The cowl caps what the ramps can buy

The last shock should be the weakest optimised a supersonic intake’s train of oblique shocks exactly and found Oswatitsch’s equal strengths for the oblique ones, a terminal normal shock about five per cent weaker, and recoveries that climb towards one as the shocks multiply: at Mach 3, 88.9 per cent with four oblique shocks and 96.4 with eight. It also found that those trains turn the flow further and further — 42.8° with four shocks at Mach 3, approaching the 49.8° that would take the flow isentropically to Mach one — and that from three shocks on, the turn passes an angle a cowl cannot follow.

That angle is the subject here. It is not a property of the ramps, which can be built to any shape, but of the free stream outside the intake, and it caps what any arrangement of ramps can do.

Why the cowl sets a limit #

The ramps turn the captured flow towards the cowl, and the cowl’s lip receives it. The lip’s inner surface must lie along the flow arriving at it, or the lip itself makes a shock inside the intake; so the lip is inclined to the free stream by the train’s total turn. The cowl has thickness, and its outer surface, which faces the free stream, is inclined at least as steeply as its inner one. The free stream meeting that outer surface is turned by it, through an oblique shock on the outside of the cowl.

An oblique shock exists only up to a maximum turn at each Mach number — 22.97° at Mach 2, 34.07° at Mach 3, 38.77° at Mach 4 — past which the wedge is too blunt and the shock stands off as a curved bow wave. A cowl whose outer surface asks for more than that pushes a detached shock ahead of its lip, the flow behind it is subsonic, and the intake spills air around its lip rather than swallowing it. The train’s total turn is therefore capped by the free stream’s detachment angle, however the turn is divided.

Recovery under the cap #

Put the cap into the optimisation and the picture changes from the third shock. At Mach 3 one and two oblique shocks are unaffected: their best trains turn 22.6° and 33.8°, inside the 34.07° cap. Three capped shocks recover 81.7 per cent against 83.6 free; four, 84.6 against 88.9; six, 86.7 against 94.2. The free curve keeps climbing towards one and the capped one bends over.

Its limit is a smooth ramp. A train of many weak shocks becomes, in the limit, a compression that costs nothing in the end — an isentropic turn, with the Mach number falling as the Prandtl–Meyer angle falls — and the most such a ramp can turn is the cap. From the Prandtl–Meyer angle at Mach 3, 49.76°, a 34.07° turn leaves 15.69°, which is Mach 1.63, and a normal shock at Mach 1.63 recovers 88.5 per cent. No external compression at Mach 3 can recover more than 88.5 per cent, whatever its ramps.

The capped optimum is solved with the cap as an equality, since it binds: the first $n - 1$ strengths are searched and the last is whatever closes the turn to exactly the cap, so the search moves along the constraint rather than bouncing off a penalty wall. The six-shock train’s turn equals the cap to $10^{-9}$ degrees and its recovery is below the isentropic ceiling, as it must be.

The terminal shock takes what the ramps cannot #

The cap reverses the shape of the best train. Free, four oblique shocks at Mach 3 are equally strong at a normal Mach number of 1.326 and the terminal shock is weaker, at 1.261. Capped, the oblique shocks must turn less, so they weaken — to about 1.26, nearly equal still, the strongest last — and the compression they can no longer do is left to the one shock that turns nothing: the terminal shock now acts at 1.59, the strongest in the train. The loss goes as the cube of strength, and the cap has concentrated the strength in exactly one place.

The pattern holds at every train length the search was run at. With three capped shocks at Mach 3 the oblique ones sit at 1.362, 1.366 and 1.373 and the terminal one at 1.568; with six, all six oblique shocks at 1.167 and the terminal at 1.613. As the oblique shocks multiply they weaken towards Mach waves, and the terminal shock climbs towards the isentropic ramp’s 1.628, which is the limit both the recovery and the terminal Mach number approach.

That is the mechanism of the ceiling, read from the inside. Equal shocks are the right answer when every shock can be made weaker by adding more of them. Under the cap, adding oblique shocks cannot weaken the terminal one, because the flow they hand it is fixed by the total turn, and the terminal shock’s loss is the floor that no ramp arrangement lifts.

