23 Wing Shapes & Nomenclature

Introduction

One of the most critical parts of an airplane is, obviously, its wings. The wings primarily create the lift to overcome the aircraft’s weight. However, they also have movable surfaces, such as ailerons, flaps, spoilers, trim tabs, etc., to develop additional aerodynamic forces to control the aircraft during its flight. Wings are geometrically defined in terms of their span (distance from wing tip to wing tip), planform, twist (pitch angle) distribution, and cross-section (i.e., airfoil section shape or profile shape). The shape of a wing must be engineered to give good aerodynamic efficiency in lift production for the minimum amount of drag, i.e., the maximization of the lift-to-drag ratio, which is one fundamental goal in aerodynamic design. However, there will always be other aerodynamic requirements that will factor into the wing shape design, including its low-speed flight and stalling characteristics.

The wing of an airplane is not only a masterful piece of engineering but also a piece of beauty. In this case, the wing sports a blended winglet (the upturned part at the tip of the wing) to help reduce drag.

In addition, the wing structure must be strong enough and stiff enough to carry all of the aerodynamic and other loads acting on the wing, so the wing must be tailored to meet all structural requirements. Examples include:

  • Minimizing the loads and deformations of the wing under the action of the aerodynamic forces and moments.
  • Avoiding the onset of adverse aeroelastic effects and flutter.
  • Carrying undercarriage loads or other point loads such as engines and various inertial loads.

The wings carry the entire weight of the aircraft, so large shear loads and bending moments are produced near the root of the wing. To this end, the wing shape typically needs to be much larger in chord and thicker in cross-section than at the wing tips to obtain the required structural strength and stiffness.

Learning Objectives

  • Know the critical geometric parameters used to define the shape of a wing.
  • Calculate the wing area and aspect ratio of an arbitrary wing planform.
  • Understand the significance and use of mean wing chords.

Geometric Definition of a Wing

Engineers use various geometric parameters to describe the shapes of wings. Terms such as span, chord, mean chord, aspect ratio, and sweep angle are used routinely in wing design, and it is essential to understand what these terms mean. Other terms used in wing design include the wing’s twist or washout, the taper ratio, the dihedral or anhedral, and the thickness-to-chord ratio.

Wing Span & Semi-Span

The span of the wing or wing span, given the symbol b, is defined as the distance from one wing tip to the other (for now, the effect of a winglet will not be considered), as shown in the figure below. Sometimes, the semi-span is used to define the wing in engineering analysis, given the symbol s and equal to b/2. Notice that the symbol \rm C_{\!\!\!\rm \Large L} means the centerline of the wing or aircraft.

The key geometric parameters used to describe the shape of a wing are span, semi-span, chord, and sweep angle. Learning the definitions of these parameters is essential.

Wing Chord and Planform

The wing chord is the distance from its leading edge to its trailing edge in the streamwise direction, i.e., parallel to the airplane’s longitudinal axis. The chord is given the symbol c. On many airplanes, the chord changes along the wing’s span, i.e., c = c(y) as in the above figure, mainly for aerodynamic reasons. A primary aerodynamic goal for the wing is minimizing drag for a given amount of lift, i.e., maximizing the lift-to-drag ratio.

The shape of the wing is defined in terms of the chord distribution along the span of the wing, and when the wing is viewed from above, the resulting shape is called the planform. If y is measured from the longitudinal centerline of the aircraft (i.e., not the wing root), as shown in the figure below, then the local value of the wing chord can be expressed as

(1)   \begin{equation*} c = {\rm function} (y) \end{equation*}

where y = 0 at the centerline of the aircraft and y = s = b/2 at the wing tip. The wing chord can also be expressed in terms of the non-dimensional span where y/s = 0 at the wing centerline and y/s = 1 at the wing tip.

Geometric definition of linear planform taper of a wing.

