below. Notice in this case that the region of supersonic flow over the leading edge region produces a more uniform low pressure terminated by an abrupt pressure recovery, 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. Needless to say, the steep adverse pressure gradient in the vicinity of the shock wave makes the boundary layer more prone to thickening and often produces flow separation.

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

Example #3 – Integrating a Pressure Distribution

The pressure coefficient C_p distribution over a two-dimensional body of chord c operating at low angles of attack is described by

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

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

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

Calculate the lift coefficient of the body.

The lift coefficient C_l is given by

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

which for this problem can be split into two integrals namely

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

Taking the first integral gives

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

and so

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

Taking the second integral gives

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

and so

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

Therefore, adding the two parts of the integral gives

    \[ C_{l} = C_{l_{1}} + C_{l_{2}} = 0.09525 + 1.307 = 1.4025 \]

Numerical Integration of Pressure

Rarely are analytic (continuous) distributions of pressure available about a body as in the foregoing 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 not possible to find the drag this way, however, at least using measurements, and usually a momentum deficiency approach is used.

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

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

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

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

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

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

The principle of numerical integration of the differential pressure distribution over a surface to find the lift, etc.
The foregoing is a satisfactory approach if the values of \Delta C_{p} are known at all of the discrete points (both on the upper and lower surfaces) along the chord. The problem is that more usually pressure points are available at different values of x/c 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

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

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

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

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

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

where now an extra \overline{x} term appears in the equation, which is a moment arm. Notice the sign on the pitching moment, which can be recalled as being positive in a nose-up sense.

Again, this latter integral would be converted to a numerical summation where the moment arms will be included in the summation, which are calculated at the mid-point of the trapezoid, i.e., giving an equation of the form

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

For completeness, then the pitching moment about 1/4-chord can be determined from

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

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

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

The axial force (or chord force) coefficient can be determined by evaluating

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

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

Example #4 – Numerically Integrating a Pressure Distribution

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

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

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

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 cannot be overlooked, 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.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • Do some research to determine 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.
  • Research shadowgraph and schlieren flow visualization systems for use in the wind tunnel. What are their relative advantages, and which one may be preferred?
  • Integration of the pressures to find the lift is usually a numerically accurate and successful process, but what is the most reliable method of finding the drag?

Other Useful Online Resources

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

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