Concept

Wedge — where it appears

A two-dimensional body with flat faces meeting at a sharp edge, the simplest shape that turns a supersonic stream through a fixed angle. It makes the whole turn at once, at an attached oblique shock, which makes it the reference a cone's gradual turn is measured against.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

A 10° wedge at Mach 2.00 has two shocks that solve it. The wedge turns the flow through a fixed angle, and the θ–β–M relation offers two shock angles that achieve it. The weak solution, drawn steeply forward, leaves the flow supersonic and is what a wedge in a free stream produces. The strong solution leaves it subsonic and appears where downstream pressure forces it. Nothing local to the wedge chooses between them.

A shock that leans

Tilt a shock and only the velocity component across it is changed — the component along it passes through untouched. That single observation turns every oblique shock into a normal shock in disguise, and it is why a wedge at Mach 2 leaves the flow supersonic while a blunt nose does not.

compressible · Oblique shock

A 20° cone at Mach 2: the shock at 37.80°, and a flow still compressing behind it. The conical flow round a cone of 20° half-angle at Mach 2, from the Taylor–Maccoll equation. The shock sits at 37.80° from the axis and turns the flow crossing it through only 8.57°, leaving it at Mach 1.693. Between the shock and the surface every ray from the apex carries its own state: the Mach number falls from 1.693 just behind the shock to 1.568 at the surface and the pressure rises from 1.586 to 1.912 times the free stream's — the rest of the turn, made without a shock. Nothing depends on the distance from the apex, so the rays are lines of constant state.

A cone finishes its turn after the shock

A wedge turns a supersonic stream all at once, at its shock. A cone of the same angle does not: its shock turns the flow only part of the way and leaves the rest to a smooth compression between the shock and the surface. Solved from Taylor and Maccoll's equation, the cone's shock is weaker, keeps more of the total pressure, carries less than half the wedge's surface pressure, and stays attached to 40.7° at Mach 2 where the wedge gives up at 23°.

compressible · Oblique shock

A cone hands the terminal shock a spread of Mach numbers. At Mach 2, the Mach number across the annulus a cowl on the conical shock captures, from the cone's surface to the shock, for cones of 15°, 22° and 28°, the last the best for recovery. Behind a wedge the flow is uniform; behind a cone it is fastest just behind the shock and slowest at the surface, 1.43 to 1.32 for the best cone. The terminal shock acts on all of it.

A cone intake's shock sees the whole capture

A cone's shock keeps far more total pressure than a wedge's at the same angle, because the cone finishes its turn in an isentropic compression after the shock. An intake then needs a terminal normal shock, and that shock meets not one Mach number but the whole spread the cone leaves across the captured annulus — fastest by the shock, where most of the air is, and slowest at the surface, where little is. Averaged over the mass, the cone's advantage over a wedge at the best angle for each is under two points of recovery, and the usual estimate from the surface Mach number nearly doubles it.

compressible · Oblique shock

Up a step the pressure rises to a stagnation at the foot and falls without limit at the lip, and the face feels both. The pressure coefficient along the wall of a right-angled forward-facing step of height h in ideal flow, against distance along the wall from the foot of the face in step heights: the floor ahead of it, the face, and the top beyond the lip (clipped at −3, where it goes to minus infinity). The face is pushed back by 0.6366 of a dynamic pressure times its height over its lower part and pulled forward by exactly as much near its lip.

A step's face is pushed at its foot and pulled at its lip, equally

Ideal flow running up a step has two corners in it, and the local theory of corners says what each does: the inside corner at the foot brings the flow to rest, the outside corner at the lip sends it round at infinite speed. Solved exactly, the step's face feels both — a push of 2/π of a dynamic pressure times its height spread over its lower four-fifths, and a pull concentrated at the lip — and they cancel exactly, at every ramp angle, because one reflection of the mapping exchanges the two corners. A step has no drag in ideal flow for the same reason a flat plate's lift has no drag: a finite force carried by an infinite speed.

inviscid · Wedge flow

Named alongside it

The objects these essays reach for when they reach for this one.

Mach cone Oblique shock Total pressure Detachment Entropy Isentropic Model validity Shock wave The θ–β–M relation Conformal map d'Alembert's paradox Ideal flow

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