2.4 Finite wing theory

Finite wing characteristics

Finite wing theory bridges the gap between the idealized infinite (2D) wing and the real wings found on aircraft. An infinite wing has no tips, so it never "leaks" pressure from bottom to top. A finite wing does have tips, and that pressure leakage changes everything: it reshapes the lift distribution, creates trailing vortices, and introduces a new type of drag called induced drag. Understanding these effects is central to practical wing design.

The planform shape, aspect ratio, taper ratio, and sweep angle all work together to determine a finite wing's aerodynamic performance.

Planform shape effects

The planform shape is the outline of the wing as seen from directly above.

• Elliptical planforms produce the most efficient lift distribution (minimum induced drag for a given lift), but they're difficult and expensive to manufacture because every rib has a different length.
• Rectangular planforms are the simplest to build. Every rib is identical. The trade-off is higher induced drag because the lift distribution is far from elliptical.
• Tapered planforms split the difference: they approximate an elliptical lift distribution reasonably well while remaining much easier to fabricate than a true ellipse.

Aspect ratio impact

Aspect ratio (AR) measures how long and slender a wing is:

$AR = b^2 / S$

where $b$ is the wingspan and $S$ is the wing planform area.

• Higher AR means the wingtips influence a smaller fraction of the total span, so tip vortices have less effect. This directly reduces induced drag.
• A high-AR wing produces a better lift-to-drag ratio, which is why gliders and long-range airliners use long, narrow wings.
• The structural cost is real, though. Long, slender wings experience larger bending moments at the root and can be susceptible to aeroelastic problems like flutter, where aerodynamic forces couple with structural flexibility to produce dangerous oscillations.

Taper ratio considerations

Taper ratio compares the tip chord to the root chord:

$\lambda = c_{tip} / c_{root}$

• A taper ratio of 1.0 means a rectangular wing (constant chord).
• A taper ratio near 0 means the wing narrows sharply toward the tip.
• Moderate taper (roughly $\lambda \approx 0.3$ to $0.5$) pushes the lift distribution closer to elliptical, reducing induced drag.
• Lower taper ratios also shift structural material inboard (where the root is thicker), reducing wing weight. But taper that's too aggressive can cause the tips to stall first, which is undesirable for handling.

Sweep angle influence

Sweep angle is the angle between the wing's leading edge and a line perpendicular to the fuselage.

• Backward sweep reduces the component of freestream velocity perpendicular to the leading edge. This lowers the effective Mach number the airfoil "sees," delaying compressibility effects (shock waves) at transonic speeds. Most jet transports use 25°–35° of sweep for this reason.
• Forward sweep can improve low-speed handling and delay root stall, but it introduces a structural tendency called divergence, where aerodynamic loads twist the wing further into the flow, requiring stiff (and heavy) construction to counteract.
• The choice of sweep depends on the target flight regime. High-subsonic and transonic aircraft benefit most from sweep; low-speed aircraft (turboprops, general aviation) typically use little or none.

Vortex system of finite wings

The aerodynamic behavior of a finite wing can be modeled using a system of vortices. This system has three components: the bound vortex, trailing vortices, and a vortex sheet. Together, they explain both lift generation and induced drag.

Bound vortex

The bound vortex is a conceptual vortex line running along the span of the wing, typically placed at the quarter-chord. It represents the circulation $\Gamma$ responsible for generating lift. The strength of this vortex varies along the span: it's typically strongest near the root (where the chord and loading are greatest) and drops toward the tips.

Trailing vortices

At the wingtips, high-pressure air from below the wing curls around to the low-pressure region above. This creates trailing vortices that stream downstream behind the aircraft. These vortices carry rotational energy away from the wing, and that energy cost shows up as induced drag.

Vortex sheet

The bound vortex doesn't jump instantly from full strength to zero at the tips. Instead, vorticity is shed continuously along the trailing edge wherever the bound vortex strength changes. This creates a vortex sheet in the wake. Far downstream, this sheet rolls up into the two concentrated tip vortices you can sometimes see in humid conditions.

Helmholtz's theorems application

Two of Helmholtz's vortex theorems are especially relevant here:

1. First theorem: The strength (circulation) of a vortex filament is constant along its entire length. So if the bound vortex weakens at some spanwise station, that "lost" circulation must be shed into the wake as trailing vorticity.
2. Second theorem: A vortex filament cannot end in a fluid. It must either form a closed loop or extend to a boundary (like infinity). This is why the bound vortex doesn't just stop at the wingtips; it continues as trailing vortices stretching far downstream.

