34 Wing Shapes & Nomenclature

Introduction

An airplane’s wings are among its most critical components. Their primary role is to generate the aerodynamic lift needed to balance the aircraft’s weight. However, they also support movable control surfaces, including ailerons, flaps, spoilers, and trim tabs, which create additional aerodynamic forces that enable the pilot to stabilize and maneuver the aircraft throughout its flight. The photograph in Figure 1 shows the wing of an airliner, a striking example of how aerodynamic shaping, structural design, and mechanical functionality must all come together in a single component. Its sweep, taper, control surfaces, and winglet are not arbitrary features; each reflects a specific design compromise between lift, drag, weight, strength, stability, and operational requirements.

The wing of an airplane is a masterful piece of engineering. This wing also features a blended winglet to reduce drag further.

Wings are geometrically defined in terms of their span or “wingspan” (distance from wing tip to wing tip), planform (their shape in outline looking down on the wing from above), twist (pitch angle) distribution, and cross-section (i.e., airfoil section shape or profile shape). The shape of a wing must be engineered to achieve high aerodynamic efficiency in producing lift while minimizing drag, thereby maximizing the lift-to-drag ratio, a fundamental goal in aerodynamic design. However, other aerodynamic requirements will always influence the wing design, including its performance at low speeds, at high angles of attack, and in stalling conditions.

In addition to the preceding aerodynamic requirements, the wing structure must be sufficiently strong and stiff to carry aerodynamic, weight, inertial, landing-gear, engine, and maneuver loads while keeping structural weight as low as practical. Therefore, the wing must be designed to:

  • Limit structural loads and deformations caused by aerodynamic forces and moments.
  • Avoid adverse bending, twisting, aeroelastic effects, and flutter.
  • Carry concentrated loads from the landing gear, engines, stores, or other attached components.

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 requires a significantly larger chord and a thicker cross-section near the fuselage than at the wing tips to achieve the required structural strength and stiffness. While airplane wings are enormously strong, they are still limited in what they can achieve aerodynamically and structurally.

Learning Objectives

  • Know the critical geometric parameters used to define the shape of a wing.
  • Be able to calculate the wing area and aspect ratio of an arbitrary wing planform.
  • Understand the significance and use of mean wing chords.
  • Gain an appreciation of winglets and how they work.

Geometric Definition of a Wing

Engineers use various geometric parameters to characterize wing shapes. 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 wash-out, the planform taper ratio, the dihedral or anhedral, and the wing section thickness-to-chord ratio.

Wing Span & Semi-Span

The length of the wing, called the 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 Figure 2. Sometimes, the semi-span is used to define the wing in engineering analysis, denoted by s and equal to b/2. The usual assumption is that the right and left wing panels are geometrically and aerodynamically mirror images of each other. Notice that the symbol \rm C_{\!\!\!\rm \Large L} in Figure 2 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. It is essential to learn their definitions.

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 denoted by {c}. On many airplanes, the chord changes along the wing’s span, i.e., c = c(y) in Figures 2 and 3, mainly for aerodynamic reasons. A primary aerodynamic goal for the wing is to minimize drag for a given amount of lift, i.e., to maximize the lift-to-drag ratio, which is an aerodynamic efficiency metric for a wing.

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. Suppose {y} is measured from the longitudinal centerline of the aircraft (i.e., not the wing root), as shown in Figure 3. In that case, the local value of the wing chord can be expressed as c = {\rm function} (y), 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. Many airplanes have been designed with linearly tapered wing shapes. They offer a good compromise between aerodynamic requirements, structural efficiency, and weight, and they can help tailor the spanwise lift distribution when used together with appropriate wing twist and airfoil-section choices.

In this case, the linear taper ratio of the wing can be defined as

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

For a linearly tapered wing, as shown in Figure 3, the chord distribution along the wing will be

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

The taper ratio can also quantify the average taper of the wing planform when the wing is not precisely linearly tapered.

Wings may not only taper in planform but also in thickness or, indeed, as combinations of taper and thickness, as shown in Figure 4. Using both taper and thickness provides considerable engineering latitude to tailor the wing’s shape to achieve a specified level of aerodynamic performance while minimizing structural loads and weight. A further aerodynamic advantage may be gained by using different airfoil sections along the span: a relatively thin airfoil at the wing tip for low drag, where structural loads are lower, and thicker airfoils further inboard, where structural load resistance is a more important consideration.

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 problems of adverse stall characteristics and a tendency to spin, common issues in the first generation of swept-wing jet aircraft. The wing’s sweepback promotes spanwise flow, making the wingtips more likely to stall first and reducing the effectiveness of the ailerons. However, besides increasing wing weight and roll inertia, 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 Figure 5; however, some lower-performance aircraft may have no sweepback. The primary aerodynamic purpose of sweepback on a wing is to delay the onset of compressibility effects and the buildup of wave drag at higher flight Mach numbers, and/or to reduce drag at a given Mach number, thereby decreasing the propulsive thrust and fuel required for flight.

