An overexpanded nozzle lets go before its shock arrives

One area, two answers worked through the sequence every textbook of gas dynamics draws for a convergent–divergent nozzle as its back pressure is raised. At the design pressure the flow leaves the exit at exactly the ambient pressure. Raise the back pressure a little and the nozzle still runs full, supersonic to the exit, and the flow adjusts outside through oblique shocks from the lip. Raise it until a normal shock can stand at the exit plane, and then further, and the shock walks up into the nozzle to the place where the subsonic flow behind it can decelerate and leave at the back pressure. The calculation is clean, the shock is a discontinuity, and the duct is inviscid. Friction moves the sonic point past the throat put friction back into the same duct and ended by naming what it still had not: every nozzle it drew ran full, and a real nozzle, overexpanded far enough, does not.

This essay takes a rocket nozzle through the same sequence twice. Once as the textbook does, inviscid; and once with the one fact the textbook leaves out, which is that the flow next to the wall is a boundary layer, and a boundary layer can climb only a modest pressure rise before it lets go. The two sequences disagree over most of the range in which a sea-level rocket engine actually works.

A rocket nozzle, three ways #

The nozzle is a bell-like divergent section — a bell rather than a cone, which sprays its jet sideways — with an exit area forty times its throat’s, the proportions of a first-stage engine meant to work from sea level upwards. The gas is combustion products, with a ratio of specific heats of 1.2. At its design point it expands to Mach 4.24 and a pressure 479 times below the chamber’s, so a chamber at a hundred bar is perfectly expanded only at about a fifth of a bar — high in the atmosphere. At lift-off it is overexpanded, and the question is what that does.

The flow is treated as quasi-one-dimensional, in three ways. Inviscid, as the textbook does: full to the exit while the outside can recompress it, then a normal shock placed inside so that mass is conserved through the jump — the shock raises the sonic area by the inverse of its stagnation pressure ratio — and the subsonic flow behind it leaves at the ambient pressure. Separated: the flow follows the isentropic expansion along the wall until the wall pressure falls to the separation pressure, and beyond that point the wall sits at ambient and carries no load, with the jet running free of it. The separation pressure is Summerfield’s criterion, 0.4 of ambient, or Schmucker’s, which falls gently with the Mach number there. Adapted: an ideal nozzle cut at every pressure ratio to expand exactly to ambient, the most any nozzle can give.

Where the flow leaves the wall #

The first figure puts the two sequences side by side. The inviscid nozzle runs full all the way down to a pressure ratio of 24.5, where a normal shock finally stands at the exit, and below that the shock walks inside: at a pressure ratio of 10 it is a fifth of the way along the divergent section, at 5 a tenth.

The separated nozzle lets go far earlier. By Summerfield’s criterion the wall flow separates at the exit when the pressure ratio falls to 192 — two-fifths of the design value — and by Schmucker’s at 138. Below that the separation point moves steadily upstream: at a pressure ratio of 100 it is a third of the way along by Summerfield’s criterion and half by Schmucker’s; at 40, about a sixth. Across the whole band from 24.5 to between 138 and 192 — a factor of six to eight in pressure ratio — the textbook’s nozzle runs full and the real one does not. And below 24.5, where the textbook’s shock is inside the nozzle, the real flow has already left the wall well upstream of where that shock would be.

For a first-stage engine the band is the one that matters. A chamber at a hundred bar at sea level is a pressure ratio of a hundred, squarely inside it.

Why a boundary layer cannot climb a normal shock #

The second figure shows the difference in its simplest form. For the inviscid shock to stand at a station, the pressure behind it must be the ambient pressure, so the pressure in front must be the ambient pressure divided by the shock’s own pressure ratio — at Mach 4, about a seventeenth of ambient. The wall flow would have to arrive at a seventeenth of ambient and then be compressed back in one step.

The flow next to the wall cannot do that. A boundary layer is slow near the wall, and how much uphill a layer can take is limited: the slow fluid near the wall has only its own momentum to push against a rising pressure, and when the rise is too steep it stops and reverses. A shock imposes the steepest rise there is. What happens instead is that the layer separates ahead of where the shock would be, the separated flow turns the main stream through a small angle, and the turn makes an oblique shock that is much weaker than a normal one — the pressure rise the layer can survive. Measured across many nozzles, separation happens when the wall pressure has fallen to between about a quarter and a half of ambient: Summerfield’s 0.4 is the round number from the 1950s, and Schmucker’s correlation lets it fall from about a half at Mach 2 to 0.3 at Mach 4. Either way, the layer separates at a pressure rise of two to four, not seventeen.