These theorems guarantee that the bound vortex, trailing vortices, and starting vortex (left behind at takeoff) form a closed vortex loop, which is the theoretical basis for the entire finite-wing vortex model.

Prandtl's classical lifting-line theory

Prandtl's lifting-line theory is the classical analytical method for predicting the lift distribution and induced drag of a finite wing. It reduces a complex 3D problem to a more tractable mathematical framework.

Fundamental assumptions

The theory rests on several simplifying assumptions:

• The wing is replaced by a single lifting line located at the quarter-chord.
• A system of trailing vortices extends from the lifting line to infinity downstream.
• The flow is inviscid, incompressible, and irrotational everywhere except at the vortex filaments.
• The wing has a high aspect ratio, so each spanwise section behaves approximately like a 2D airfoil responding to its local effective angle of attack.

These assumptions work well for conventional subsonic wings with $AR \gtrsim 4$ or so. For low-AR or highly swept wings, more advanced methods (like vortex-lattice or panel methods) are needed.

Bound vortex strength distribution

The circulation along the span, $\Gamma(y)$, is the central unknown. Prandtl expressed it as a Fourier sine series:

$\Gamma(\theta) = 2bV_\infty \sum_{n=1}^{N} A_n \sin(n\theta)$

where $\theta$ is a transformed spanwise coordinate ($y = -(b/2)\cos\theta$). The coefficients $A_n$ are found by enforcing the condition that each spanwise section operates at the correct angle of attack, accounting for the local geometric angle, the zero-lift angle, and the induced angle.

Induced angle of attack

Trailing vortices create a downwash velocity component $w$ at the wing. This tilts the local relative wind downward, reducing the effective angle of attack each section experiences:

$\alpha_{eff} = \alpha - \alpha_i$

The induced angle of attack is:

$\alpha_i = \arctan(w / V_\infty) \approx w / V_\infty$

(the small-angle approximation is valid for typical flight conditions). Because the effective angle of attack is reduced, the wing produces less lift than an equivalent infinite wing at the same geometric angle.

Downwash velocity calculation

The downwash at any spanwise station is calculated by applying the Biot-Savart law to the entire trailing vortex sheet. Each infinitesimal trailing vortex filament contributes to the downwash at every other point along the span. The result is an integral expression:

$w(y_0) = \frac{1}{4\pi} \int_{-b/2}^{b/2} \frac{(d\Gamma/dy)}{y_0 - y} \, dy$

Downwash is generally strongest near the tips (where trailing vorticity is concentrated) and weaker near midspan.

Induced drag determination

Because the local lift vector is tilted backward by the induced angle $\alpha_i$, a component of the lift acts in the drag direction. This is induced drag. For an elliptical lift distribution (the ideal case), the induced drag coefficient is:

$C_{D,i} = \frac{C_L^2}{\pi AR}$

For non-elliptical distributions, a span efficiency factor $e$ (where $0 < e \leq 1$) accounts for the penalty:

$C_{D,i} = \frac{C_L^2}{\pi e \, AR}$

An elliptical distribution gives $e = 1$. Any deviation from elliptical increases $e^{-1}$ and therefore increases induced drag. This single equation shows why both high AR and near-elliptical loading are so important for efficiency.

Lift distribution along finite wing

The way lift is spread across the span has a direct impact on induced drag, structural loads, and stall behavior. Getting the lift distribution right is one of the most important tasks in wing design.

Elliptical lift distribution

An elliptical lift distribution produces the minimum induced drag for a given total lift and wingspan. The lift per unit span follows an ellipse:

$L'(y) = L'_0 \sqrt{1 - (2y/b)^2}$

where $L'_0$ is the lift per unit span at the wing root. This distribution produces a uniform downwash across the span, which is why it minimizes induced drag. Achieving it exactly requires an elliptical planform (like the Spitfire's wing) or careful use of twist on a non-elliptical planform.

Non-elliptical lift distributions

Most real wings don't achieve a perfectly elliptical distribution. Rectangular wings, for instance, produce more lift per unit span near the root than an ellipse would, and less near the tips. Tapered wings can overshoot or undershoot depending on the taper ratio.