Using a sweepback angle on a wing can help mitigate the onset of compressibility effects and the increase in supersonic drag caused by shock waves.

Aerodynamically, the component of the flight Mach number perpendicular to the wing’s leading edge primarily affects lift and drag, assuming that the component parallel to the leading edge does not, a principle often referred to as the independence principle. Therefore, for aerodynamic analysis, the freestream Mach number, that is, the aircraft’s flight Mach number, 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, forward-swept wings are prone to aeroelastic divergence because aerodynamic loads tend to twist the wing in a direction that increases the local angle of attack unless the structure is sufficiently stiff or aeroelastically tailored. For this reason, forward-swept wings are rarely used.

Aircraft designed for sustained supersonic flight inevitably have much higher sweep angles than subsonic aircraft; the optimal planform for supersonic flight approaches a classic “Delta” shape. However, sweepback can have additional effects on the aerodynamics of the wing and the airplane, including adverse stall characteristics and low-speed handling; therefore, as little sweepback as possible is usually used. Sweepback also tends to increase the airplane’s lateral stability. Some aircraft, such as the B-1 bomber, may have variable sweepback to optimize flight aerodynamics. Still, there is a significant structural weight penalty associated with such “swing-wing” designs, at the expense of useful load, i.e., fuel and/or 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 for the inboard wing panel (typically the smaller angle) and another for the outboard wing panel (typically the larger angle).

Wing Twist

Wings may also be slightly twisted in their angle relative to the flow along their span; see Figure 6. One purpose of wing twist is to help achieve the desired distribution of aerodynamic forces over the span. In practice, most wings are twisted in some form, often subtly. It is well established in aerodynamic theory and practice that the spanwise distribution of lift is crucial 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 to tailor the wing lift distribution to obtain the desired aerodynamic effects.

Most wings are twisted from root to tip to improve aerodynamic efficiency. Note: For clarity, the twist angles in these figures are exaggerated for illustration purposes.

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 (i.e., nose down) toward the wing tip. This effect is called “washing out” the wing twist, and is therefore referred to as wash-out. Typical wash-out values on a wing are between 0 and 10 degrees nose-down; anything more than this is somewhat unusual. The wash-out may vary in value along the wingspan, i.e., a more significant wash-out at the wing tip than at its root.

Although very unusual for an airplane, it is called a wash-in if a wing is twisted nose-up by increasing its twist from the root to the tip along its span. One notable example was the X-29, which featured a wash-in twist in its forward-swept wings to mitigate adverse stall characteristics. Interestingly, some helicopter blades use wash-out and wash-in techniques, with the wash-in twist component applied at the blade tip to prevent 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 using geometric twist to change the local pitch angle of the wing relative to the flow.

Airfoil Sections

Wings can use various types and distributions of airfoil sections to suit the aircraft’s application. By cutting a slice out of an airplane wing and viewing it from the side, the wing’s cross-section is obtained, which is usually referred to as an airfoil section or profile. An example is shown in Figure 7. The shape of airfoil sections can be described using camber, thickness, and nose radius as primary geometric parameters. Outside the U.S., wing sections are typically 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 minimal thickness, inspired by birds’ wings, to sophisticated shapes with camber and thickness that provide high lift and low drag. Airfoils used on subsonic airplanes are typically relatively thick and feature camber, as illustrated in Figure 8. Meanwhile, the airfoils used in high-speed or supersonic aircraft are thinner, have a small leading-edge radius, and exhibit slight camber. Notice that “thickness” usually means the section’s maximum thickness-to-chord ratio. Hence, 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 indicates an airfoil with a thickness of 12% of the chord.

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 blended progressively to create an overall wing design superior to what could be achieved with a single airfoil. This latter approach is often necessary with large commercial aircraft. As shown in Figure 9, the need for significant thickness to carry bending and shear loads at the wing’s root makes the airfoil design for low drag at higher flight Mach numbers particularly challenging, especially when the wing operates in transonic flow. The designer can utilize thinner airfoils, better suited to high-speed flight and transonic flow conditions, at the wing tips, where bending moments and structural stresses are lower. Even on low-performance airplanes, there can be significant aerodynamic and performance advantages to using different airfoils at the wing root compared to the wing tips.

The wing must carry considerable bending and shear loads and requires a thicker root section to prevent excessive structural bending displacements.

