Subsonic and Supersonic Inlets

Capture area and spillage, pressure recovery, the pitot inlet, external compression and Oswatitsch's rule, mixed compression and unstart

Unstart

The SR-71 cruised at Mach 3.2 at 85,000 feet. In front of each engine, a conical spike that sat forward below Mach 1.6 moved aft as the aircraft accelerated, holding a carefully placed system of shocks inside the inlet. If the pressure behind it grew too high or the spike was in the wrong place, the shock system became unstable and shot forward to a stable position outside the cowl — an inlet unstart. It often blew out that engine's afterburner, and the thrust of the other engine yawed the aircraft violently while the stability system, autopilot and pilot fought to regain control (the SR-71). Every engine in this course so far started at station 0 with air delivered smoothly to its fan. At Mach 3, delivering it is the hardest job on the aircraft — and an inlet that does it badly can throw away a third of the engine's thrust before the compressor turns.

An inlet has three jobs. It must slow the air from flight speed to what the fan can accept — about Mach 0.5–0.6 at the fan face; keep as much stagnation pressure as possible; and deliver the flow uniformly, because a fan fed with distorted flow can stall (Lesson 6-2). It must do all three from take-off to top speed.

The figure of merit is the pressure recovery,

\pi_d = p_{t2}/p_{t0}

. Every point of recovery lost costs about a point of thrust and fuel (Lesson 4-2 found +0.01 of recovery worth 1.1 % of TSFC).

💡Capture area and spillage

The capture stream tube

A_0

is the free-stream area of the air the engine actually swallows. It is set by the engine, not the inlet:

In cruise it is usually a little smaller than the inlet's throat: the extra air spills around the lip, which costs spillage drag.
At take-off the engine swallows more than the lip area: the stream tube is larger than the inlet, and air is sucked in from the sides.

A subsonic inlet is a lip and a diffuser sized for the cruise flow, with a lip rounded enough to survive take-off without separating.

📐Sizing an airliner inlet

The turbofan of Lesson 4-2 swallows 374 kg/s in cruise at Mach 0.8 and 10.7 km (Lab 3):

Capture stream tube: $A_0 = 4.15$ m².
Throat, sized for Mach 0.75: 4.25 m² — so the stream tube is 2 % smaller and a little air spills.
Fan face at Mach 0.6: an annulus of 4.78 m², a diffuser area ratio of 1.12.
With a hub-to-tip ratio of 0.3, a fan tip diameter of 2.59 m.

\pi_d = \frac{p_{t2}}{p_{t0}}, \qquad \eta_{r,\,\mathrm{MIL}} = 1 - 0.075\,(M_0 - 1)^{1.35} \quad (1 < M_0 < 5)

Inlet pressure recovery, and the supersonic recovery that the US military specification MIL-E-5007D used as its standard of a good inlet.

Supersonic inlets. The simplest supersonic inlet is a pitot inlet: a subsonic inlet with a normal shock standing in front of its lip. It works well up to about Mach 1.6 — 93 % recovery at Mach 1.5 — and then collapses: 72 % at Mach 2, 33 % at Mach 3 (Lesson 2-2). Faster aircraft compress the air first through oblique shocks from ramps or cones, then finish with a weak normal shock at the lip: external compression.

💡Oswatitsch's rule

For a given number of shocks, which system keeps the most stagnation pressure? In the 1940s Klaus Oswatitsch showed that the best one gives its oblique shocks equal strength — equal normal Mach numbers — with the terminal normal shock a little weaker. At Mach 2.5 with two ramps (Lab 2): the best ramps are 13.35° and 15.73°, both oblique shocks have a normal Mach number of 1.441, the terminal shock 1.362, and the recovery is 0.868 — against 0.499 for a bare normal shock.

Derivation

Why equal shocks are best

For a weak shock the entropy rise grows as the cube of its strength:

\frac{s_2 - s_1}{R} \approx \frac{2\gamma}{(\gamma + 1)^2}\,\frac{(M_n^2 - 1)^3}{3}.

Each shock's stagnation-pressure loss follows its entropy rise,

p_{t2}/p_{t1} = e^{-\Delta s/R}

, so the total recovery depends on the sum of the cubes of the shock strengths

x_i = M_{n,i}^2 - 1

. Doing a fixed amount of compression with several shocks, the sum

\sum x_i^3

is smallest when the

x_i

are all equal:

x^3

is convex, so any unequal split costs more than an equal one. Strong shocks are not weak, so the exact optimum (Lab 2) is slightly different for the last, normal shock — but the logic is the same: never let one shock do the work of two.

📐Inlet recovery across the Mach range

Pitot, best single ramp, best two ramps and the military standard (Lab 1):

Mach 2: 0.721, 0.903, 0.957, 0.925.
Mach 2.5: 0.499, 0.751, 0.868, 0.870.
Mach 3: 0.328, 0.581, 0.744, 0.809.

Two ramps beat the standard at Mach 2 and just meet it at 2.5. At Mach 3 they are not enough — which is why the SR-71's inlets used a moving spike and mixed compression, finishing the compression inside the duct.

💡Mixed compression and unstart

A mixed-compression inlet does part of its compression inside the duct, with oblique shocks reflecting between its walls and the terminal shock standing just downstream of an internal throat. It gives high recovery at high Mach number and low drag — but the terminal shock sits in an unstable position. If the engine suddenly needs less air, or the pressure behind the shock rises, the shock moves forward past the throat and is expelled: an unstart, with a sudden loss of recovery and flow, a jump in drag and violently distorted flow at the fan. Mixed-compression inlets therefore need active control: moving spikes or ramps and bleed doors.

If you are looking for perfect safety, you will do well to sit on a fence and watch the birds; but if you really wish to learn, you must mount a machine and become acquainted with its tricks by actual trial.

– Wilbur Wright, speech to the Western Society of Engineers, 18 September 1901 (Wikiquote)

⚠Distortion

A fan does not just need enough stagnation pressure, it needs it evenly spread. Separation on a lip in a crosswind, a shock that has moved, or the wake of the fuselage ahead of a buried inlet leave a region of low stagnation pressure at the fan face. The blades passing through it are pushed toward stall each revolution — the inlet–engine compatibility problem every new aircraft has to clear (Lesson 6-2).

💡Where you will use this

Lesson 6-2 shows how distorted flow eats a compressor's surge margin. Lesson 8-1 treats the inlet as one of the matched components. Ramjets and scramjets (Lesson 3-3) are mostly inlet: at Mach 5 the inlet does all of the compression.

📚Further reading

J. Seddon and E. L. Goldsmith, Intake Aerodynamics (2nd ed., Blackwell, 1999).
Jack D. Mattingly and Keith M. Boyer, Elements of Propulsion: Gas Turbines and Rockets (2nd ed., AIAA, 2016), on inlets.
The SR-71 on Wikipedia, the section on its inlets and unstarts.

↓But there is a problem

The inlet slows the air and squeezes it; the core adds energy; at the other end, the nozzle must turn all of it back into velocity, matching the jet's pressure to the air around it. Watch an afterburning fighter take off at night and you can see how well it manages: a row of bright diamonds stands in its plume.