An intake at Mach 2.5, both ways #

The cap binds most usefully in the middle of the range, where it is a design choice rather than a wall. At Mach 2.5 the free stream’s detachment angle is 29.8°. Two oblique shocks, at their best, turn the flow 29.1° and recover 86.8 per cent: inside the cap, and the answer either way. Three free shocks would turn 33.0° for 92.3 per cent; capped, they turn 29.8° and recover 91.5. Four free shocks would turn 35.0° for 95.0; capped, 93.3. Each capped train hands its terminal shock a little more Mach number than the free one — 1.38 with three shocks, 1.39 with four — and each extra capped shock buys about a point and a half where the free one bought two or three.

So at Mach 2.5 the cap costs a point or two on a three- or four-ramp intake, and a designer can choose between living with it and moving some compression inside. That is the speed at which both kinds of intake have been built. What a shock costs gave the general reason for many small shocks; at Mach 2.5 the cowl starts to ration them.

Why the cap falls behind #

The reason the ceiling collapses above Mach 3 is a race between two angles. The detachment angle grows with Mach number, but slowly and towards a limit: 23.0° at Mach 2, 34.1° at 3, 38.8° at 4, 44.4° at 10 and 45.5° at 50, approaching 45.6° for a perfect gas of $\gamma = 1.4$ at infinite speed. The Prandtl–Meyer angle, which measures how much turning an isentropic compression needs to bring the flow to Mach one, grows much faster: 26.4° at Mach 2, 49.8° at 3, 65.8° at 4, and towards 130.5° in the limit. The cowl lets the ramps remove a share of that angle that falls from 87 per cent at Mach 2 to 68 at 3 and 59 at 4, and the rest has to be removed by the terminal shock in one step — the most expensive way there is. An expansion could undo any amount of turning for nothing, as turning the other way is free showed; a compression is rationed by the one angle at which the free stream will still accept it.

That race is the whole content of the ceiling, and it is set by the gas rather than the hardware. A gas with a lower ratio of specific heats has a larger Prandtl–Meyer angle and a larger detachment angle both, and which grows faster would decide whether its external intakes do better or worse; the calculation here is for air. A shock that leans is where both angles come from: a shock turns the flow by changing only the velocity component across it, and the most it can turn is reached when that component is as large as it can be made subsonic behind.

The ceiling, at every Mach number #

Swept over Mach number, the ceiling has a shape worth knowing. It is 0.993 at Mach 2, where the cap of 23.0° takes the flow from a Prandtl–Meyer angle of 26.4° to one of 3.4°, Mach 1.19, at which a normal shock costs almost nothing. It is 0.955 at Mach 2.5, 0.885 at Mach 3, 0.798 at 3.5 and 0.710 at 4. The detachment angle grows slowly with Mach number — it approaches 45.6° only at infinite speed — while the Prandtl–Meyer angle that has to be removed grows quickly, so the turn the cowl permits covers a falling share of the compression the flow needs.

A common design requirement for military intakes, $1 - 0.075(M-1)^{1.35}$, sits below the ceiling throughout, and the margin closes from 6.8 points at Mach 2 to under five near Mach 3.7. Real ramps lose some of that margin to their boundary layers and to the bleed that controls them, and some to spillage. Above Mach 3 a purely external intake that meets the requirement inviscidly may not meet it in metal.

What the terminal shock is handed #

The same arithmetic says what the throat sees. An isentropic ramp turned to the cap hands the terminal shock Mach 1.19 at a free stream of Mach 2, 1.41 at 2.5, 1.63 at 3 and 2.02 at 4; capped trains of real shocks hand it about the same, a little higher for short trains. A normal shock at Mach 1.3 or so is commonly taken as the strongest a turbulent boundary layer survives without separating, and the ceiling’s terminal shock passes it at Mach 2.25. Past that, the terminal shock of an external intake sits on a separated layer at its best operating point, which costs recovery the inviscid ceiling does not show and makes the shock’s position unsteady — the instability intake designers call buzz has its roots there.

The cowl’s own price #

Even inside the cap the cowl is not free. Its outer surface, inclined at the train’s total turn, carries the pressure behind its own oblique shock on an area that faces forward, and the pressure coefficient there rises with the angle — to 0.93 just short of detachment at Mach 2 and 1.18 at Mach 3. That is wave drag on the outside of the intake, paid for by the recovery gained inside it, and it rises steepest just where the cap is approached. A designer choosing ramps therefore does not run the train to the cap: the last degrees of turn buy the least recovery and cost the most cowl drag, and the best installed intake stops short of the inviscid optimum on both counts. Pricing that trade needs the engine’s thrust as well as the intake’s recovery, and is the calculation this one sets up rather than makes.