The departure from elliptical loading is captured by the span efficiency factor $e$. Typical values for well-designed wings range from about 0.85 to 0.95.

Lift slope comparison

The 2D (infinite wing) lift curve slope for a thin airfoil is $2\pi$ per radian. For a finite wing, the effective lift curve slope is reduced because of the induced angle of attack:

$a = \frac{a_0}{1 + a_0 / (\pi AR)}$

where $a_0$ is the 2D section lift slope. This means a finite wing needs a larger angle of attack to reach the same $C_L$ as an infinite wing. Lower aspect ratios make this effect more pronounced.

Stall progression

How stall develops across the span matters enormously for safety and handling.

• An elliptical wing tends to stall along the entire span nearly simultaneously, which can cause an abrupt and dangerous loss of lift with little warning.
• A rectangular wing tends to stall at the root first. This is favorable because the ailerons (located outboard) keep working, and the turbulent wake from the stalled root buffets the tail, giving the pilot a natural stall warning.
• A highly tapered wing can stall at the tips first, which is the worst case: you lose aileron effectiveness and roll control right when you need it most.

Wing twist (washout) and other design features are used to ensure that stall begins inboard and progresses outboard, regardless of planform.

Wingtip vortices

Wingtip vortices are an unavoidable consequence of generating lift with a finite wing. They represent the single largest source of drag at low speeds and also create significant hazards for other aircraft.

Formation mechanism

The pressure difference between the wing's lower surface (high pressure) and upper surface (low pressure) drives airflow around the wingtip from bottom to top. This spanwise flow, combined with the freestream, creates a rotating flow structure that trails behind each wingtip. The stronger the lift, the stronger the vortices.

Vortex core structure

The vortex core is a tight region of intense rotation and very low pressure. Tangential velocity is highest just outside the core and decreases with radial distance (following roughly an inverse relationship). The core diameter depends on wing geometry, angle of attack, and Reynolds number. In humid conditions, the low pressure in the core can cause water vapor to condense, making the vortices visible.

Induced drag contribution

The kinetic energy stored in the rotating flow of the trailing vortices has to come from somewhere, and that source is the aircraft's thrust. This energy cost is induced drag. As noted earlier:

$C_{D,i} = \frac{C_L^2}{\pi e \, AR}$

At cruise, induced drag might account for 30–40% of total drag for a transport aircraft. During climb at high $C_L$, it can dominate.

Wake rollup process

Immediately behind the wing, the vortex sheet contains distributed vorticity shed from the entire trailing edge. Within a few wingspans downstream, this sheet rolls up into two concentrated counter-rotating vortices (one from each tip). These rolled-up vortices can persist for several minutes and extend miles behind a large aircraft. The resulting wake turbulence is hazardous to following aircraft, which is why air traffic control enforces separation standards based on aircraft weight class.

Wing twist effects

Wing twist is one of the most practical tools designers have for shaping the spanwise lift distribution. It's used on nearly every production aircraft.

Geometric vs aerodynamic twist

• Geometric twist means the physical chord line is rotated to different angles of incidence at different spanwise stations. You can literally see the twist if you sight along the wing.
• Aerodynamic twist means using different airfoil sections along the span, each with a different zero-lift angle of attack. The wing might look untwisted, but the aerodynamic properties still vary from root to tip.

Both types shift the local angle of attack and therefore reshape the lift distribution. In practice, many wings use a combination of both.

Washout vs washin

• Washout reduces the angle of incidence from root to tip. This is by far the more common type. It unloads the tips, pushing the lift distribution inboard and ensuring the root stalls before the tips.
• Washin increases the angle of incidence toward the tip. It's rarely used on straight wings but sometimes appears on swept wings to compensate for the tendency of swept wings to stall at the tips first (due to spanwise boundary-layer flow migrating outboard).

Stall characteristics improvement

Washout is the primary tool for controlling stall progression. By reducing the local angle of attack at the tips, washout ensures the inboard sections reach their critical angle first. This keeps the ailerons effective during the initial stages of stall and provides natural buffet warning as the separated flow from the stalled root hits the horizontal tail.

Lift distribution optimization

By carefully choosing the twist distribution, a designer can push a non-elliptical planform's lift distribution much closer to elliptical, reducing induced drag. For example, a rectangular wing with several degrees of washout can achieve a span efficiency factor significantly higher than the same wing without twist. The optimal twist schedule depends on the planform, the design $C_L$, and structural constraints.