Along the wall #

The third figure follows the wall at a pressure ratio of 40, well inside the band. Running full, the flow expands to 0.08 of ambient at the exit: the outer part of the nozzle is a surface held at a twelfth of the outside pressure, and the air outside recompresses the jet through shocks beyond the lip. Separated, the wall pressure falls only to the criterion — 15 per cent of the way along by Summerfield’s, 18 by Schmucker’s — and beyond that point the wall sits at ambient. From there on the nozzle’s bell is a shroud round a jet that is no longer using it.

In a real separated nozzle the wall pressure after separation does not jump straight to ambient. It rises through a short interaction region to a plateau somewhat below ambient, and recovers slowly towards the exit as the recirculating air, drawn in from outside, feeds the region between the jet and the wall. The model’s step is the simplest version of that and the one Summerfield’s criterion was built for.

The jet after it lets go #

Once the flow has left the wall it is a free jet, and its boundary is no longer the bell but the ambient air. At the separation point the jet is still supersonic and still below ambient pressure — at the criterion, four-tenths of it — and it adjusts the way any overexpanded jet does, through a shock that leans from the separation line inwards, which turns the jet’s edge away from the wall and raises its pressure towards ambient. Inside the bell the jet is therefore narrower than the nozzle, bounded by a separated shear layer, with slowly recirculating air between it and the wall that has been drawn in from outside past the lip.

Beyond the exit, the jet carries the pattern of shocks and expansions that any imperfectly expanded jet carries — the diamonds in a rocket’s plume — starting from the separation line rather than from the lip. That is why the plume of an engine at lift-off can look much narrower than its bell: the flow the eye sees left the wall some way inside it.

Separation is what keeps the thrust #

The fourth figure is the consequence, and it reverses the usual feeling that flow separation is a loss. The thrust coefficient — thrust over chamber pressure times throat area — is set by the momentum and pressure of the flow where it leaves the nozzle. The adapted nozzle, cut to expand exactly to ambient at each pressure ratio, is the most any nozzle can do. The fixed nozzle running full loses thrust quickly below its design point, and the reason is the wall in the third figure: every part of the bell at a pressure below ambient is pulled backwards by the air outside, and at a pressure ratio of 40 that pull takes the thrust coefficient down to 0.88. With the textbook’s shock inside, below a pressure ratio of 24.5, the thrust would collapse almost entirely — 0.21 at a pressure ratio of 20 — because the flow behind the shock is subsonic and leaves slowly.

The separated nozzle simply stops using the wall where the wall would pull it back. At a pressure ratio of 40 its thrust coefficient is 1.47, two-thirds more than the full-running nozzle’s and within four per cent of the adapted nozzle’s 1.52. At 100 it is 1.60 against the adapted 1.64. Separation turns a fixed nozzle into something close to an adapted one, cut automatically at about the right place, because the criterion that decides where the flow leaves the wall is close to the condition that decides where the wall stops helping. At the moment separation begins the two thrusts are exactly equal, and below it the separated one is always larger.

Put as a loss against the adapted nozzle, the difference is stark. At a pressure ratio of 40 the full-running nozzle gives up 0.64 of the adapted thrust coefficient — forty-two per cent of it — to the part of its bell that sits below ambient. The separated nozzle gives up 0.06. The familiar formula for an overexpanded nozzle’s penalty, the pressure mismatch times the exit area, is correct while the flow is attached and overstates the loss tenfold once it is not, because the part of the exit area it charges is no longer carrying the flow.

Twice the size it expands to #

The fifth figure turns the result into a design rule. For any pressure ratio, the adapted nozzle has the area ratio that expands the flow exactly to ambient; the largest nozzle that still runs attached has the area ratio at which the exit pressure is 0.4 of ambient. The second is about twice the first at every pressure ratio: at a pressure ratio of 100, an adapted area ratio of 12 and a largest attached one of 24; at 200, 20 and 41.

