Convergent vs Convergent-Divergent Nozzles: Choking, Expansion and Thrust
The nozzle has no moving blades, no combustion, no controls on most airliners — just a shaped hole. Yet it is where the engine's entire pressure investment gets cashed out as velocity, and it obeys the most counterintuitive rule in gas dynamics: once the flow goes supersonic, squeezing it slows it down. That one fact splits the world of propulsion nozzles in two.
The convergent nozzle: simple, light, and hard-limited
A convergent nozzle narrows toward the exit. Subsonic flow accelerates as the area shrinks — the continuity equation at work. But there is a ceiling: the exit can reach Mach 1 and no further. Beyond that point the nozzle is choked: mass flow per unit area is fixed by upstream total pressure and temperature, and raising the pressure ratio no longer increases exit velocity.
Choking begins at a specific total-to-static pressure ratio:
pt / p* = ((γ+1)/2)γ/(γ−1) ≈ 1.893 (air) · 1.85 (hot exhaust)Above that ratio, a convergent nozzle exits at exactly Mach 1 with p9 > p_ambient. The leftover pressure doesn't disappear — it shows up in the thrust equation as the pressure term:
F = ṁ(V9 − V0) + (p9 − p0)A9So the engine still gets thrust, but the expansion finishes outside the nozzle, in uncontrolled expansion fans, instead of as organized momentum inside it. As we detailed in the exhaust plume analysis, that unexpanded pressure is exergy the engine paid for and only partially recovers.
The convergent-divergent nozzle: past Mach 1
The de Laval nozzle adds a divergent section after a throat. At the throat the flow reaches Mach 1; downstream, now supersonic, it obeys the reversed rule — increasing area means increasing velocity. Pressure and temperature keep falling while kinetic energy climbs, and the exit can reach Mach 2, 3 or beyond. The area-Mach relation governs the whole geometry:
A/A* = (1/M) · [ (2/(γ+1)) · (1 + (γ−1)M²/2) ](γ+1)/(2(γ−1))This is why every rocket engine is a bell and every afterburning fighter has a divergent petal section: when the available pressure ratio is large, only a divergent section can turn it into supersonic momentum rather than external waste.
When each one wins
| Convergent | Convergent-divergent | |
|---|---|---|
| Exit Mach | ≤ 1 (choked) | > 1, design-dependent |
| Best pressure ratio | Low–moderate NPR | High NPR, large expansion |
| Weight / complexity | Minimal, often fixed | Heavy; variable petals on fighters |
| Typical users | Subsonic airliner turbofans | Rockets, afterburning fighters, SST concepts |
| Main off-design risk | Under-expansion | Over-expansion shocks and separation |
The airliner choice surprises people: cruise nozzle pressure ratios on high-bypass turbofans are modest, the fan stream barely chokes, and a divergent section would add weight, length and cooling area to recover very little. The convergent nozzle is not a compromise — at those conditions it is the optimum. Military engines live at the other extreme: high T4, afterburning (see what reheat costs), and nozzle pressure ratios that would waste enormous pressure exergy through a purely convergent exit.
The design-point trap: a C-D nozzle is only perfectly expanded at one ambient pressure. Climb, throttle back or light the afterburner and the exit pressure drifts off ambient in both directions — which is why the fighter nozzle that looks so elegant on the ground needs variable geometry to stay efficient in the air.
Off-design: the two ways to miss perfect expansion
- Under-expanded (p9 > p0). Exit pressure is too high; expansion fans finish the job outside. You keep the pressure-thrust term but lose the additional momentum a larger divergent section could have captured. Typical of convergent nozzles at high NPR and rockets at sea level vs vacuum.
- Over-expanded (p9 < p0). Exit pressure is too low; the atmosphere squeezes the jet, and in severe cases oblique shocks and flow separation appear inside the divergent section. Shock losses are directly irreversible — stagnation pressure drops across them, and the associated exergy destruction is immediate and unrecoverable.