Airfoil section shapes may also introduce an aerodynamic twist along the wing span. Different airfoils can have different zero-lift angles of attack, so airfoils with different camberline shapes can be used to aerodynamically twist the wing. The effects are typically combined with a geometric twist to achieve the desired spanwise lift distribution, thereby meeting aerodynamic performance and other objectives.

It is also known that using wing twist is particularly helpful in controlling the stall developments on the wing, especially in preventing the wing tips from stalling. Therefore, tailoring the wing twist distribution and, to some extent, the airfoil sections allows the designer some latitude in meeting the airplane’s low-speed handling requirements. It is not unusual for newly designed airplanes to have the airfoils over the outer wing panels changed[1] after their first flights to meet the stall and handling qualities requirements needed for certification. However, other methods may be required to control the airplane’s stall characteristics.

Dihedral & Anhedral

The dihedral angle is the upward angle the wing panels make relative to the aircraft’s reference axis, as shown in Figure 10. Its primary purpose is to improve the aircraft’s lateral (roll) stability; typically, only a few degrees are needed to achieve a significant improvement. The horizontal tail may also exhibit some dihedral, particularly on larger aircraft, which contributes somewhat to lateral stability. Some 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.

The photograph in Figure 11 illustrates a dihedral angle on a Boeing 737. Notice that the main wing and horizontal tail both have a notable amount of dihedral. Good lateral stability is desirable for most aircraft, especially airliners, as it enables passengers to experience a smooth and comfortable ride 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 tailplane is apparent on this Boeing 737-800. This airplane also features a dual-winglet design with upper and lower sections.

A downward wing angle is called anhedral and is somewhat less common on airplanes without wing sweepback or a high-wing design, as it reduces roll stability. However, airplanes with swept wings may employ anhedral to offset the increase in roll stability associated with sweepback. Airplanes with high-mounted wings often exhibit increased lateral stability because the fuselage and wing geometry produce a stabilizing rolling moment in sideslip. The effect is sometimes described qualitatively as “pendular stability,” although it does not arise simply because the center of gravity lies below the wing. The restoring tendency depends primarily on the aerodynamic sideforce and rolling-moment response of the complete aircraft. In this case, anhedral on the wing may be necessary to prevent lateral stability from being too strong, which would make the aircraft difficult to turn and maneuver.

Gravitational pendular stability can be mitigated by incorporating anhedral into the wing design.

An example of a wing with anhedral is found on the C-5 Galaxy military transport aircraft, as shown in Figure 13. The C-5 uses a high-mounted wing to help keep the fuselage and loading deck close to the ground while providing clearance for the engines and landing gear. The anhedral angle helps offset the strong lateral stability associated with the high-mounted swept wing. While lateral stability is desirable, excessive lateral stability can reduce maneuverability and agility, resulting in deteriorating overall handling qualities; i.e., the aircraft becomes sluggish in response to control inputs.

The military C-5 Galaxy transport aircraft features anhedral wings to counter the inherent pendular stability of its 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 the wing chord along the span from one wing tip to the other, i.e.,

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

where c = c(y), as shown in Figure 14.

Calculating the wing reference area requires 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

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

A dilemma of sorts arises here. Does the wing area refer to the projected reference area of the wing, or to the actual surface area exposed to the airflow? In the above equation, the projected planform area has been assumed, and the value of the area, S, is called the planform area or wing reference area. It is important not to confuse the wing area (capital “S“) with the semi-span of the wing (lowercase “s“). However, there can be circumstances when the actual surface area exposed to the flow needs to be known, called the wetted area. The wetted area is not the same as the projected planform area and must be defined and calculated separately. Therefore, it is essential to verify the actual definition(s) of wing area used in various engineering analyses.

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.,

(5)   \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 crucial in aerodynamic analysis because a higher aspect ratio generally reduces the induced drag coefficient for a given lift coefficient.

However, typical values range from 5 to 10 for a small general aviation aircraft, from 9 to 15 for a commercial transport aircraft, and from 30 or more for a glider or sailplane. As a point of reference, the Rutan Voyager airplane in 1986 flew around the world without refueling,[2] 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

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

where b = 2 s and so

(7)   \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 wingspan divided by 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, so the aspect ratio must be calculated by using the wing chord distribution and resulting area, i.e., using the formula

(8)   \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 uses the wetted wing area rather than the reference wing area; however, this is rare.

Check Your Understanding #1 – Finding the planform area & aspect ratio of a wing

The wing of a Supermarine Spitfire has an approximately elliptical wing planform with a centerline, or maximum, chord c_0 of 100 inches, or 2.54 m, and a wingspan b of 445 inches, or 11.30 m. Calculate the planform area and aspect ratio of its wing.

Show solution/hide solution.