That factor is why first-stage engines look oversized for sea level. An engine that must work from lift-off to the thin air of forty kilometres gains thrust all the way up from a larger bell, and the largest bell it can carry is set by what separates at lift-off. Designers push to the edge of the attached region, and several engines run with their exit pressure close to Summerfield’s line at sea level for exactly that reason. The same logic drives the designs that try to be adapted at every altitude — extendible nozzles, dual-bell nozzles that separate deliberately at a designed step, and plug nozzles whose jet boundary is the ambient air.

Where the textbook’s sequence does happen #

The inviscid sequence is not wrong everywhere, and it is worth saying where it holds. It needs the pressure rise across the shock to be taken by the whole stream, and that happens when the flow is confined and the shock does not have to meet a free boundary layer at a much lower pressure: in a supersonic wind tunnel starting up, where the throat stops listening to the pressure behind it and the starting shock must be swallowed through the test section, or in a duct run backwards as a diffuser, where the shock stands near a second throat. There the pressure rises across the shock are modest, the boundary layers are thin compared with the channel, and a shock inside the duct — with a lambda-shaped foot where it meets each wall — is what is seen.

A rocket nozzle is the opposite case. Its exit pressure at lift-off is a small fraction of ambient, the pressure rise the textbook’s shock would need is large, and the boundary layer at the wall, at the lowest pressure in the nozzle, is the weakest part of the flow. It lets go first.

What was checked #

The sixth figure is the ledger. The inviscid shock placed inside the nozzle satisfies its two conditions — the sonic area rises by the inverse of the shock’s stagnation-pressure ratio, and the subsonic exit is at ambient — to $2\cdot10^{-13}$ at three pressure ratios. Its arrival at the exit happens at a pressure ratio of 24.55, the closed form: the design pressure ratio divided by a normal shock’s pressure ratio at the exit Mach number. The full-running thrust coefficient agrees with the textbook’s closed-form expression to $4\cdot10^{-14}$. And the separated thrust joins the full-running thrust continuously where separation begins, to $3\cdot10^{-8}$.

What a one-dimensional nozzle cannot show #

Two ways to be separated. A nozzle that separates can do so in two patterns. In free-shock separation the flow leaves the wall and never comes back. In restricted-shock separation the separated flow closes into a bubble and reattaches further down, and the wall pressure downstream of the bubble can rise above ambient. Which pattern appears depends on the nozzle’s contour and on the internal shock system of the bell, and a nozzle can switch between them as its chamber pressure rises during start-up. The switch is asymmetric and violent, and it is the source of the side loads that have broken nozzle extensions on test stands; none of it is visible in one dimension.

The interaction region. The step from the separation pressure to ambient is really a short region of oblique shock, separated shear layer and recirculation, with a plateau pressure below ambient; the criteria are correlations of where it begins, fitted to experiments, and they carry a scatter of their own.

Asymmetry. A real separation line is not a circle: it wanders round the nozzle’s circumference and in time, and the net sideways force that results is not zero even when the average separation point is steady.

The criteria are borrowed. Summerfield’s and Schmucker’s criteria are measured, not derived here. The calculation’s contribution is what they imply once the nozzle’s own expansion is put against them.

The convention the numbers depend on #

The nozzle pressure ratio is chamber pressure over ambient pressure. Positions are fractions of the divergent section’s length, from the throat. The thrust coefficient is thrust divided by the chamber pressure times the throat area. The gas has $\gamma = 1.2$ and the nozzle an exit-to-throat area ratio of 40; the separation criteria are applied to the isentropic wall pressure, as they were fitted.

Who found it, and when #

Summerfield, Foster and Swan reported in 1954 that overexpanded rocket nozzles separate when the wall pressure falls to about 0.4 of ambient, long before the inviscid shock would reach the exit, and the criterion carries Summerfield’s name. Schmucker’s correlation of 1973 added the Mach-number dependence. Restricted-shock separation was identified in the J-2S engine’s tests of the early 1970s and studied in detail by Frey and Hagemann in the 1990s on Europe’s Vulcain engine, whose side loads it explained.

Still open: the pattern that reattaches #

The separated nozzle here separates once and stays separated, which is free-shock separation. The other pattern needs something one dimension does not have: a bell contour whose internal shock — formed where the contour turns the flow back towards the axis — meets the separation shock and closes the separated region into a cap. The next calculation marches the bell’s supersonic flow by the method of characteristics, finds that internal shock, and asks at which pressure ratio its meeting with the separation shock can close a bubble; that is the pressure ratio at which a start-up switches between the two patterns, and the one at which the side loads peak.