Lesson 5 · 30 min

Wing Sweep

Module 1 told the story: the swept wing, studied in Germany during the war, gave the F-86 and the Boeing B-47 their speed and became the shape of every jet transport since. This lesson puts numbers on it. Sweep is chosen for the cruise Mach number, together with the airfoil's thickness, and it has costs that the rest of the planform has to manage.

Learning objectives

Explain simple sweep theory and compute the normal Mach number.
Choose sweep and thickness together for a cruise Mach number.
Find the leading-edge sweep that keeps a supersonic wing's leading edge inside the Mach cone.
List what sweep costs: lift-curve slope, maximum lift, tip stall and pitch-up, weight, aeroelastic effects.

Simple sweep theory

Split the flow over an infinitely long swept wing into a component normal to the leading edge and one along it. The flow along the edge sees a wing that does not change in that direction, so it does nothing to the pressures: the airfoil only feels the normal component. Its Mach number is lower than the flight Mach number:

Normal Mach number

\[ M_n = M\cos\Lambda \]

The shock, and the drag rise, are delayed until \(M_n\) reaches what the airfoil can take unswept. In simple theory the critical Mach number rises by \(1/\cos\Lambda\). Real wings gain less: the root and the tip do not behave like an infinite swept wing.

Sweep also makes the airfoil effectively thinner and less loaded: measured normal to the sweep, the chord is shorter and the lift coefficient higher. That is where the \(\cos^2\Lambda\) and \(\cos^3\Lambda\) of the Korn equation (Lesson 3) come from.

Example 5.1 — Sweep for the business jet

(a) Find the normal Mach number of the business jet's wing, swept \(25^\circ\), at Mach 0.80. (b) With the Korn equation (\(c_l = 0.30\)), compare \(\MDD\) for a 12% supercritical airfoil and a 12% conventional one at \(0^\circ\), \(25^\circ\) and \(35^\circ\) of sweep.

Show solution

\[ \text{(a)}\ M_n = 0.80\cos 25^\circ = 0.80(0.9063) = 0.725 \]

(b) With \(\MDD = \kappa_A/\cos\Lambda - 0.12/\cos^2\Lambda - 0.30/(10\cos^3\Lambda)\):

Drag-divergence Mach number of a 12% airfoil at \(c_l = 0.30\)
Sweep	Supercritical (\(\kappa_A = 0.95\))	Conventional (\(\kappa_A = 0.87\))
\(0^\circ\)	0.800	0.720
\(25^\circ\)	0.862	0.774
\(35^\circ\)	0.926	0.829

A conventional 12% wing needs about \(35^\circ\) of sweep to cruise at Mach 0.80 with a margin, as the jet transports of the 1960s had. A supercritical airfoil needs only about \(15^\circ\); the business jet's \(25^\circ\) leaves room to cruise faster, or to make the root thicker. The \(35^\circ\) supercritical value shows the limit of the simple theory: it overestimates what large sweeps buy.

Sweep and Mach number in practice

Figure 5.1 Wing sweep against Mach number for the jets in Module 1's data (points, from Loftin's NASA survey; transports at their cruise Mach number, fighters at their maximum). The blue curve is the sweep that holds the normal Mach number at the chosen value; the dashed curve is the sweep that puts a supersonic leading edge inside the Mach cone. Hover over a point to name the aircraft.

The subsonic transports follow the normal-Mach curve closely: they cruise with \(M_n\) between 0.65 and 0.76, 0.72 on average. The supersonic fighters sit well below the Mach-cone curve, because they spend most of their time subsonic and use thin airfoils, not sweep alone, for their top speed. The unswept F-104 is the extreme: a wing only 3.4% thick.

Supersonic leading edges

Above Mach 1 a disturbance spreads within a cone of half-angle \(\mu = \arcsin(1/M)\), the Mach angle. A leading edge swept behind the Mach cone is a subsonic leading edge: the flow normal to it is subsonic, so it can be round and generate suction like a subsonic airfoil. Swept ahead of the cone, it is a supersonic leading edge and must be sharp, with higher wave drag.

Leading-edge sweep for a subsonic leading edge

\[ \Lambda_{LE} > 90^\circ - \arcsin\frac{1}{M} \]

Mach 1.6: \(\Lambda_{LE} > 51.3^\circ\). Mach 2: \(\Lambda_{LE} > 60^\circ\), the sweep of the delta wings of the 1960s.

This is simple sweep theory again. Since \(90^\circ - \arcsin(1/M) = \arccos(1/M)\), the condition is the same as \(M\cos\Lambda_{LE} < 1\): the Mach number normal to the leading edge is subsonic exactly when the edge lies inside the Mach cone from its apex (Figure 5.2).

Figure 5.2 A wing in supersonic flight, seen from above with the nose up, and the Mach cone from its apex (violet), half-angle \(\mu = \arcsin(1/M)\). The oncoming flow \(M\) at the leading edge splits into a component along the edge, \(M_t = M\sin\Lambda_{LE}\), which the wing does not feel, and one normal to it, \(M_n = M\cos\Lambda_{LE}\). Raise the sweep until the leading edge passes inside the cone: \(M_n\) drops below 1 and the edge becomes subsonic (green). Raise the Mach number and the cone narrows, so the same edge turns supersonic (red).

What sweep costs

Lift. The lift-curve slope falls (the \(\tan^2\Lambda\) term of Lesson 4), and so does the maximum lift, roughly in proportion to \(\cos\Lambda\). A swept wing needs larger flaps and slats, and lands at a higher angle of attack.
Tip stall and pitch-up. On an aft-swept wing the boundary layer drifts outboard and thickens near the tips, which also carry more load (Lesson 6). The tips stall first. They are aft of the center of gravity, so losing their lift pitches the nose up, deeper into the stall. Twist, taper, fences, notches and vortex generators are the remedies (Lesson 7).
Weight. The structural span is \(b/\cos\Lambda\), and the swept box must also carry torsion. Swept wings are heavier for the same span and area.
Aeroelasticity. When an aft-swept wing bends up, its tips twist nose-down, which unloads them: good for the loads, bad for aileron effectiveness. A forward-swept wing twists the other way and can diverge, which is why only a few, built with tailored composites, have flown.
Lateral stability. Sweep adds a dihedral effect: a swept wing in a sideslip rolls away from the slip, as if it had dihedral (Lesson 7).

Forward sweep has the opposite stall pattern, root first, and keeps the ailerons working. It is used where a structural layout needs the wing box behind the cabin, as on some business jets, more often than for aerodynamics.

A propeller aircraft below about Mach 0.6 gains nothing from sweep and pays all its costs: it uses none, or a few degrees only to place the aerodynamic center or improve visibility.

Check your understanding

Key takeaways

The airfoil of a swept wing feels \(M_n = M\cos\Lambda\); choose sweep and thickness together for the cruise Mach number.
Supercritical airfoils need much less sweep than conventional ones for the same speed.
Supersonic: a subsonic leading edge needs \(\Lambda_{LE} > 90^\circ - \arcsin(1/M)\), the same as \(M\cos\Lambda_{LE} < 1\): the edge inside the Mach cone.
Sweep costs lift-curve slope, maximum lift, structure, aileron effectiveness, and brings tip stall and pitch-up.
Next, Lesson 6 tapers the wing and works out its geometry.