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 chord and s = b/2 is the semi-span. 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

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

In USC units, the planform area of this wing is

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

Equivalently, in SI units,

    \[ S = \frac{\pi c_0 s}{2} = \frac{ \pi \times 2.54 \times (11.30/2)}{2} = 22.54~\mbox{m${^{2}}$} \]

The aspect ratio, A\!R, is dimensionless, so either unit system gives the same value, i.e.,

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

or equivalently,

    \[ A\!R = \frac{11.30^2}{22.54} = 5.67 \]

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 Figure 15; this example also maintains a constant wing area. 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 wing’s aspect ratio increases, it becomes more challenging to design it to meet stiffness and strength requirements without increasing its weight. Consequently, a longer wing is inevitably more flexible, unless some additional structure is used to stiffen it.

A higher-aspect-ratio wing has a greater span and a narrower chord, producing lower induced drag because the tip vortices are farther from most of the wing.

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 substantial aerodynamic advantages to using a wing with the highest possible aspect ratio, provided that all structural, weight, flutter, and other airplane requirements are met.

Airplanes come in many shapes and sizes, utilizing wings with varying aspect ratios, as illustrated in Figure 16. The wing design is inevitably a compromise that depends on what the aircraft does, in terms of performance, weight, and cost. General aviation aircraft, for example, tend to have relatively modest aspect ratios as a compromise between adequate aerodynamic performance and low structural weight.

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

Sailplanes, which are high-performance gliders, typically have wings with a very high aspect ratio compared to those of powered aircraft. Thus, they can achieve high lift-to-drag ratios and glide long distances by design. The DG-800 shown in Figure 17 exemplifies a modern sailplane with an aspect ratio of just over 27. These types of sailplanes can glide more than 50 miles (in still air) from an altitude of only 5,000 feet.

The DG-800 is an exemplary contemporary sailplane, featuring an aspect ratio of 27 and a lift-to-drag ratio exceeding 50.

Calculation of Mean Chords

Other derived geometric parameters relevant to wings include the standard mean chord (SMC) and the aerodynamic mean chord (AMC), also known as the mean aerodynamic chord (MAC). A mean chord provides a standardized length scale that can be applied across different wing geometries, shapes, and sizes. Mean chords are used in aerodynamic analyses as a reference length, such as to calculate pitching moment coefficients. For example, in aerodynamics, the moment coefficient about some point a can be expressed as

(9)   \begin{equation*} C_{M_{a}} = \displaystyle{ \frac{M_a}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 \, S \, c_{\rm ref}} } \end{equation*}

where  c_{\rm ref} is a reference length. For a wing, c_{\rm ref} can be defined as either the SMC or the MAC, with the MAC preferred. Mean wing chords are also used as reference lengths in other aeronautical disciplines, where certain lengths and length scales may be non-dimensionalized by using the value of the mean wing chord.

Standard Mean Chord

The SMC is a purely geometric quantity defined as the wing reference area divided by the wing span. However, its interpretation as the chord of an equivalent rectangular wing producing the same lift introduces a hidden aerodynamic assumption, namely that the sectional lift coefficient is uniform across the span, i.e., C_l(y)=C_L.

Definition

The standard mean chord (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}}{b} = \frac{ \displaystyle{\int_{0}^{s} c \, dy}}{s} \end{equation*}

where b = 2\,s. It will be apparent from Figure 18 that the SMC is the chord of an equivalent rectangular wing with the same area and span that produces the same lift force. It should be noted that the wing shape is not restricted to a linearly tapered planform, as shown in the figure, and Eq. 10 is valid for any wing planform.

Definition of the standard mean chord (SMC). Although a geometric definition, its derivation depends on certain aerodynamic assumptions.

The work required to obtain the SMC value depends on the planform’s complexity, ranging from simple rectangular to more complex tapered designs. It may also involve regions with different tapers and geometrically distinct wing panels, possibly with a sweep angle. In these latter cases, the integral can be evaluated by summing the contributions from each part of the wing.

Derivation

The derivation of the equation for SMC proceeds by considering the elemental lift on a two-dimensional strip of the wing of area c \, dy. Assuming that both wing panels have mirror geometric and aerodynamic symmetry, the derivation can be written over the full span, or equivalently, over one half-span and then doubled. The lift on the element is given by the usual formula, i.e.,

(11)   \begin{equation*} dL = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \, C_l \, c \, dy \end{equation*}

where c = c (y) is the local chord and C_l = C_l (y) is the local sectional lift coefficient. Written over the full span, the total lift on the wing is then

(12)   \begin{equation*} L = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \int_{-s}^{s} \, C_l \, c \, dy \end{equation*}