A cone does better, and why the planar case is the one to state #

A conical spike is the axisymmetric version of the ramp, and a cone intake’s shock sees the whole capture found its compression continuing behind its shock. Its cowl faces a free stream that the spike has turned less near the lip, and a cone’s own detachment angle is larger than a wedge’s at every Mach number, so an axisymmetric intake’s cap is looser and its ceiling higher. The planar numbers here are therefore the conservative bound — the ceiling a two-dimensional intake of the kind fighters carry faces, and below the one a nose spike faces. Both fall above Mach 3, for the same race between two angles, and both are the reason the compression eventually has to move inside the duct.

External, mixed, internal #

The ceiling is the reason intakes for different speeds look different. Below Mach 2 a pitot intake, a plain hole, loses a few per cent and nothing more is worth building. Around Mach 2 an external intake with two or three ramps reaches the high nineties, inside a cap it barely touches. Around Mach 3 the cap binds hard: an external intake cannot pass 88 per cent and hands its terminal shock Mach 1.6. The way out is to move some of the compression inside the duct, behind the cowl lip, where the turned flow can be turned back by the duct’s own walls rather than by the free stream: mixed compression. Inside, the duct contracts to slow the supersonic flow — the duct that works backwards is the reason a narrowing passage decelerates a supersonic stream — and the cowl’s outer surface no longer has to follow the whole turn, because the duct’s walls take some of it back. Its price is a duct that must swallow its own starting shock, a problem Busemann’s biplane has in two dimensions and every mixed intake has in three, and the unstart that follows when the shock is pushed out.

How the capped trains were checked #

The six-shock capped train at Mach 3 turns to the detachment angle to $10^{-9}$ degrees and recovers 0.98 of the isentropic ceiling, below it as any train of finite shocks must be. With two shocks at Mach 3, where the free optimum turns 33.8° and the cap is 34.1°, the capped and free searches agree to $10^{-15}$: the cap is not binding and the calculation knows it. The detachment angles are the tables’ 22.97° and 34.07°, from the same θ–β–M relation as every shock here.

The convention: the cap is the free stream’s #

The cap is the free stream’s detachment angle, which assumes the cowl’s outer surface is inclined at exactly the train’s total turn — a cowl of no thickness. A real lip with an included angle of a few degrees reaches detachment a few degrees sooner, and every capped number here is then an upper bound. Recovery is the product of total-pressure ratios to the throat; the isentropic ceiling is a ramp whose waves do not coalesce before the cowl, which a real ramp’s do, focusing into a shock near the lip.

What the planar cap leaves out #

Axisymmetric intakes. A cone’s compression continues behind its shock, and a spike’s outer cowl faces a free stream that has not been turned, so the cap there is set by a cone’s detachment rather than a wedge’s, which is larger; the planar numbers are the conservative case. Viscous effects: the boundary layer on the ramps thickens the effective ramp and turns the flow further, eating into the cap from the inside. Off design. The cap is the free stream’s at the design Mach number, and below it the cap is smaller and the same ramps detach the cowl’s shock: an intake designed to its cap at Mach 3 spills at Mach 2.8. And the cowl’s drag is computed and not traded against anything here.

Who drew the line #

That a cowl’s outer surface limits external compression is in every intake design text, usually as a rule that external compression is economical up to about Mach 2.5 and mixed compression above. The Concorde’s intakes, at Mach 2, were external with variable ramps; the SR-71’s, at Mach 3.2, were mixed, with a translating spike. The numbers here put a ceiling under that rule, from the same θ–β–M relation and Prandtl–Meyer function the rest of the subject uses, and the ceiling’s fall above Mach 3 — 88 per cent at 3, 71 at 4 — is the rule’s reason written as a curve.

Still open: compression behind the lip #

Mixed compression puts some of the turn inside the duct, where the cowl’s inner wall turns the flow back towards the axis through a shock reflected off the ramp side, and only the external part of the turn faces the cap. The next calculation splits a train between an external part capped by the cowl and an internal part bounded instead by the duct’s ability to start — the largest internal contraction whose starting shock the duct can swallow — and asks what recovery the best split reaches at Mach 3 and 4, how much of the compression the best split puts inside, and at what Mach number a mixed intake’s ceiling first passes the external one by more than its boundary layers can be expected to take back.