Perfect expansion (p9 = p0) is the thrust maximum at a given nozzle inlet state, and variable-geometry nozzles exist to stay near it across the envelope. Everything else is a trade between pressure thrust, shock loss and mechanical simplicity.
The exergy view: a good nozzle destroys almost nothing
Here is the part that surprises students of component exergy destruction: a well-designed nozzle is one of the best components in the engine. Isentropic efficiencies of 0.95–0.98 are routine, and internal irreversibility is small — friction near the walls, little else. The nozzle's losses mostly aren't destruction inside it at all. They are exergy handed over at the exit plane: unexpanded pressure, residual swirl, and off-design shocks. That makes nozzle design less about fighting entropy and more about bookkeeping — making sure the pressure the compressor and combustor built actually leaves as ordered kinetic energy, at the right pressure, pointed the right way. In the turbojet vs turbofan comparison, this is precisely where the turbojet's high-velocity jet demands a C-D nozzle while the turbofan's gentle fan stream does not.
Frequently asked questions
What is the difference between a convergent and a convergent-divergent nozzle?
A convergent nozzle only narrows: it accelerates subsonic flow up to Mach 1 at the exit and can go no faster, no matter how high the upstream pressure. A convergent-divergent (de Laval) nozzle narrows to a throat where the flow reaches Mach 1, then widens again — and in the divergent section the now-supersonic flow keeps accelerating, converting pressure into velocities far above the speed of sound. That is why rocket engines and afterburning military jets use C-D nozzles while most subsonic airliner engines use simple convergent nozzles.
Why can't a convergent nozzle produce supersonic flow?
In subsonic flow, squeezing the area raises velocity; in supersonic flow the physics reverses and acceleration requires increasing area. A purely convergent nozzle therefore accelerates flow only until the exit reaches Mach 1 — the choked condition. Raising upstream pressure beyond the critical ratio (about 1.85–1.89 of ambient for typical exhaust gases) no longer raises exit velocity; the extra pressure leaves as unexpanded pressure thrust instead.
What is nozzle choking and at what pressure ratio does it occur?
Choking means the throat or exit of a nozzle has reached Mach 1, fixing the mass flow per unit area regardless of downstream pressure. It begins when the total-to-static pressure ratio reaches ((γ+1)/2)^(γ/(γ−1)): about 1.893 for cold air (γ=1.4) and about 1.85 for hot exhaust gas (γ≈1.33). Most jet engines operate choked at takeoff, which is why their mass flow is essentially set by nozzle throat area and upstream total pressure and temperature.
What are under-expanded and over-expanded nozzle flows?
A nozzle is under-expanded when its exit pressure exceeds ambient: the jet still carries pressure potential and finishes expanding outside the engine through expansion fans, wasting some thrust. It is over-expanded when exit pressure falls below ambient: the jet is squeezed by the atmosphere, potentially driving shocks into the divergent section and separating the flow. Maximum thrust occurs at perfect expansion, where exit pressure exactly equals ambient — the condition variable-geometry nozzles chase across the flight envelope.
Why do fighter jets use variable convergent-divergent nozzles?
An afterburning fighter sees nozzle pressure ratios and required exit areas swing enormously between dry cruise, full reheat and supersonic flight. A fixed C-D nozzle would be perfectly expanded at only one condition and shock-laden or under-expanded everywhere else. Hinged petal actuators vary throat and exit area independently, keeping the nozzle near perfect expansion and the engine unchoked across the envelope — the weight and complexity buy back more thrust than they cost.
Compute it — don't guess it
Set a nozzle pressure ratio and flight condition, then watch choking, exit pressure mismatch and net thrust respond. The moment the exit pressure term starts carrying real thrust is the moment a divergent section starts paying rent — and you can find that moment numerically instead of by rule of thumb.
Sweep nozzle pressure ratio and altitude on a real engine model — choked mass flow, exit pressure mismatch, thrust and the full exergy budget in about a minute.
Run a free exergy analysis →