The total lift on the equivalent rectangular wing can also be written as

(13)   \begin{equation*} { L = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \, C_L \, \overline{c} \, b = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \, C_L \, S } \end{equation*}

where S = \overline{c} \, b, i.e., the areas of the actual wing and the equivalent rectangular wing are equal, and the equivalent rectangular wing is assigned the same total lift coefficient C_L. Remember that s is the semi-span of the wing, and S is the wing area. Equating Eqs. 12 and 13 gives

(14)   \begin{equation*} C_L \, \overline{c} \, b = \int_{-s}^{s} \, C_l \, c \, dy \end{equation*}

Making the aerodynamic assumption that C_l(y) = C_L, i.e., the local lift coefficient is constant at all sections over the wing span and equal to the total lift coefficient, then

(15)   \begin{equation*} \rm SMC = \overline{c} = \frac{\displaystyle{\int_{-s}^{s} c \, dy}}{b} = \frac{\displaystyle{\int_{0}^{s} c \, dy}}{s} \end{equation*}

which was previously defined in Eq. 10.

Therefore, although the SMC definition is ultimately geometric, it assumes that the wing comprises two-dimensional airfoils with a uniform lift coefficient across the span. However, despite arguments about the validity of the aerodynamic assumptions, it does not matter, as the SMC is simply a reference standard. To this end, the SMC is useful for many purposes because it enables comparison of wings and their aerodynamic characteristics on a common basis. For this reason, all engineers subscribe to the same definition of the SMC.

Because the lift per unit span is

(16)   \begin{equation*} \frac{dL}{dy} = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \, C_l \, c \end{equation*}

then it will be apparent that, under this assumption, the local lift force per unit span (as opposed to the lift coefficient) is proportional to the chord length. Other than for the region near the wing tip, where there are three-dimensional effects, including those from the wing tip vortex, Eq. 16 provides a first approximation to the spanwise lift distribution on a wing, as shown in Figure 19.

Planform effects on the span loading on a wing are based on two-dimensional theory with a constant lift coefficient.

Mean Aerodynamic Chord

The mean aerodynamic chord (MAC), also known as the aerodynamic mean chord (AMC), is a term frequently encountered in aerodynamics. The MAC can be easily confused with the SMC, but they are distinct quantities and generally have different numerical values. Like the SMC, the MAC is a purely geometric quantity. However, its interpretation as the chord of an equivalent rectangular wing having the same lift and pitching-moment behavior introduces aerodynamic assumptions, including a uniform sectional lift coefficient across the span, i.e., C_l(y)=C_L, and a constant aerodynamic-center location as a fraction of the local chord.

Definition

The mean aerodynamic chord (MAC) is the chord of an equivalent rectangular wing that has the same reference area and gives the same pitching-moment reference behavior under the simplifying assumptions used in the derivation below. The MAC is defined as

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

This is another type of “average” aerodynamic chord definition for the wing. It is more frequently used because, in this definition, the moment reference length is chosen to preserve the relevant chordwise moment contribution of the wing planform. Again, the work required to determine the MAC depends on the complexity of the wing planform.

Derivation

The derivation of the MAC equation follows the same steps as that for the SMC. In this case, however, a pitching moment must be calculated. In Figure 20, which illustrates an unswept wing for simplicity, the locus of the aerodynamic centers of the wing sections lies along a line parallel to the {y}-axis. Notice that for a thin airfoil in an incompressible flow, the aerodynamic center is at the quarter-chord point, i.e., x_{\rm ac} = c /4; the aerodynamic center is a fixed point independent of C_l. Assuming that both wing panels have mirror geometric and aerodynamic symmetry, the derivation can be written over the full span, or equivalently, over one half-span and then doubled. For each two-dimensional strip, taking pitching moments about the spanwise {y} axis that runs through the apex of the wing, then

(18)   \begin{equation*} dM_y = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \, C_l \, x_{\rm ac} \, c \, dy \end{equation*}

The principle of the mean aerodynamic chord (MAC) is to create an equivalent rectangular wing of the same area with the same lift and pitching moment.

The total pitching moment produced by the entire wing is then

(19)   \begin{equation*} M_y = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \int_{-s}^{s} \, C_l \, x_{\rm ac} \, c \, dy = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \int_{-s}^{s} \, C_l \, \left( \dfrac{x_{\rm ac}}{c} \right) c^2 \, dy \end{equation*}

The total moment on the equivalent rectangular wing can also be written as

(20)   \begin{equation*} M_y = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \, C_L \, S \, \left( \dfrac{x_{\rm ac}}{c} \right) \overline{\overline{c}} \end{equation*}

where the aerodynamic center location has been written as the same fraction of the local chord for both the actual wing sections and the equivalent rectangular wing. For example, for a thin airfoil in incompressible flow,

