Lesson 4 · 30 min

Wing Area and Aspect Ratio

The planform starts with two numbers: how much wing, and how slender. The area is set by the wing loading, which Module 5 chooses from the takeoff, landing and maneuvering requirements. The aspect ratio is chosen here, and it is a trade between aerodynamics, which wants a long slender wing, and structure, which does not.

Learning objectives

Define the reference wing, its span, mean chord and aspect ratio.
Explain what aspect ratio does to induced drag, \(\LDmax\), the lift-curve slope and the wing weight, and compare Helmbold's and Prandtl's lift-curve slopes.
Estimate a first aspect ratio from statistics for propeller aircraft, jets and sailplanes.
Judge the statistics against real aircraft, and recognize when a design falls outside them.

The reference wing

The wing that appears in the equations is not the wing you can see. The reference wing is the straight-tapered trapezoid that matches the real wing outboard, extended through the fuselage to the centerline. Its area \(S\) (the \(\Sref\) of Module 3) includes the part hidden inside the fuselage, and its span \(b\) is measured tip to tip. Every coefficient (\(C_L\), \(\CDz\)) and every wing loading uses this area.

Span, mean chord and aspect ratio

\[ A = \frac{b^2}{S} = \frac{b}{\bar c_g}, \qquad \bar c_g = \frac{S}{b} \]

\(\bar c_g\) is the mean geometric chord: the chord of a rectangular wing with the same span and area. A high aspect ratio wing is long and slender; a low one is short and broad.

What aspect ratio buys and costs

The induced drag of Module 1, \(C_{D_i} = KC_L^2\) with \(K = 1/(\pi Ae)\), falls in inverse proportion to the aspect ratio: a longer span moves the same lift with a weaker downwash. That is the main reason to choose a high aspect ratio, and it shows in \(\LDmax\), which grows as \(\sqrt A\) (Module 3, Lesson 3).

The lift-curve slope grows with aspect ratio too, because the tip vortices of a short wing reduce the angle of attack its sections feel:

Wing lift-curve slope (Helmbold, with compressibility and sweep)

\[ \CLa = \frac{2\pi A}{2 + \sqrt{4 + \dfrac{A^2\beta^2}{\eta^2}\left(1 + \dfrac{\tan^2\Lambda}{\beta^2}\right)}} \]

per radian, with \(\beta^2 = 1 - M^2\), \(\eta \approx 0.95\) (the airfoil's slope as a fraction of \(2\pi/\beta\)) and \(\Lambda\) the sweep of the line of maximum thickness. As \(A \to \infty\) it tends to the airfoil's \(2\pi\); a low aspect ratio wing needs a much higher angle of attack for the same lift.

At low speed and without sweep, Helmbold's formula reduces to \(\CLa = 2\pi A/\big(2 + \sqrt{4 + A^2/\eta^2}\big)\). The classical result it improves on is Prandtl's lifting-line theory, which models the wing as a single line of vortices along its span. For an elliptic wing it gives:

Wing lift-curve slope (Prandtl's lifting line, elliptic wing)

\[ \CLa = \frac{a_0}{1 + \dfrac{a_0}{\pi A}}, \qquad \CLa = \frac{2\pi A}{A + 2}\ \ \text{for}\ a_0 = 2\pi \]

per radian, where \(a_0\) is the airfoil's lift-curve slope. Other planforms lose a little more, which lifting-line theory accounts for with a factor \(1 + \tau\) on the \(a_0/(\pi A)\) term, with \(\tau\) a few percent.

A single line of vortices is a good model of a long, slender wing and a poor one of a short, broad wing. With the same airfoil slope \(a_0 = 2\pi\eta\), the two formulas agree within about 4% above \(A = 6\), but at low aspect ratio Prandtl's is too high: about 40% at \(A = 1\), where it tends to \(\pi A\). Helmbold's formula tends instead to \(\pi A/2\), the value from slender-wing theory for very short wings, so it covers the whole range.

Figure 4.1 puts these effects on the two curves a designer uses most. On the lift curve, a higher aspect ratio and a higher Mach number steepen the slope, and sweep flattens it. Sweep also lowers the maximum lift, roughly as \(C_{L,\max} \approx 0.9\,c_{l,\max}\cos\Lambda\) for a wing of moderate to high aspect ratio, and a wing with a shallower slope reaches its stall at a higher angle of attack. On the drag polar \(C_D = \CDz + KC_L^2\), a higher aspect ratio or Oswald factor shrinks \(K\) and narrows the parabola, which raises \(\LDmax = 1/(2\sqrt{\CDz K})\) and the lift coefficient \(C_L^{*} = \sqrt{\CDz/K}\) at which it occurs.