Wings are often linearly tapered in planform, for good engineering reasons, with different values of the root chord c_0 and the tip chord c_T. In this case, the linear taper ratio of the wing is defined as

(2)   \begin{equation*} \lambda = \frac{c_T}{c_0} \end{equation*}

For a linear form of wing taper, as shown in the figure above, the chord distribution along the wing will be

(3)   \begin{equation*} c (y/s) = c_0 \bigg( 1 - (1 - \lambda) \left( \frac{y}{s} \right) \bigg) \end{equation*}

The taper ratio may also be used in cases where the wing is not precisely linearly tapered to quantify the average taper of the wing planform.

Wings may not only taper in planform but also in thickness or, indeed, as combinations of taper and thickness, as shown in the figure below. Using both taper and thickness together gives considerable engineering latitude in tailoring the shape of the wing to meet a given level of aerodynamic performance and minimizes structural loads and weight. A further aerodynamic advantage may be gained by using different airfoil sections along the span, e.g., using a relatively thin airfoil section at the wing tip for low drag where the structural loads are lower.

Illustration showing the geometric effects of planform taper and thickness taper.

The use of inverse wing taper is unusual, i.e., the chord increases toward the wing tip. However, it has been used to address the problem of adverse stall characteristics and a tendency to spin, a common issue in the first generation of jet aircraft with swept wings. The sweepback of the wing encourages a spanwise flow, making the wing tips more likely to stall first and reducing the effectiveness of the ailerons. However, this latter approach was unsuccessful, and other (and simpler) methods were more effective in mitigating the stall problem with swept wings.

Sweepback

Today, many airplanes have wings with sweepback, as shown in the figure below, but many lower-performance airplanes will have no sweepback. The primary aerodynamic purpose of sweepback on a wing is to delay the onset of compressibility effects and the build-up of wave drag to a higher flight Mach number and/or to reduce drag at a given Mach number, decreasing the propulsive thrust and fuel required for flight.

Using sweepback on a wing can help mitigate the onset of compressibility effects and the rise of supersonic drag.

Aerodynamically, the flight Mach number component perpendicular to the wing’s leading edge primarily affects the lift and drag, assuming that the Mach number parallel to the leading edge makes no contribution, often referred to as the independence principle. Thus, for aerodynamic analysis, the free-stream Mach number, that is, the flight Mach number of the aircraft, is resolved into components normal and parallel to the wing’s leading edge based on the local sweep angle. Wings can also be swept forward to obtain the same effect. However, a problem with a forward-swept wing is that it is aeroelastically unstable and tends to twist nose-up under the action of aerodynamic loads, so forward sweep is rarely used.

Aircraft designed for sustained supersonic flight inevitably have much higher sweepback angles than subsonic airplanes; the best planform shape for supersonic flight approaches a classic “Delta” shape. However, the use of sweepback can have other effects on the aerodynamics of the wing and the airplane, including adverse stall and low-speed handling characteristics, so usually, as little as possible sweepback is used. Sweepback also tends to increase the lateral stability of the airplane. Some aircraft may have variable sweepback to optimize the flight aerodynamics, such as the B-1 bomber. Still, there is a significant structural weight penalty with such “swing-wing” designs, which will be at the expense of some useful load, i.e., fuel and payload.

The sweepback angle is often defined by the angle made by the location of the 1/4-chord points along the span of the wing, i.e., \Lambda_{1/4} or it may be alternatively defined by the leading edge and trailing edge angles, i.e., by the values \Lambda_{\rm LE} and \Lambda_{\rm TE}, respectively. Wings may also be designed in parts with two different sweepback angles: one sweepback angle for the inboard wing panel (usually the smaller angle) and another (larger) angle for the outboard wing panel.

Wing Twist or Washout

Wings may also be slightly twisted along their span; the primary purpose of wing twist is to help give the desired distribution of aerodynamic forces over the span. In practice, most wings are twisted in some form, often subtly. It is known from aerodynamic theory and practice that the spanwise form of the lift distribution is critically important for minimizing induced drag. While the spanwise lift distribution is strongly affected by wing planform (i.e., the wing chord distribution), the additional use of wing twist can help tailor the wing lift distribution to obtain the desired aerodynamic effects.

Most wings are twisted from root to tip to improve aerodynamic efficiency. Note: The amount of twist is exaggerated in these figures for clarity.

