Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Oblique shock — where it appears

A shock standing at an angle to the flow, which turns it as well as compressing it. For a given turn there are two possible angles, and which one a body produces depends on the conditions downstream rather than on the wedge.

Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.

The lift curve that made flight look impossible. Lift coefficient against incidence, by Newton's impact theory and by thin-aerofoil theory. One is quadratic in the angle and the other linear, so at small incidence — which is where aircraft fly — they differ by more than an order of magnitude. Newton's version says a wing large enough to carry a man would need an engine nobody could build, and for a century that arithmetic was taken as settling the question.

The theory that forbade flight

Newton treated air as a hail of particles that give up their normal momentum on impact, and got a lift coefficient of 2sin²α cos α. At five degrees that is a thirty-sixth of what a wing actually makes, and the quadratic is why powered flight looked arithmetically impossible for two centuries. The same formula is exact at Mach twenty.

misconceptions · Newtonian 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 Past 23.0° at Mach 2.00 there is no attached shock. The same wedge at two half-angles. On the left the θ–β–M relation has a root and the shock sits on the nose. On the right it has none, and the solver throws rather than returning the nearest thing — which matters, because a solver that quietly clamped to the maximum would draw a neat attached shock on a body that cannot carry one. The bow shock on the right is indicative: its shape is not solved here.

When the wedge is too blunt

Every curve of shock angle against deflection has a maximum. Past it there is no attached shock at any angle — the solver has no root to return and must say so, rather than quietly handing back the nearest thing and drawing a picture that cannot exist.

compressible · Oblique shock The two ways a shock can meet a wall. Left: the incident shock from the wedge reaches the wall and a second shock turns the flow back parallel to it, meeting at a point. Right: at a larger wedge angle no reflected shock can turn the flow that far, and the intersection lifts off the wall into a triple point with a nearly normal Mach stem standing on the surface and a slip line trailing downstream. The two configurations are drawn at the angles the solver returns, with every shock angle computed rather than sketched.

When a shock cannot bounce

A shock reflects off a wall until the reflected shock runs out of turning, which happens at a wedge angle well below the free stream's own limit. Between the two boundaries both configurations exist, both are stable, and which one appears depends on which direction the experiment came from.

compressible · Mach reflection A curved shock, and the entropy each streamline picks up crossing it. A parabolic bow shock ahead of a blunt nose at Mach six, with the streamlines drawn arriving horizontally and a marker at each crossing whose size is the total pressure lost there. The streamline through the nose crosses a normal shock and keeps three per cent of its total pressure; one four nose radii out crosses at fourteen degrees and keeps ninety-four per cent. Every streamline gets a different entropy, and the stagnation enthalpy is the same on all of them.

The spin a shock leaves behind

A curved shock gives every streamline a different entropy rise and the same stagnation enthalpy. Crocco's theorem then forces vorticity into a flow with no viscosity anywhere — and it scales as the inverse of the shock's radius of curvature, exactly, so a straight shock makes none.

compressible · Crocco So splitting a turn into N ramps costs one over N squared. A twelve-degree compression at Mach 3, done in one ramp and in up to sixty-four. The entropy is N times a cube of one Nth, so it falls as exactly the inverse square of the number of ramps — the measured exponent is −2.00 — and sixty-four ramps cost a two-hundred-and-fifty-sixth of what one costs.

A compression that costs nothing in the end

Turning a supersonic stream away from itself is free and turning it into itself is not. But the price of a compression is the cube of its strength, so splitting one turn into N turns costs one over N squared — and in the limit the compression is free too.

compressible · Expansion fan 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 The shaded face's share of the force is set by K, and it is not small until K is. The fraction of a flat plate's normal force carried by its leeward face against K = M sin α, at Mach 3, 5, 10, 20, with the hypersonic small-disturbance value and the share the leeward face would have at vacuum (dashed). Newtonian theory puts it at zero. The curves collapse on K: the small-disturbance share is 35.7 per cent at K = 0.5, 24.2 at 1, 11.2 at 2 and 3.4 at 4, and the vacuum bound is 28.8, 11.4 and 3.4 per cent at 1, 2 and 4. The zero is a good approximation only where K is large — which is also the only place the Newtonian windward pressure is itself accurate.

The face Newton left in shadow

Newtonian theory gives a surface turned away from the stream a pressure coefficient of exactly zero, and at hypersonic speed the rest of the theory is nearly right. The shaded face is not. Computed exactly on a flat plate, its share of the force depends on the similarity parameter K = M sin α rather than on the Mach number, it is a quarter of the force at K = 1, and it moves a hypersonic plate's best lift-to-drag ratio from 5 to 7 at Mach 10.

misconceptions · Newtonian The worst jet amplifies the stagnation pressure by about the Mach number. The largest amplification of the stagnation pressure, over every incident turn, against the free-stream Mach number, on a logarithmic axis: the type IV jet, the best single turning shock followed by a normal shock, and the lossless ceiling. The jet's peak runs close to the line equal to the Mach number itself, from 3.5 at Mach 4 to 12.4 at Mach 12. The ceiling grows as the Mach number to the power of three and a half and is never approached. The estimate with one turning shock falls further behind the jet as the Mach number rises.

The spot a local theory cannot see

Newtonian theory gives every panel of a hypersonic vehicle a pressure set by its own angle to the stream, and no panel more than the stagnation pressure behind a normal shock. Let a shock from one part cross the bow shock of another and a supersonic jet forms that reaches the surface through weaker shocks. At Mach 8 it stagnates at 8.6 times the ceiling — and the worst amplification at every Mach number is close to the Mach number itself.

misconceptions · Newtonian

Named alongside it

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

Shock waveTotal pressureEntropyModel limitThe θ–β–M relationHypersonicMisconceptionNewtonian impactPressure coefficientDetachmentIsentropicLift coefficient

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