One of the most critical parts of an airplane is, obviously, the wings. The wings primarily create the lift to overcome the aircraft’s weight. However, they also have movable surfaces, such as ailerons, flaps, spoilers, etc., to develop additional aerodynamic forces to control the aircraft during its flight. Wings are geometrically defined in terms of their span, planform, twist, and cross-section (i.e., airfoil section shape or profile shape). The shape of a wing is engineered to give good aerodynamic efficiency in terms of 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.
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.
- Know about the key 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.
Geometric Definition of a Wing
Wing Span and Semi-Span
Wing Chord and Planform
The wing’s 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 . On many airplanes, the chord changes along the wing’s span, i.e., c = c(y) in the above figure, mainly for aerodynamic reasons. A primary aerodynamic goal is the minimization of profile and induced drag, i.e., the maximization of 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 is measured from the longitudinal centerline of the aircraft (i.e., not the wing root), as shown in the figure below, then the wing chord can be expressed as
where at the centerline of the aircraft and at the wing tip.
Wings are often linearly tapered in planform, for good engineering reasons, with different values of the root chord and the tip chord . In this case, the linear taper ratio of the wing is defined as
For a linear form of wing taper, as shown in the figure above, the chord distribution along the wing will be
The taper ratio may also be used in cases where the wing is not exactly linearly tapered as a means of quantifying 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. The use of 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 also minimizes structural loads and weight. A further aerodynamic advantage may be gained with the option of 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.
Sweepback of a Wing
Aerodynamically, the component of the flight Mach number that is perpendicular to the leading edge of the wing primarily affects the lift and drag. 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, but there is a significant structural weight penalty with such “swing-wing” designs.
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., or it may be alternatively defined by the leading edge and trailing edge angles, i.e., by the values and , 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
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 being used over the blade tip region to keep the blade 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 for zero lift, which manifests similarly to what would be obtained by changing the pitch angle of the wing.
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.
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. 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. Out toward the wing tips, where the bending moments and structural stresses are lower, the designer can use much thinner airfoils better suited to high-speed flight and transonic flow conditions. 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.
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 different 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
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. For an airliner, good stability about all three flight axes is desirable so that the passengers experience a smooth and comfortable ride, especially through turbulence.
A downward angle is called anhedral and is somewhat less common to find on airplanes without wing sweepback 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 lift. Wing anhedral may also be used in this case, a good example being on the C-5 Galaxy military transport aircraft, as shown in the image below. While stability is generally good, too much stability can make the aircraft less agile and maneuverable, and overall handling qualities can suffer. Sweepback can also introduce certain undesirable flight dynamic characteristics, which can be offset using anhedral.
Calculation of Wing Area
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
Calculation of Wing Aspect Ratio
The aspect ratio of a wing is important in aerodynamic analysis because a wing with higher aspect ratio generally is more aerodynamically efficient because it will have lower values of drag.
which is just the ratio of the wing span to the chord. Therefore, a wing with a higher span and a narrower chord will, in general, have a higher aspect ratio, so numerical value of the aspect ratio becomes a measure of the slenderness of the wing.
The chord for an elliptical wing planform shape can be expressed as
where is the root (centerline) chord. The area of the wing is
and substituting for the chord gives
This is a standard integral so
Therefore, the area of this wing is
and the aspect ratio is
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 shown in the photograph below is exemplary of 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.
It will be apparent then that the SMC is the average chord of an equivalent rectangular wing with the same area and span. For various reasons, including the fact that the SMC is purely a geometric quantity, the SMC is rarely used in practical aerodynamics.
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.
Linearly Tapered Wing
and the aspect ratio is
For the SMC then
and after integration and some algebra gives
For the MAC then
and after integration and some algebra gives
Calculate the span and area of this wing.
The area of the wing is
The chord is
and substituting for the chord distribution gives
noting that . The aspect ratio is
Therefore, solving for the span for an aspect ratio of 12 gives
The wing area is
Summary & Closure
The geometrical parameters that define the shape of a wing include its span, chord distribution, aspect ratio, washout or other 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. 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. Using winglets in increasingly innovative forms has led to significant drag reductions that can save significant amounts of fuel, especially for an airliner.
- Do some research to determine what types of powered airplanes typically have the highest aspect ratio wings.
- For a large commercial airplane, think about some of the non-engineering reasons 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 been flown successfully with a different left wing versus a right wing?
- What might be the relative advantages of a conventional winglet versus a blended winglet?