Most wings are twisted nose-down from root to tip, i.e., the pitch angles change from wing section to section and become increasingly negative toward the wing tip. This effect is called “washing” the wing twist, and so is called washout. Typical washout values on a wing are between 0 and 7 degrees nose-down, anything more than this being somewhat unusual.

Although very uncommon for an airplane, if a wing is twisted nose-up along its span by increasing the wing twist from root to tip, it is called washin. Interestingly, some helicopter blades use both washout and washin, the washin component being used over the blade tip region to keep the tips from producing negative lift at higher forward airspeeds. Aerodynamic twist can also be incorporated into the wing design by changing the shape of the airfoil section, i.e., the angle of attack at which the section produces zero lift, which manifests similarly to what would be obtained by changing the pitch angle of the wing.

Airfoil Sections

Wings can use various types and distributions of airfoil sections to suit the application of the aircraft. By cutting a slice out of an airplane wing and viewing it from the side, the wing cross-section is obtained, or what is usually called just the airfoil section or airfoil profile, with an example shown below. It is possible to describe the shape of airfoil sections by using camber, thickness, and nose radius as primary geometric parameters. Outside of the U.S., wing sections are usually called “aerofoils.”

The key geometric parameters that define the shape of an airfoil.

In the evolution of wings, airfoil sections have progressed from simple curved plate-like shapes with little thickness inspired by birds’ wings to sophisticated shapes with camber and thickness to give high lift and low drag. Airfoils used on subsonic airplanes are usually relatively thicker and have camber, as shown in the figure below. Whereas for high-speed or supersonic aircraft, the airfoils are thinner with a small leading-edge radius and have slight camber. Notice that the term “thickness” usually means the section’s maximum thickness-to-chord ratio, so it measures how thick the wing section is relative to its chord, i.e., thickness/chord, which is usually quoted in percent. Therefore, a thickness-to-chord ratio of 0.12 would mean an airfoil that was 12% thick.

Airfoil sections comprise the cross-section of a wing. Subsonic airfoils are generally fairly thick with camber, whereas supersonic wings are very thin and have mild or no camber.

Most wings use different airfoils along their span, which can be progressively blended together to give an overall wing design that is better than could be obtained using a single airfoil. This latter approach is often necessary with large commercial aircraft. As shown in the figure below, the need for significant thickness to carry the bending and shear loads at the wing’s root makes the airfoil design for low drag at higher flight Mach numbers rather challenging, especially when the wing operates in transonic flow. The designer can use thinner airfoils better suited to high-speed flight and transonic flow conditions toward the wing tips, where the bending moments and structural stresses are lower. Even on low-performance airplanes, there can be significant aerodynamic and performance advantages in using different airfoils at the wing root compared to the wing tip sections.

The wing must carry considerable bending and shear loads and needs to be structurally thicker at the wing root to prevent excessive wing tip bending displacements.

Airfoil sections may also be used to introduce an aerodynamic twist along the wing span. Different cambered airfoils inevitably have different zero-lift angles of attack (although they may differ only by a degree or two at most), so other airfoils with different camber can be used to effectively twist the wing aerodynamically. The effects obtained are usually combined with a geometric twist to achieve the desired spanwise lift distribution to meet aerodynamic performance and other goals.

It is also found that using wing twist is particularly helpful in controlling the stall developments on the wing, especially in preventing the wing tips from stalling, allowing the designer to have some latitude in satisfying the low-speed handling qualities of the airplane. It is not unusual for newly designed airplanes to have the airfoils over the outer wing panels changed after their first flights to meet stalling and handling qualities requirements needed for certification and the award of a certificate of airworthiness.

Dihedral & Anhedral

The dihedral angle is the upward angle that the wing panels make relative to a reference axis for the aircraft, as shown in the figure below. The primary purpose of using dihedral is to improve the aircraft’s lateral (roll) stability; usually, only a few degrees are needed to enhance stability significantly. The horizontal tail may also have some dihedral, especially on larger aircraft, contributing somewhat to the lateral stability. Some amount of dihedral will come from wing-bending structural displacements.

Using a dihedral angle on a wing improves the aircraft’s lateral or “roll” stability. Anhedral, sometimes used on specific aircraft types, has the opposite effect.

