Influencing the Critical Mach Number

Wing sweep

Sweep works by a geometric argument. An aerofoil section responds to the flow crossing its chord, accelerating it over the curvature, building the suction that leads to the first sonic point. Resolve the oncoming flow at a swept wing into two components: one perpendicular to the quarter-chord line, crossing the section the way its designer intended, and one along the span, sliding along the wing and producing no lift (not that it is idle everywhere: it is what drags the boundary layer outboard, the tip-stall story below). Only the perpendicular component counts, and sweep shrinks it: the section effectively flies at

M \cos \Lambda

, a fraction of the flight Mach number set by the sweep angle Λ. The wing's shocks are governed by that reduced effective Mach number, so the flight Mach number can go correspondingly higher before the first local sonic point arrives: sweep raises the critical Mach number. An equivalent way to say it: measured along the actual flow direction, the swept wing's section is stretched: its effective thickness-to-chord ratio is reduced, and a slimmer effective section accelerates the flow less. With 30° of sweep the perpendicular component is about 87% of the flight Mach number, a 13% reduction, which in principle lets the flight Mach number go about 15% higher (1/cos 30° ≈ 1.15) before the first sonic point arrives, though root, tip and fuselage effects take a good part of that back on a real wing.

The price of sweep

Sweep is paid for at low speed, and the exam asks for each part of that price:

Maximum lift falls — only the perpendicular component of the flow generates lift, so the swept wing lifts less at every angle of attack and its $C_{L\,max}$ is markedly lower than the same wing unswept: take-off and landing speeds rise.
High-lift devices work less well (and are needed more) — a flap deflected across a swept hinge line changes the effective camber less than on a straight wing, so each device delivers a smaller $C_{L\,max}$ increment exactly when the wing needs a bigger one. This is why swept-wing transports carry the full set of the subsonic module's high-lift lessons: multi-slotted Fowler flaps AND slats, working together.
Pitch-up at the stall — on a swept wing the boundary layer drifts outboard along the span, thickening toward the tips, so the tips tend to stall first. The tips of a swept wing sit well AFT, losing their lift shifts the centre of pressure forward, and the aircraft pitches nose-UP into the stall, deepening it instead of breaking it. The stall module returns to this dangerous behaviour and the design fixes for it.

Sweep and the drag curve

On drag, sweep does at the high-speed end exactly what it was hired for: the whole drag-rise curve of the previous lesson (the creep past $M_{CRIT}$ , the steep divergence at $M_{DD}$ ) is shifted to higher Mach numbers, with a lower and later peak. The transonic drag penalty is postponed and softened, which is the whole case for accepting the low-speed price above.

Thin sections, less camber

The second lever is the section itself. The first sonic point arrives where the flow is accelerated hardest, and how hard a section accelerates the flow is set by how much it forces the streamlines to deviate: its thickness and its camber. A thin aerofoil with little camber accelerates the flow gently, keeps the local Mach numbers close to the flight Mach number, and so reaches its critical Mach number later. The price is the subsonic module in reverse: less camber and thickness mean a lower $C_{L\,max}$ (higher low-speed stalling speeds), a shallower spar (a heavier structure for the same strength), and less internal volume for fuel exactly where an airliner wants to carry it.

The supercritical aerofoil

The modern answer refuses that trade. The supercritical aerofoil is designed not to avoid supersonic flow but to carry it cheaply: its purpose is to let the wing cruise beyond the critical Mach number with only a weak shock and a late drag divergence. The shape tells the story: a relatively blunt nose and a strikingly flat upper surface spread the acceleration out, producing a long, shallow supersonic region at nearly constant local Mach instead of a violent peak. A shallow supersonic region ends in a weak shock, far aft, with little separation behind it. The lift that the flattened top no longer produces is recovered at the back: the lower surface curves up and then cusps downward near the trailing edge, the rear-loading camber that loads the aft chord where the upper surface cannot.

The design trade, as the exam wants it:

Advantages — a higher drag-divergence Mach number for a given thickness; or, read the other way, the SAME cruise Mach from a thicker wing with less sweep, and a thicker, straighter wing is a lighter one, holds more fuel, allows a higher aspect ratio (the induced-drag lesson's favourite number), and behaves better at low speed.
Disadvantages — the rear loading produces a strong nose-down pitching moment, which must be trimmed away continuously (trim drag, heavy tail loads); the aft chord carries high loads where flaps and structure must live; and the finely balanced upper-surface flow is sensitive to being flown, or built, off its design point.

Vortex generators against the shock

The last device is already familiar. The subsonic module's vortex generators (small vanes whose tip vortices stir fast outside air into the boundary layer) have a second job on transonic wings, and it is the same trick aimed at a different pressure rise. The pressure jump across a shock is exactly the kind of sudden adverse gradient a tired boundary layer cannot climb; a row of vortex generators ahead of the shock's position re-energises the boundary layer so that it can take the jump and stay attached. Shock-induced separation is avoided or confined: less drag from the separation half of wave drag, buffet postponed, and the ailerons behind the protected stretch keep their bite — the previous lesson's control problem, patched for the price of a small permanent parasite-drag penalty.