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 14 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 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 Mark's criterion lets the wall gas through above Mach 6.5; the gas inside the layer never gets through. The stagnation pressure of the incident shock's boundary-layer gas in the reflected shock's frame, over the reservoir pressure p₅, against the incident shock's Mach number in air: for the gas at the wall, which is Mark's criterion, and for the lowest anywhere in the layer. Below one the gas cannot pass into the reservoir and the shock bifurcates. The wall gas is refused from Mach 1.32 to 6.5; the layer's lowest from 1.32 on, levelling near 0.69. At the tailored helium condition, Mach 3.41, both give 0.51; at hydrogen's, Mach 6, the wall gas gives 0.89 and the layer 0.64.

The gas a reflected shock refuses is inside the layer

A reflected shock in a shock tube bifurcates when the gas in the wall's boundary layer cannot be pushed into the reservoir behind it: its stagnation pressure, in the shock's frame, is below the reservoir's. Mark's criterion asks that of the gas at the wall, and in air it says the bifurcation stops above an incident Mach number of 6.5. But the gas that fails at high Mach numbers is not at the wall. It is inside the layer, heated by friction until it meets the shock too slowly for its sound speed, and on that measure a reflected shock in air bifurcates at every Mach number above 1.3.

compressible · Shock tube Each oblique shock added before the normal one buys less than the last. The best total-pressure recovery of a train of oblique shocks ending in a normal shock, against the number of oblique shocks, at Mach 2, 2.5, 3 and 4. At Mach 3 one oblique shock recovers 58.1%, two 74.4%, four 88.9% and eight 96.4%; a normal shock alone keeps 32.8%.

The last shock should be the weakest

An intake slows a supersonic stream through a train of oblique shocks and one normal shock, and the rule for laying out the train is that its shocks should be equally strong. Solved exactly rather than quoted, the rule is right for the oblique shocks to four figures and wrong for the last one: the terminal normal shock should be about five per cent weaker than the others, and the turns that give equal strengths are not equal turns but grow along the train.

compressible · Oblique shock At Mach 3 the cowl caps external compression at 88 per cent. Best recovery at Mach 3 against the number of oblique shocks, free and with the total turn capped at the free stream's detachment angle, 34.07°, which is as far as the cowl's outer surface can turn. Free, six shocks reach 94.2%; capped, 86.7%, and no number of shocks, nor a continuous isentropic ramp turning the same angle, can pass 88.5%.

The cowl caps what the ramps can buy

Every ramp in front of a supersonic intake turns the flow towards the cowl, and the cowl's outer surface has to turn the free stream at least as far. Past the free stream's own detachment angle it cannot, and that one angle caps every external intake: at Mach 3 no number of ramps, nor a smooth ramp, recovers more than 88 per cent, and the shock that is left at the throat is the strongest in the train. Above Mach 3 the cap falls fast, which is why fast intakes compress inside.

compressible · Oblique shock Behind the lip the ceiling is set by starting, not by the cowl. The best total-pressure recovery with every compression isentropic but the terminal normal shock, against free-stream Mach number: an external ramp turning to the free stream's detachment angle; the same ramp followed by the largest internal contraction a fixed duct can start through, Kantrowitz's limit; and the same ramp followed by a contraction to a throat Mach number of 1.2, which only a duct that opened its throat to start can hold. At Mach 3 the three are 88.5%, 95.1% and 99.3%; at Mach 4, 71%, 82.5% and 99.3%.

Compression behind the lip is bought by opening the throat

The cowl caps how far an intake's ramps can turn the flow, and moving some of the compression inside the duct escapes the cap. It does not escape the duct's other limit: a duct that contracts a supersonic stream has to swallow the shock that stands at its lip while it starts. With fixed walls that allows a contraction of about a tenth, and at Mach 3 six shocks with three inside recover exactly what six outside do. A duct that opens its throat to start and closes it after recovers 94 per cent — the internal compression is paid for by the moving wall, not by the walls being inside.

compressible · Oblique shock

Named alongside it

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

Total pressureModel limitEntropyNormal shockShock waveDetachmentMach numberOptimisationThe θ–β–M relationHypersonicIntakeIsentropic

All concepts