An example of a Boeing 737 is shown in the photograph below, where it can be seen that both the main wing and horizontal tail have a notable amount of dihedral. Good stability about all three flight axes is desirable for most aircraft, especially an airliner, so the passengers experience a smooth and comfortable ride, especially through turbulence.

A blue and white Boeing 737-800 aircraft facing the camera, moving down the airport runway. Water and low mountains in the background.
The dihedral of the main wing and the tailplane is apparent on this Boeing 737-800. This aircraft also sports a dual winglet with both upper and lower sections.

A downward wing angle is called anhedral and is somewhat less common to find on airplanes without wing sweepback or a high wing design because it decreases roll stability. However, airplanes with swept wings may use anhedral to offset the increase in roll stability from using sweepback. Airplanes with high-mounted wings also tend to have significant pendular lateral stability because the center of gravity lies below the center of the lift, as shown in the figure below. The center of lift can be assumed to be the aerodynamic “pivot” point. In this case, anhedral on the wing is needed so that the lateral stability is not too strong to make the aircraft difficult to turn and maneuver.

 

Gravitational pendular stability can be offset by using anhedral on the wing design.

A good example of a wing with anhedral is found on the C-5 Galaxy military transport aircraft, as shown in the photograph below. The C-5 needed to have the fuselage and loading deck as close to the ground as possible, so the only design choice for this aircraft was a high wing with anhedral. While lateral stability (and stability, in general) is generally good, too much stability can make the aircraft less maneuverable and agile, and overall handling qualities can suffer, i.e., the aircraft becomes sluggish in response to control inputs. Sweepback on the wing, which is used to delay the onset of compressibility effects, can also introduce certain undesirable flight dynamic characteristics, which can be offset using anhedral.

The military C-5 Galaxy transport aircraft has anhedral on the wings to balance the inherent pendular stability caused by a high-wing design.

Calculation of Wing Area

Several derived geometric characteristics of wings, including the wing area, aspect ratio, and mean chord are important in engineering analysis. The planform area of the wing, which is given the symbol S, is obtained by integrating the distribution of wing chord along the span from one wing tip to the other, i.e.,

(4)   \begin{equation*} S = \int_{-s}^{s} c \, dy = \int_{-b/2}^{b/2} c \, dy \end{equation*}

where c = c(y), as shown in the figure below.

Calculating the wing reference area requires a knowledge of the wing chord distribution followed by an area integration.

If both wings are of the same geometry, i.e., symmetrically disposed with respect to the longitudinal axis along the fuselage, and so are mirror images of each other (which is the case on nearly all airplanes), then

(5)   \begin{equation*} S = 2 \int_{0}^{s} c \, dy = 2 \int_{0}^{b/2} c \, dy \end{equation*}

It is important not to confuse the wing area (capital “S“) with the semi-span of the wing (lowercase “s“). Of course, there can be a dilemma here, and a question to ask is whether the wing area is the actual area of the wing exposed to the airflow or something else. In the above equation, the former has been assumed, and the value of the area is called the planform or wing reference area. However, there can be circumstances when the actual area of the wing exposed to the flow needs to be known, called the wetted area, in which case the lower limit of integration would be adjusted accordingly to start from the side of the fuselage. Therefore, it is essential to verify the actual definition(s) of the wing area used in different types of engineering analysis.

Calculation of Aspect Ratio

The aspect ratio of the wing, which is given the symbol A\!R, is defined as the ratio of the square of the wing span to the wing reference area, i.e.,

(6)   \begin{equation*} A\!R = \frac{{\rm span}^2}{\rm area} = \frac{b^2}{S} = \frac{4s^2}{S} = \frac{4s^2}{2 \displaystyle{\int_{0}^{s} c \, dy}} \end{equation*}

The aspect ratio of a wing is important in aerodynamic analysis because a wing with a higher aspect ratio is generally more aerodynamically efficient and will have lower drag.

Every aircraft design will have a somewhat different wing aspect ratio. However, typical values range from 5 to 10 for a small general aviation aircraft to 9 to 15 for a commercial transport aircraft, and 30 and higher for a glider (sailplane). As a point of reference, the Voyager aircraft that in 1986 flew around the World without refueling had a wing aspect ratio of 34, which is unusually high for an aircraft other than a sailplane.