High-lift devices for finite wings

High-lift devices temporarily increase the wing's maximum lift coefficient so the aircraft can fly slower during takeoff and landing. Without them, runway lengths would need to be much longer, or wings would need to be impractically large for cruise efficiency.

Leading-edge devices

Leading-edge devices re-energize or redirect the flow near the leading edge to delay separation at high angles of attack.

• Slats are small airfoil sections that deploy forward and slightly downward from the leading edge, creating a slot. The accelerated flow through the slot energizes the boundary layer on the main wing, allowing it to remain attached to higher angles.
• Krueger flaps are panels that hinge out from the lower surface of the leading edge, effectively increasing the leading-edge camber and radius.

Both devices primarily increase the stall angle rather than shifting the entire lift curve upward.

Trailing-edge flaps

Trailing-edge flaps increase both camber and (in some designs) wing area when deployed.

• Plain flaps simply hinge the trailing-edge portion downward. Simple but limited in effectiveness.
• Split flaps deflect only the lower surface, creating high drag along with moderate lift increase.
• Slotted flaps open a gap that channels high-energy air onto the flap's upper surface, delaying separation on the flap itself. Much more effective than plain flaps.
• Fowler flaps translate aft before deflecting, increasing both camber and wing area. These produce the largest lift increments and are standard on transport aircraft.

Lift coefficient enhancement

The combination of leading-edge and trailing-edge devices can increase $C_{L,max}$ dramatically. A clean wing might have $C_{L,max} \approx 1.5$, while a fully deployed high-lift system can push this to 2.5–3.5 or higher on modern transports. The exact gain depends on the device type, size relative to the chord, and deflection angle.

Stall angle increase

Leading-edge devices are especially effective at increasing the stall angle. A clean airfoil might stall around 12°–16° angle of attack. With slats deployed, the stall angle can increase to 20°–25° or more. This larger usable angle-of-attack range provides a greater safety margin during approach and landing, where the aircraft operates at high $C_L$ and low speed.

Finite wing design considerations

Designing a wing involves balancing competing demands. Aerodynamic efficiency is only one piece of the puzzle; structural weight, manufacturing cost, stability, and mission requirements all constrain the final design.

Lift-to-drag ratio optimization

The lift-to-drag ratio $L/D$ is the single best measure of aerodynamic efficiency. Maximizing $L/D$ at the cruise condition minimizes fuel burn for a given payload and range. The key levers are:

• High aspect ratio (reduces induced drag)
• Near-elliptical lift distribution (maximizes span efficiency)
• Appropriate airfoil selection (minimizes profile drag at the design $C_L$)
• Wing twist tailored to the cruise condition

Every improvement in $L/D$ translates directly to longer range or lower fuel consumption.

Structural constraints

Aerodynamic ideals often conflict with structural realities. A very high aspect ratio wing is aerodynamically efficient but experiences large bending moments at the root, requiring heavier spar caps and skin panels. The internal structure (spars, ribs, stringers) must handle not just cruise loads but also gust loads, maneuver loads, and fatigue over the aircraft's lifetime. Aeroelastic effects like flutter and divergence set upper limits on how flexible the wing can be, which in turn constrains the aspect ratio and sweep angle.

Stability and control requirements

The wing's geometry affects the aircraft's stability in all three axes.

• Sweep contributes to directional (yaw) stability and also affects lateral (roll) stability through dihedral effect.
• Dihedral angle (upward tilt of the wings) provides positive lateral stability, helping the aircraft return to wings-level after a disturbance.
• Aileron sizing and placement must provide adequate roll authority across the flight envelope, including at low speeds near stall.

The wing's position relative to the center of gravity also determines the aircraft's longitudinal (pitch) stability characteristics.

Mission-specific adaptations

Different missions demand different wing designs:

• Long-range transports use high-AR, moderately swept wings with advanced high-lift systems to balance cruise efficiency with acceptable takeoff and landing performance.
• Fighter aircraft use low-AR, highly swept (or delta) wings for high maneuverability and supersonic capability, accepting higher induced drag.
• General aviation aircraft often use moderate-AR, unswept wings for simplicity, low cost, and good low-speed handling.
• STOL (Short Takeoff and Landing) aircraft prioritize large, effective high-lift devices and may use lower wing loading to achieve the required field performance.

The wing is always a compromise shaped by the aircraft's primary mission.