Compression behind the lip is bought by opening the throat

The cowl caps what the ramps can buy found that every ramp in front of a supersonic intake turns the flow towards the cowl, that the cowl’s outer surface must turn the free stream at least as far, and that the free stream’s own detachment angle therefore limits the whole external turn: at Mach 3 no external intake recovers more than 88.5 per cent of its total pressure, and at Mach 4 no more than 71. It ended on the way intakes for those speeds actually go: they compress inside. The cowl’s lip can be turned back towards the axis, and the reflected shocks inside the duct can do the compressing the ramps were not allowed to finish. The cowl’s outer surface then only has to follow the part of the turn that happens outside.

That is true, and it moves the limit rather than removing it. A duct that slows a supersonic flow by narrowing has a condition of its own to meet, and it is met or not at start-up, before any of the design point’s shocks exist.

The cap does not reach inside #

The cap comes from the outside of the cowl. Inside, the flow turned by the ramps meets the cowl’s inner surface and is turned back by a shock that the duct’s walls make, and the duct can turn the flow as far as its own geometry asks without the free stream having any say. The compression an intake does inside therefore does not count against the detachment angle, and the question becomes how much compression the inside can hold.

The cowl’s cap was a wedge’s — the angle at which a wedge becomes too blunt for an attached shock, read on the cowl’s outer face — and nothing inside the duct is a wedge in the free stream. An internal shock can turn the flow by any angle its own upstream Mach number allows, and that Mach number is the duct’s, not the flight’s.

The answer depends on what the duct does to the flow’s area. Supersonic flow slows as its passage narrows — the duct that works backwards is the reason — so internal compression is a contraction, from the lip’s area to a smaller throat. Treated one-dimensionally, with mass conserved and the area per unit mass flux proportional to $f(M)/p_0$, where $f = A/A^*$, a compression from the lip’s Mach number $M_L$ to the throat’s $M_t$ through shocks of total-pressure ratio $r_i$ contracts the duct by

$$\frac{A_L}{A_t} = r_i\,\frac{f(M_L)}{f(M_t)}.$$

The more compression inside, the smaller the throat. And a small throat is exactly what cannot be started.

A duct has to swallow its own shock #

When an intake accelerates through Mach one, or when its flow is first established at speed, there is no oblique-shock system inside it yet. There is a normal shock standing in front of the lip, with subsonic flow behind it at a lower total pressure, and the duct must pass that flow through its throat. If the throat is too small, the subsonic flow behind the shock chokes there, the shock cannot be drawn in, and the intake stays unstarted: a bow shock ahead of the lip, most of the air spilled round it, recovery of a single normal shock at the flight Mach number. The largest contraction that still lets the shock through is the one that passes the post-shock flow at Mach one in the throat,

$$\left(\frac{A_L}{A_t}\right)_{\max} = f(M_2),$$

with $M_2$ the Mach number behind a normal shock at $M_L$ — Kantrowitz and Donaldson’s starting limit of 1945. Computed two ways, as $f(M_2)$ and as $f(M_L)$ times the normal shock’s total-pressure ratio, it agrees to the last digit, as it must.

The allowance is small where it matters. At the Mach number an external intake at Mach 3 hands its lip, 1.63, a fixed duct may contract the flow by 12.6 per cent. The same duct, once started, could hold a contraction of 23.6 per cent and still keep its throat at Mach 1.2. At a lip Mach number of 2.5 the gap is 1.32 against 2.56. The starting rule allows a fraction of what the started duct could use, and the fraction falls as the lip’s Mach number rises — which is where internal compression was supposed to help most.

Busemann’s biplane met the same rule in two dimensions: its channel’s contraction was right at the design Mach number and could not be reached from below, because the shock that formed on the way up could not be swallowed. Every fixed-geometry supersonic diffuser has that problem, and every one solves it either by staying under Kantrowitz’s limit or by changing shape.

Three ceilings #

With every compression isentropic but the last, the ceilings come out in closed form.

At Mach 3 the external ceiling is the cowl essay’s 88.5 per cent. A fixed duct that adds the largest startable contraction behind the lip reaches 95.1: the throat now receives Mach 1.43 rather than the lip’s 1.63, and the normal shock there is that much weaker. A duct that was started open and then closed to a throat Mach number of 1.2 reaches 99.3 per cent — and it reaches 99.3 at every Mach number, because its ceiling is no longer set by the cap or by starting but only by the margin it keeps in its throat. At Mach 4 the three are 71.0, 82.5 and 99.3.