It will be apparent that the aspect ratio of a wing is a physical measure of the geometric slenderness of the wing. The easiest way to understand this is to assume a wing of a constant chord, i.e., c(y) = c = constant. In this case, then

(7)   \begin{equation*} S = 2 \int_{0}^{s} c \, dy = 2 c s \end{equation*}

and

(8)   \begin{equation*} A\!R = \frac{b^2}{S} = \frac{4s^2}{S} = \frac{4s^2}{2 c s} = \frac{2s}{c} = \frac{b}{c} \end{equation*}

which is just the ratio of the wing span to the chord. Therefore, a wing with a higher span and a narrower chord will have a higher aspect ratio. Hence, the numerical value of the aspect ratio becomes a measure of the slenderness of the wing.

However, the aspect ratio must be calculated using the exact wing planform; most wings will not have rectangular planforms and so the aspect ratio must be calculated by using the wing chord distribution and resulting area, i.e., using the formula

(9)   \begin{equation*} A\!R = \frac{4s^2}{2 \displaystyle{\int_{0}^{s} c \, dy}} \end{equation*}

In some cases, the wetted aspect ratio may be specified, which will use the wetted wing area rather than the reference wing area, but this is rare.

Worked Example #1 – Finding the planform area & aspect ratio of a wing

The wing of a Supermarine Spitfire has an elliptical wing planform with a root chord c_0 of 100 inches and a span s of 445 inches. Calculate the planform area and aspect ratio of this wing.

The chord for an elliptical wing planform shape can be expressed as

    \[ c(y) = c_0\sqrt{ 1 - \left( \frac{y}{s} \right)^2 } \]

where c_0 is the root (centerline) chord. The planform area of the wing, S, is

    \[ S = 2 \int_{0}^{s} c \, dy \]

and substituting for the chord distribution gives

    \[ S = 2 \int_{0}^{s} c_0\sqrt{ 1 - \left( \frac{y}{s} \right)^2 } \, dy = 2 c_0 \int_{0}^{s} \sqrt{ 1 - \left( \frac{y}{s} \right)^2 } \, dy \]

This is a standard integral, so

    \[ S = 2 c_0 \left[ \frac{1}{2} \left( y \sqrt{1 - \frac{y^2}{s^2} } + s \sin^{-1} \left( \frac{y}{s} \right) \right) \right]_0^s \]

or

    \[ S = c_0 \left[ y \sqrt{1 - \frac{y^2}{s^2} } + s \sin^{-1} \left( \frac{y}{s} \right) \right]_0^s = \frac{\pi c_0 s}{2} \]

Therefore, the planform area of this wing is

    \[ S = \frac{\pi c_0 s}{2} = \frac{ \pi \times (100.0/12.0) \times (445/2/12)}{2} = 242.71~\mbox{ft$^2$} \]

and the aspect ratio, A\!R, is

    \[ A\!R = \frac{b^2}{S} = \frac{4s^2}{S} = \frac{4 \times (445/2/12)^2}{242.71} = 5.67 \]

The aspect ratio of a wing on an airplane can be increased by increasing the wing span and decreasing the wing chord, as shown in the figure below; this example also holds the wing area constant. The aerodynamic advantage in doing so is a significant reduction in induced drag because the wing tip vortices (the source of this type of drag) are further away from more of the wing. However, as the aspect ratio of a wing increases, it becomes more challenging to design the wing to have sufficient stiffness and strength without increasing its weight. As a result, a more extended wing is inevitably more flexible unless some additional structure is used to stiffen the wing.

A higher aspect ratio wing has a higher span and a narrower chord. Because the wing tip vortices are further away from more of the area of a high aspect ratio wing, it has lower “induced” drag.

In airplane design, the final selection of the wing aspect ratio is inevitably a compromise between aerodynamic, structural, weight, and aeroelastic considerations. Longer wings are also heavier and more susceptible to flutter problems because they are inevitably more flexible. Nevertheless, there are tremendous aerodynamic advantages in using a wing of the highest possible aspect ratio if all of the structural, weight, flutter, and other requirements for the airplane can be satisfied.