(21)   \begin{equation*} \dfrac{x_{\rm ac}}{c} = \frac{1}{4} \end{equation*}

Equating Eqs. 19 and 20 gives

(22)   \begin{equation*} \int_{-s}^{s} \, C_l \, \left( \dfrac{x_{\rm ac}}{c} \right) c^2 \, dy = C_L \, S \, \left( \dfrac{x_{\rm ac}}{c} \right) \overline{\overline{c}} \end{equation*}

Using Eq. 21 and making the same aerodynamic assumption as previously used for the SMC, namely that {C_l(y)} = {C_L}, then

(23)   \begin{equation*} \int_{-s}^{s} c^2 \, dy = S \, \overline{\overline{c}} \end{equation*}

so that

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

which was previously defined in Eq. 17. It will also be apparent that the lift produced on both the actual wing and the equivalent rectangular wing based on the standard mean chord is equal because

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

In general, the MAC is written as

(26)   \begin{equation*} {\rm MAC} = \overline{\overline{c}} = \frac{\displaystyle{\int_{-s}^{s} c^2 \, dy}}{\displaystyle{\int_{-s}^{s} c \, dy}} \end{equation*}

which is based solely on the wing geometry, despite the aerodynamic assumptions used in its derivation. Again, as with the SMC, the physical justification of the aerodynamic assumptions is irrelevant, since the MAC is simply a standardized definition that everyone subscribes to.

As a corollary to the preceding, the spanwise location of the resultant lift on one wing panel can also be determined by taking moments about the {x} axis. This location is often described as the spanwise center of lift of the wing panel. In this case, it is clearer to work with one half of the wing, so

(27)   \begin{equation*} M_x = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \int_{0}^{s} \, C_l \, y \, c \, dy \end{equation*}

The total moment for the same half-wing panel can also be written as

(28)   \begin{equation*} M_x = \frac{1}{2} \varrho_{\infty} V_{\infty}^2 \, C_L \, y_L \int_{0}^{s} c \, dy \end{equation*}

Therefore, using Eqs. 27 and 28, and with the same assumption that {C_l(y)} = {C_L}, gives

(29)   \begin{equation*} y_L = \dfrac{\displaystyle{\int_{0}^{s} y \, c \, dy}}{\displaystyle{\int_{0}^{s} c \, dy}} \end{equation*}

or, in non-dimensional form,

(30)   \begin{equation*} \dfrac{y_L}{s} = \dfrac{\displaystyle{\int_{0}^{s} y \, c \, dy}}{\displaystyle{s \int_{0}^{s} c \, dy}} \end{equation*}

The value of {y_L} is the effective spanwise location of the resultant lift on the wing panel, assuming a uniform sectional lift coefficient. Notice that in the case of a rectangular wing with {c} = constant, then

(31)   \begin{equation*} \dfrac{y_L}{s} = \dfrac{\displaystyle{c \int_{0}^{s} y \, dy}}{\displaystyle{s \int_{0}^{s} c \, dy}} = \dfrac{c s^2/2}{c s^2} = \frac{1}{2} \end{equation*}

which is exactly at the mid-semi-span location, as expected for a uniformly spanwise-loaded wing panel.

Notice that, in principle, the SMC and the MAC can be determined for any wing, including horizontal and vertical stabilizers. However, in all cases, the evaluation will involve spanwise integration, either analytically or numerically.[3]

The addition of a constant sweepback angle, {\Lambda}, on the wing, for example, will move the locus of the section aerodynamic-center locations farther aft relative to the reference moment axis, adding a term proportional to y \tan \Lambda to the chordwise moment arm in Eq. 19. The value of the MAC itself is still determined by the spanwise chord distribution and so is independent of sweep for a given planform. However, the chordwise location of the MAC, and hence the corresponding aerodynamic-center reference point for the wing, must be located separately from the wing geometry and will generally move aft as sweepback is increased.

It is important to remember that mean (average) wing chords are used as reference lengths in aerodynamics and other related aeronautical disciplines. For example, as mentioned previously, the aerodynamic pitching moment coefficient for a finite wing about some point a is defined as

(32)   \begin{equation*} C_{M_{a}} = \displaystyle{ \frac{M_a}{\frac{1}{2} \varrho_{\infty} V_{\infty}^2 \, S \, c_{\rm ref}} } \end{equation*}

where usually {c_{\rm ref} = \overline{\overline{c}}}, although in some cases {c_{\rm ref} = \overline{c}} may be used. It is important to note that the specific definition used in any application can vary.[4] In fields such as flight dynamics, the values of the quantified parameters are measured relative to a datum point or reference axis, such as the neutral point, center of gravity, etc., and are often quoted as a fraction of a mean chord.