The ceilings say where the room is, and they flatter the fixed duct. Isentropic internal compression has no shocks to pay for; a real duct turns its flow back with shocks reflected between the cowl and the ramp, and those shocks cost total pressure and change how much contraction they need.

With shocks, the fixed duct gains nothing #

Give the design six oblique shocks and let it put each one outside or inside, under the cap and under one starting rule or the other, optimising every strength together. At Mach 3 all six outside recover 86.7 per cent. Three outside and three inside a duct that must start itself recover 86.7 per cent, the same to a part in five thousand, and the third internal shock adds nothing at all: the contraction sits exactly on Kantrowitz’s limit, 1.110, and an extra shock inside could only spend total pressure that the limit gives no room to use. Three and three in a duct started open recover 94.1.

The fixed duct’s internal compression is not worthless; it is worth exactly as much as more ramps would be. Three capped external shocks alone recover 81.7 per cent at Mach 3, the cowl essay’s figure. Add one shock inside a self-starting duct and the recovery rises to 86.7, the throat now receiving Mach 1.37 instead of the lip’s 1.57 — and a second and third internal shock add nothing, because that first one has already used the whole contraction the starting limit allows. One internal shock does everything a fixed duct’s inside can do, and what it does is equal, to the fourth figure, to what three more ramps under the cap would have done. Both are spending the same thing: the room between the Mach number the cap leaves at the lip and the Mach number a normal shock can be made to act at.

The isentropic ceiling above says a perfectly smooth fixed duct could do better, 95.1 per cent at Mach 3, by turning its external ramp all the way to the cap and contracting inside without shocks at all. That is the compression that costs nothing carried inside a duct, and it is as fragile there as outside: a smooth internal contraction focuses its Mach waves into a shock as soon as the flight Mach number moves off its design point, and the reflections a real duct’s walls produce are shocks from the start. The ceiling says where the room is; the design with shocks says how little of it a fixed duct can reach.

Across the range the pattern holds. The self-starting mixed design never passes the all-external one; above Mach 3.5 it falls behind it, because three external shocks under the cap hand the lip a flow the startable contraction can do little with. The started design pulls away: 97.0 against 94.5 at Mach 2.5, 83.5 against 67.7 at Mach 4. What makes mixed compression worth building is not that some of the turn is inside; it is that the duct’s throat can change.

Where the shocks go when the throat can move #

The started design rearranges the whole train. With three shocks outside and none inside, the ramps turn the full 34.1° the cap allows and hand the lip Mach 1.56, and the recovery is 80.5 per cent. With three inside as well, the ramps turn only 20.2° and hand the lip Mach 2.10 — a faster flow than before — and the internal shocks take it down to the throat’s 1.20. Half of the Mach number’s fall from 3 to the throat happens inside the cowl. The external ramps are no longer pressed against the cap at all; the cap stops binding the moment the duct can take the compression the ramps were straining to do.

That is the opposite of what the external optimum taught. Under the cap, the best train turned as far as it was allowed and left the rest to the terminal shock. With internal compression available, the best train turns less outside, accepts a faster lip, and spends its strength inside where no cap applies — the cowl’s own outer surface then meets the free stream at twenty degrees rather than thirty-four, which is also a smaller cowl drag.

The margin’s price #

A throat at exactly Mach one would recover the most and would unstart at the first disturbance: a small rise in the engine’s back pressure moves the terminal shock forward, through the throat, and out of the lip, and the intake’s flow collapses to the unstarted state the start-up had to escape. The throat margin is insurance against that. At Mach 3 a margin of 1.1 recovers 94.2 per cent and 1.2 recovers 94.1 — the shocks rearrange to keep the cost small — but by 1.4 the recovery has fallen to 90.2. The dial shows the price rising with the flight Mach number, because the shock the throat must hold back becomes stronger as the whole system does.

When the walls pay for themselves #

The started design’s lead over the all-external one is 2.5 points at Mach 2.5 and 7.4 at Mach 3, crossing five near Mach 2.8, and 15.8 at Mach 4. Against it stands everything the inviscid model leaves out: a longer wetted duct, the boundary-layer bleed every mixed intake needs to keep its internal shocks from separating the layer they reflect from, and the bleed air’s own drag. Those are a few points of recovery or of thrust-equivalent drag, and the crossover of five points near Mach 2.8 is where the internal walls start to pay for them clearly. The military requirement, $1 - 0.075(M - 1)^{1.35}$, says the same thing from the other side: six external shocks meet it to about Mach 4.3, so below Mach 3 an external intake with movable ramps is enough, and above it the margin to the requirement is what a mixed intake buys.