Comparison of the aspect ratio of a sailplane with that of a general aviation (GA) airplane.

Sailplanes, which are high-performance gliders, typically have very high aspect ratio wings compared to powered aircraft, so they can achieve high lift-to-drag ratios and can glide long distances by design. The DG-800 in the photograph below exemplifies a modern sailplane with an aspect ratio of just over 27. These types of sailplanes may be able to glide more than 50 miles (in still air) from an altitude of only 5,000 feet.

This DG-800 is exemplary of a contemporary sailplane, with an aspect ratio of 27 and a lift-to-drag ratio of over 50.

Winglets

Winglets can be used to increase the effective aspect ratio of the wing without substantially increasing the wingspan. Winglets were designed by Richard Whitcomb at NASA Langley to help move the wing tip vortices away from more of the wing and reduce the induced drag. Winglets do nothing to reduce the strength of the wing tip vortices. The effects are often apparent from the natural flow visualization, as shown in the photograph below, where it can be seen that the wing tip vortex is trailed from the very tip of the winglet. However, winglets also add some wetted area, increasing skin friction and overall profile drag, yet there is still a net reduction in total aircraft drag.

A Boeing A330 Aircraft flying away from the camera on a clear sky background. White water vapor is trailing off the wings and the winglet tips.
Natural flow visualization in the form of water vapor about a wing shows a vortex trailing from the tip of each winglet.

As shown in the images below, many different variations of winglets have been used on commercial airliners. Today, the trend is toward using more blended winglets with smooth chord variations in the wing-to-winglet transition area, which helps minimize the profile drag while maximizing induced drag reductions.

Many different winglet designs have been used in airplanes. The winglets help increase the effective aspect ratio of the wing and reduce the induced drag from the wing tip vortices.

More recent variations of the winglet appear on aircraft such as the Boeing 737 MAX, which is designed further to reduce the induced drag from the lifting wing and improve the aircraft’s flight efficiency and range. Increasing the length (height) of a traditional winglet from the wing’s top surface helps to further reduce the induced drag, at least to a point, but it also increases the wing’s overall structural weight. Adding another winglet pointing downwards realizes the aerodynamic benefits without as much weight penalty. It seems unlikely, however, that further reductions in induced drag can be realized by yet more permutations of the basic winglet design.

Boeing’s Advanced Technology Winglet is claimed to reduce fuel consumption on 737 MAX by up to 1.5%. While this may seem like a slight drag reduction, the potential fuel savings are significant over the aircraft’s operational life.

Mean Chords

Other derived geometric parameters relevant to wings are the standard mean chord (SMC) and the aerodynamic mean chord or mean aerodynamic chord (MAC). Mean chords are used in aerodynamic analyses as a reference length, such as to calculate pitching moment coefficients. The idea of using a mean chord is to provide a standardized measure that can be applied across different wing geometries and sizes. Mean chords can also be used as reference lengths when quantifying the stability and control characteristics of an aircraft.

Standard Mean Chord

The SMC, which is given the symbol \overline{c}, is defined as the ratio of the wing area to the wing span, i.e.,

(10)   \begin{equation*} {\rm SMC} = \frac{\rm area}{\rm span} = \overline{c} = \frac{2 \displaystyle{\int_{0}^{s} c \, dy}}{2s} = \frac{ \displaystyle{\int_{0}^{s} c \, dy}}{s} \end{equation*}

It will then be apparent that the SMC is the chord of an equivalent rectangular wing with the same area and span. The SMC is rarely used in practical aerodynamics for various reasons, including the fact that the definition of the average chord is purely geometric.

Mean Aerodynamic Chord

The mean aerodynamic chord (MAC) is defined as the chord of an equivalent wing that would experience aerodynamic forces identical to those of the actual wing. This is essentially another type of representative or “average” aerodynamic chord of the wing, the basic principle of which is shown in the figure below.

The principle behind using a mean aerodynamic chord (either the SMC or MAC) is to create an equivalent rectangular wing of average chord.