SMC & MAC of a Linearly Tapered Wing

A linearly tapered wing is one of the most common wing planforms, as shown in Figure 21, which is also useful for instructional purposes. It has a chord distribution that can be expressed as

(33)   \begin{equation*} c(y) = c_0 - k y \end{equation*}

where k is a constant with units of chord per unit span. Notice that when y = 0 then {c = c_0}, the root chord, and when y = s then c = c_0 - k s = c_T, the tip chord. Therefore,

(34)   \begin{equation*} k = \frac{c_0 - c_T}{s} \end{equation*}

Geometry of a linearly tapered wing panel.

Analysis for the SMC

The area of this wing is given by

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

Performing the integration gives

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

Therefore,

(37)   \begin{equation*} \overline{c} = \dfrac{s \left( 2 c_0 - k s \right)}{2s} = c_0 - \dfrac{k s}{2} \end{equation*}

Furthermore, because c_T = c_0 - k s, then

(38)   \begin{equation*} S = s \left( c_0 + c_0 - k 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 sum of the areas of the two trapezoids. Therefore,

(39)   \begin{equation*} \overline{c} = \dfrac{b \left( \dfrac{c_0 + c_T}{2} \right)}{b} = \dfrac{c_0 + c_T}{2} \end{equation*}

which is simply the average chord over the span.

Specific Case

Consider a specific case where k = 0.5 \, c_0/s = c_0/(2s), so that c_T = c_0/2. This is a trapezoidal wing planform with a taper ratio of

(40)   \begin{equation*} \lambda = \frac{c_T}{c_0} = 0.5 \end{equation*}

The wing area is then

(41)   \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,

(42)   \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,

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

which is the average of the root and tip chords. For \lambda = 0.5, where k = c_0/(2s),

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

as shown in Figure 22.

Interpretation of standard mean chord (SMC) and mean aerodynamic chord (MAC) for a linearly tapered wing.

Notice that because

(45)   \begin{equation*} c(y) = c_0 - \left( \frac {c_0}{2s} \right) y \end{equation*}

the value of {y} when c = \overline{c} = 3 c_0/4 is

(46)   \begin{equation*} y (\overline{c} ) = \dfrac{2 s \left( c_0 - \dfrac{3}{4} \, c_0 \right) }{c_0} = \dfrac{s}{2} \end{equation*}

which is at the midpoint of the semi-span. For any uniformly linearly tapered wing panel, the SMC equals the local chord at the mid-semi-span.

Notice also that

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

confirming that the equivalent rectangular wing based on the standard mean chord has the same area as the linearly tapered wing.

Geometric Interpretation of the SMC

The value of the SMC can also be obtained geometrically or graphically, as shown in Figure 23 for a linearly tapered wing. This approach may provide insights not immediately evident from numerical calculations alone and can be helpful in some contexts. The process visually represents how chord lengths are distributed along the wing and graphically finds the average geometric chord.

Geometric interpretation of the standard mean chord (SMC).

Using the principles of analytic geometry, the spanwise location and value of the SMC are determined by finding the intersections of straight construction lines. Adding the root chord length, c_0, to the leading and trailing edges of the tip chord, as shown, gives two auxiliary points at the extended wing tip. Straight construction lines are then drawn between the corresponding root and tip auxiliary points. The intersections of these construction lines with the leading and trailing edges of the wing panel give the spanwise location and value of the SMC.

While a visual interpretation of the mean chord is useful, its proper value would typically be obtained analytically or numerically. Again, it should be remembered that for any uniformly linearly tapered wing panel, the SMC is always the value of the local chord at the mid-semi-span location.

Analysis for the MAC

The MAC for this linearly tapered wing panel, which takes some more work to determine, is given by

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

After integration, then

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

Because S = s \left( 2 c_0 - k s \right), this result can also be written as

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

Again, take the wing planform with a taper ratio of \lambda = 0.5, for which k = c_0/(2s), then

(51)   \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*}

Check Your Understanding #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 wingspan and planform area of this wing.

    \[ c(y) = \left[ 1 - 0.32 \left( \frac{y}{b} \right) \right]~\mbox{m} \]

Show solution/hide solution.

The planform area of the wing, S, is

    \[ S = 2 \int_{0}^{s} c \, dy \]

The chord distribution is

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

where the chord is in meters. 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 = 2s. The aspect ratio, A\!R, is

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

Therefore, solving for the semi-span, s, for an aspect ratio of 12 gives

    \[ s = \frac{1.84}{4} A\!R = \frac{1.84}{4} \times 12 = 5.52~\mbox{m} \]

and so the wingspan is

    \[ b = 2s = 2 \times 5.52 = 11.04~\mbox{m} \]

The wing’s planform area is

    \[ S = 1.84 s = 1.84 \left( \frac{b}{2} \right) = 1.84 \times 5.52 = 10.16~\mbox{m${^{2}}$} \]

Winglets

Winglets are geometric extensions of the wing tip, usually canted upward or downward from the main wing plane. Their primary geometric effect is to increase the wing’s effective aspect ratio without substantially increasing its span. Aerodynamically, they modify the wing-tip flow, move the dominant trailing vortex system away from the main wing, and reduce induced drag. The effects are often apparent in natural-flow visualization, as shown in Figure 24, where the visible vortex trails from the tip of the winglet.