Aircraft built for those speeds fall on both sides of the line in the order the curve predicts. Fighters designed for Mach 2 to 2.5 use external compression with ramps that move. The Lockheed SR-71, cruising above Mach 3, used an axisymmetric mixed-compression intake whose spike translated aft as it accelerated — a throat that changed with speed — and its unstarts, when the terminal shock was expelled, were violent enough to yaw the aircraft. The XB-70, built for Mach 3, used a two-dimensional mixed intake with a variable throat. Neither is the self-starting duct; both are the started one, and both paid in mechanism for the margin the curve shows.

What the cowl gains when the ramps turn less #

One consequence of the rearrangement is outside the duct. The cowl’s outer surface meets the free stream at the external turn, and the pressure on it is a drag the intake pays whatever happens inside. The capped external intake at Mach 3 turns 34 degrees, and the cowl essay found its outer pressure coefficient above one just short of detachment: a cowl that is nearly a bow-shock generator. The started mixed design turns 20 degrees outside, and an oblique shock at Mach 3 turning 20 degrees has a pressure coefficient of about 0.4 behind it. The internal compression is therefore paid for twice over by the variable throat: once in recovery, and once in the cowl drag that the smaller external turn removes. Neither is available to a duct that must start itself, whose best design still presses its ramps against the cap.

The convention: what is counted and where #

Recovery is the ratio of total pressures from the free stream to just behind the terminal shock, the quantity an engine’s thrust depends on. The external train is planar, its oblique shocks exact; the internal part is quasi-one-dimensional, its shocks carried as jumps in Mach number and total pressure with the area following from mass conservation, so the turning that a real reflected shock does inside the duct is not tracked. Kantrowitz’s limit is applied to the internal contraction from lip to throat for the fixed duct, and a minimum throat Mach number of 1.2 for the started one. The flow is inviscid, the gas perfect with $\gamma = 1.4$, and every optimum is over the strengths of all six shocks at once, started from a grid of designs and built up one shock at a time, so that adding a shock can never lose recovery to a poor start.

What the inviscid duct leaves out #

Boundary layers and bleed, which this whole argument prices only by comparison. Shock reflections in two dimensions: a real internal shock reflects from a wall whose boundary layer it thickens or separates, and the reflected shock’s strength is not the one-dimensional jump’s. Unstart and restart: a duct with a throat that moves has hysteresis, and the conditions for restarting after an unstart are different from those for the first start. Buzz, the oscillation an intake running close to its unstart can fall into, which is a shock that will not stand still in a duct rather than on a wing. Three dimensions, and the axisymmetric intakes most fast aircraft used, whose internal compression is around a centrebody rather than between two plates; a cone’s shock sees the whole capture differently from a ramp’s, and its internal passage is an annulus whose area changes with radius as well as with gap.

Who drew the limit #

Oswatitsch gave the rule for external trains in 1944 — equal strengths, because a shock’s loss grows as the cube of its strength, refined in the essay that made the last shock the weakest. Kantrowitz and Donaldson, studying supersonic diffusers at the NACA, derived the starting limit in 1945, and the two together are the frame every supersonic intake has been designed in since: compress outside up to the cowl’s cap, compress inside up to the starting limit, and pay for anything beyond with a wall that moves. The military specification of intake recovery dates from the late 1950s, when the first Mach 3 designs were drawn. The numbers here are those rules optimised exactly against each other, and the conclusion they give — that a fixed duct’s internal compression is nearly worthless and a variable one’s is decisive — is the reason the fast intakes that were built all moved.

Still open: how hard a disturbance must push #

The margin of Mach 1.2 is a convention, and it is the one number in this essay chosen rather than computed. A designer who picks 1.1 gains a tenth of a point at Mach 3 and narrows the room the terminal shock has to move in; one who picks 1.4 gives up four points for room the engine may never need. The right margin is a property of the disturbances the intake meets — a gust’s change of incidence, a compressor’s surge, a change of engine demand. Each raises the pressure behind the terminal shock, and the shock moves forward until the duct’s own area change balances the rise, or until it passes the throat and the intake unstarts. The next calculation follows the terminal shock through the duct for a stated rise in back pressure, finds the rise that pushes it out of the throat at each margin, and asks how much recovery a designer must trade for each per cent of back-pressure tolerance — which would turn the margin from a number chosen by habit into one chosen by the disturbances an engine produces.