The MAC is defined as

(11)   \begin{equation*} {\rm MAC} = \overline{\overline{c}} = \frac{2 \displaystyle{\int_{0}^{s} c^2 dy}}{S} \end{equation*}

Finding the MAC of a wing is equivalent to finding the location of the aerodynamic center of pressure of the wing (i.e., where the resultant lift can be assumed to act) and the corresponding value of the chord at that location.

Notice that, in principle, the SMC and the MAC can be determined for any wing, including the horizontal and vertical stabilizers. However, in all cases, their evaluation will involve spanwise integration, either analytically or numerically. Mean wing chords are also used as reference lengths in other disciplines, such as flight dynamics, where the values of parameters such as neutral point, the center of gravity, etc., are often quoted as a fraction of the mean chord,

SMC & MAC of a Linearly Tapered Wing

A linearly tapered wing is one of the most common types of wing planforms. It has a chord distribution that can be expressed as

(12)   \begin{equation*} c = c_0 - \lambda y \end{equation*}

where \lambda is a constant. Notice that when y = 0 then c = c_0 (i.e., the root chord) and when y = s then c = c_0 - \lambda s = c_t (i.e., the tip chord). For example, if \lambda = 0.5 c_0/s = c_0/2s then c_t = c_0/2, which is a trapezoidal wing planform with a 2:1 taper ratio.

The wing area is given by

(13)   \begin{equation*} S = \int_{-s}^s c dy = 2 \int_0^s c dy = 2 \int_0^s \left( c_0 - \lambda y \right) dy \end{equation*}

Performing the integration gives

(14)   \begin{equation*} S = 2 \left[ c_0 y - \lambda \frac{y^2}{2} \right]_0^s = 2 s \left( c_0 - \frac{\lambda s}{2} \right) = s \left( 2 c_0 - \lambda s \right) \end{equation*}

or because c_t = c_0 - \lambda s, then

(15)   \begin{equation*} S = s \left( c_0 + c_0 - \lambda s \right) = s \left( c_0 + c_t \right) = 2s \left( \dfrac{c_0 + c_t}{2} \right) = b \left( \dfrac{c_0 + c_t}{2} \right) \end{equation*}

which is the combined area of two trapezoids. For a wing with a 2:1 taper, then c_t = c_0/2 so

(16)   \begin{equation*} S = b \left( \dfrac{c_0 + \dfrac{c_0}{2} }{2} \right) = \dfrac{3}{4} b \, c_0 \end{equation*}

For the SMC, then

(17)   \begin{equation*} \overline{c} = \dfrac{\displaystyle{\int_{-s}^{s} c dy}}{b} = \dfrac{S}{b} =\dfrac{\displaystyle{\int_{0}^{s} c dy}}{s} \end{equation*}

Therefore,

(18)   \begin{equation*} \overline{c} = s \left( \dfrac{c_0 + c_t}{2 s} \right) = \left( \dfrac{c_0 + c_t}{2} \right) \end{equation*}

which is the average of the root and tip chords. For a 2:1 taper where \lambda = c_0/2s, then

(19)   \begin{equation*} \overline{c} = \left( \dfrac{c_0 + \dfrac{c_0}{2} }{2} \right) = \dfrac{3}{4} \, c_0 \end{equation*}

as shown in the figure below. Notice that

(20)   \begin{equation*} \overline{c} \, b = \dfrac{3}{4} c_0 \, b = S \end{equation*}

confirming the wing with the standard mean chord has the same area as the linearly tapered wing.

Illustration showing the interpretation of the standard mean chord (SMC) and the mean aerodynamic chord (MAC) for a linearly tapered wing.