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 around a wing reveals a vortex trailing from the tip of each winglet.

One approach to quantify this effect is to use an aspect ratio correction of the form

(52)   \begin{equation*} A\!R_{\rm eff} = A\!R \left( 1 + K_{\rm wl} \, \frac{h_{\rm wl}}{s} \right) \end{equation*}

where A\!R_{\rm eff} is the “effective” or corrected aspect ratio, h_{\rm wl} is the length or height of the winglet, as shown in Figure 25, and K_{\rm wl} is a coefficient that depends on the winglet design. For a classic Whitcomb-type winglet, a preliminary value of K_{\rm wl} = 1 is often used. While winglets increase the wetted area, thereby increasing skin friction and overall profile drag, they still result in a net reduction in total aircraft drag.

A winglet increases the effective aspect ratio of the wing with little or no increase in wing span, thereby reducing induced drag.

As shown in the photos in Figure 26, commercial airliners have employed various winglet designs. Today, the trend is toward using winglets with smooth chord variations in the wing-to-winglet region, further minimizing drag.

A variety of winglet designs have been employed in airplanes. Winglets help increase the wing’s effective aspect ratio and reduce the drag induced by the wing tip vortices.

More recent winglet variants are featured on aircraft such as the Boeing 737 MAX, as illustrated in Figure 27. They are designed to reduce induced drag further and enhance the aircraft’s flight efficiency and range. Increasing the length or height of a traditional upward-pointing winglet can increase its aerodynamic benefit, at least to a point, but it also increases structural weight and bending loads at the wing tip. Some designs use both upward- and downward-pointing winglet elements to obtain additional aerodynamic benefit while managing the geometric, structural, and weight penalties.

Boeing’s Advanced Technology Winglet is claimed to reduce the 737 MAX’s fuel consumption by up to 1%.

Summary & Closure

The geometrical parameters that define the shape of a wing include its span, chord distribution, aspect ratio, wash-out 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, including its lift-to-drag ratio at normal flight conditions and off-design conditions, such as stall. The aerodynamics, however, also need to be balanced against the wing’s structural design in terms of strength and stiffness, amongst other requirements such as avoiding flutter. The use of increasingly innovative winglet designs has led to significant drag reductions on wings, saving substantial fuel, particularly for airliners.

In airplane design, wing area and aspect ratio are essential in determining flight performance characteristics. The planform area, which represents the projected wing area, directly influences lift generation and, consequently, the aircraft’s payload capacity, maneuverability, and efficiency across various airspeeds. The aspect ratio, defined as wingspan squared divided by wing area, strongly affects lift efficiency and induced drag; higher aspect ratios generally reduce induced drag, although the final choice must also account for structural weight, aeroelasticity, compressibility effects, and the aircraft’s intended mission. These parameters are tailored to the aircraft’s intended mission, i.e., commercial airliners favor higher wing aspect ratios for efficiency, fighter jets favor lower ratios for high airspeed and agility, and sailplanes favor very high aspect ratios for superior soaring performance. Therefore, engineers must be able to calculate wing areas, aspect ratios, and mean chord lengths, which are essential quantities for characterizing, designing, and comparing aircraft performance.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • Research to determine the types of powered airplanes that typically have the highest aspect ratio wings.
  • Consider some non-engineering reasons why a large commercial airplane may have a limited wingspan.
  • Why do sailplanes have extremely high aspect ratio wings? Explain carefully.
  • Why do many fighter jet airplanes use sweptback wings with anhedral? Are there any commercial airliners that use anhedral wings?
  • Why might an airplane use a forward-swept wing rather than an aft-swept wing? Have any airplanes been built with a forward-swept wing?
  • Have any airplanes been successfully flown with different left and right wings?
  • 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:

  1. Usually with the addition of nose camber. ↵
  2. This officially sanctioned flight was the first successful aerial nonstop, non-refueled circumnavigation of the Earth. ↵
  3. The effort involved depends mainly on one's proficiency with geometry rather than aerodynamics. ↵
  4. The ability to reconcile different results for pitching moment coefficients often comes down to the realization that definitions and reference lengths can vary. ↵

License

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

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

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