Figure 4.1 The lift curve (left) and drag polar (right) of a wing (blue), starting from the business jet of Module 3. The gray dashed curves are a reference wing that you can pin to compare against, and the violet dashed line is the airfoil's own lift curve. The straight part of the lift curve comes from Helmbold's formula with \(\alpha_{L0} = -2^\circ\). Its rounded top is schematic: \(C_{L,\max} = 0.9\,c_{l,\max}\cos\Lambda\) with \(c_{l,\max} = 1.6\), lowered further at high subsonic Mach numbers, where shocks separate the flow sooner. Prandtl's slope (orange, with its button) is for low speed and no sweep, so compare it with the wing at \(M = 0\) and \(\Lambda = 0\). The polar runs up to that maximum lift, with the tangent from the origin marking \(\LDmax\) (green). Sweep is taken to be the same along every chord line, as on an untapered wing. The polar does not include the wave drag and loss of Oswald factor that high Mach numbers and sweep add, which a later module estimates.

Against these gains:

Weight. A longer span carries the same lift farther out, so the root bending moment and the wing weight rise. For the same area and \(t/c\), the chords are also shallower.
Fuel volume. Slender chords hold less fuel for the same area.
Handling. A long span damps roll: fighters and aerobatic aircraft prefer short spans.
Span limits. Airport gates and taxiways (the ICAO codes of Module 2) cap the span of airliners.
Supersonic flight. Above Mach 1 the wing's wave drag, not its induced drag, dominates, and a short, thin, highly swept wing is better.

Example 4.1 — The business jet's wing

The business jet has \(S = 28\ \text{m}^2\) and \(A = 8\) (Module 3), with \(e = 0.75\). Find its span, mean geometric chord and induced-drag factor, and its lift-curve slope at low speed (unswept, \(\eta = 0.95\)) and in cruise at Mach 0.80 with \(22^\circ\) of sweep at mid-chord.

Show solution

\[ b = \sqrt{AS} = \sqrt{8(28)} = 14.97\ \text{m}, \qquad \bar c_g = \frac{28}{14.97} = 1.871\ \text{m} \] \[ K = \frac{1}{\pi(8)(0.75)} = 0.0531 \] \[ \CLa = \frac{2\pi(8)}{2 + \sqrt{4 + 64/0.9025}} = \frac{50.27}{2 + 8.656} = 4.72\ \text{per rad} \]

In cruise, \(\beta^2 = 0.36\) and \(\tan^2 22^\circ = 0.1632\):

\[ \CLa = \frac{50.27}{2 + \sqrt{4 + \dfrac{64(0.36)}{0.9025}(1.453)}} = \frac{50.27}{2 + 6.41} = 5.98\ \text{per rad} \]

At low speed the wing reaches 75% of the airfoil's \(2\pi\). Compressibility raises the slope in cruise, and sweep takes some of it back.

Statistics for a first choice

As with the empty weight, history gives a starting point. For propeller aircraft the aspect ratio depends mostly on the class; for jets it falls with the maximum Mach number, because faster aircraft need shorter, thinner, more swept wings:

Statistical aspect ratios (source: Raymer, Table 4.1; check the values against the edition you use)
Propeller aircraft	\(A\)

Jets: \(A = a\,M_{\max}^{\,C}\)
Jet aircraft	\(a\)	\(C\)	\(A\) at \(M_{\max} = 0.9\)	at 2.0

Sailplanes are a class of their own: their whole design is a search for \(L/D\), and their aspect ratio follows from the \(L/D\) they aim for, \(A \approx 0.19\,\LDmax^{1.3}\). A 40:1 sailplane needs an aspect ratio of about 23.

Figure 4.2 Aspect ratio against Mach number for the jets in Module 1's data (points, from Loftin's NASA survey) and the statistical trends (lines). Hover over a point to name the aircraft. The transports cluster tightly around 7.5; the fighters scatter widely, and the two variable-sweep fighters, listed at their unswept span, sit far above every trend.

The statistics describe what has been built, not what must be. A new design should start near them and depart only for a reason: today's airliners, with composite wings and folding tips, have aspect ratios of 9 to 11, well above the 7.5 of the 1960s to 1980s transports in the data.

Check your understanding

Key takeaways

The reference wing extends through the fuselage to the centerline; \(A = b^2/S\), \(\bar c_g = S/b\).
Higher \(A\): less induced drag, \(\LDmax \propto \sqrt A\), a steeper lift curve; but a heavier wing, less fuel volume and slower roll.
Helmbold's \(\CLa\) covers every aspect ratio; Prandtl's lifting line, \(2\pi A/(A + 2)\), agrees above \(A \approx 6\) but is too high for short wings.
Statistics: propeller aircraft 6 to 9; jets \(A = aM_{\max}^C\) (transports about 7.5); sailplanes \(0.19\,\LDmax^{1.3}\).
Next, Lesson 5 adds sweep, the jet's answer to the speed of sound.