For the MAC, which takes more work to determine, then

(21)   \begin{equation*} \overline{\overline{c}} = \frac{ \displaystyle{\int_{-s}^{s} c^2 \, dy}}{S} = \frac{ 2 \displaystyle{\int_{0}^{s} \left( c_0 - \lambda y \right)^2 \, dy}}{S} = \dfrac{ \displaystyle{ 2 \left[ c_0^2 y - c_0 \lambda y^2 +\dfrac{\lambda^2 y^3}{3} \right]_0^s }}{S} \end{equation*}

After integration, then

(22)   \begin{equation*} \overline{\overline{c}} = \dfrac{2 \left( c_0^2 - c_0 \lambda s + \dfrac{\lambda^2 s^2}{3} \right)}{\left( 2 c_0 - \lambda s \right)} \end{equation*}

Again, take the wing planform with a 2:1 taper ratio where \lambda = c_0/2s, then

(23)   \begin{equation*} \overline{\overline{c}} = \dfrac{2 \left( c_0^2 - c_0 \left( \dfrac{c_0}{2 s} \right) s + \left( \dfrac{c_0}{2 s} \right)^2 \dfrac{s^2}{3} \right)}{\left( 2 c_0 - \left( \dfrac{c_0}{2 s} \right) s \right)} = \dfrac{2 c_0^2 - c_0^2 + \dfrac{c_0^2}{6}} {\dfrac{3}{2} c_0} = \dfrac{7}{9} c_0 \end{equation*}

Worked Example #2 – Finding wing parameters

A wing has an aspect ratio of 12. According to the equation below, the wing’s chord varies smoothly and continuously outward from the aircraft’s centerline. Calculate the span and planform area of this wing.

    \[ c(y) = 1 - 0.32 \left( \frac{y}{b} \right)~~\mbox{in units of meters} \]

The planform area of the wing, S, is

    \[ S = 2 \int_{0}^{s} c \, dy \]

The chord is

    \[ c(y) = 1 - 0.32 \left( \frac{y}{b} \right) = 1 - 0.32 \left( \frac{y}{2s} \right) = 1 - 0.16 \left( \frac{y}{s} \right) \]

and substituting for the chord distribution, c(y), gives

    \[ S = 2 \int_{0}^{s} c \, dy = 2 \int_{0}^{s} \left( 1.0 - 0.16 \left( \frac{y}{s} \right) \right) dy \]

so that

    \[ S = 2 \left[ 1.0 y - \frac{0.08}{s} y^2 \right]_0^s = 2(s - 0.08s) = 1.84s \]

noting that b = 2 s. The aspect ratio, A\!R,  is

    \[ A\!R = \frac{b^2}{S} = \frac{4s^2}{S} = \frac{4s^2}{1.84s} = 2.174 s \]

Therefore, solving for the span, b, for an aspect ratio of 12 gives

    \[ b = 2s = 2 \left( \frac{A\!R}{2.174} \right) = \frac{2 \times 12}{2.174} = 11.04~\mbox{m} \]

The wing’s planform area is

    \[ S = 1.84 s = 1.84/b/2= 1.84 \times 5.52= 10.157~\mbox{m$^2$} \]

Summary & Closure

The geometrical parameters that define the shape of a wing include its span, chord distribution, aspect ratio, washout or another type of twist, and airfoil section shape. Wings may also have winglets, which help reduce overall wing drag. The overarching design requirement is to engineer the wing for good aerodynamic efficiency in terms of its lift-to-drag ratio at normal flight conditions, as well as off-design conditions such as stall. The aerodynamics, however, also need to be balanced against the structural design of the wing in terms of its strength and stiffness, amongst other requirements such as flutter avoidance. Using winglets in increasingly innovative forms has led to significant drag reductions on wings that can save substantial amounts of fuel, especially for an airliner.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • Do some research to determine what types of powered airplanes typically have the highest aspect ratio wings.
  • Consider some non-engineering reasons for a large commercial airplane that may limit the wing span.
  • Why do all sailplanes have high aspect ratio wings? Explain carefully.
  • Why do many fighter jet airplanes use sweptback wings with anhedral? Are there any commercial airliners that use wings with anhedral?
  • Why might an airplane use a forward-swept wing rather than an aft-swept wing? Have there been any aircraft built with a forward-swept wing?
  • Have any airplanes successfully flown with a different left versus right wing?
  • What might be the relative advantages of a conventional winglet versus a blended winglet?

Other Useful Online Resources

For more in-depth information on wings and wing shapes:

License

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Introduction to Aerospace Flight Vehicles Copyright © 2022, 2023, 2024 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n20