Over - under expanded nozzle flows: Worked examples

This page hunts down every case a nozzle-pressure problem can throw at you: too-low exit pressure, too-high exit pressure, the exact match, the degenerate "no expansion" nozzle, the limiting "vacuum" case, and the real-world climbing rocket. First we map the territory, then we conquer each cell with a full worked example.

For the machinery (the two master equations) see the parent Over - under expanded nozzle flows, and the building blocks Isentropic Flow Relations, Area-Mach Number Relation, and Choked Flow & the Throat (M=1).

Building the two tools we reuse everywhere

Before any example, let me re-state — in plain words — the only two formulas we need, so no symbol appears unearned.

The verdict rule, stated once:

The scenario matrix

Every nozzle-pressure problem is one of these cells. The worked examples below are tagged [Cell n].

#	Case class	What is special	Example
1	$p_{e} < p_{b}$ (over-expanded, mild)	oblique shocks at lip	Ex 1
2	$p_{e} < p_{b}$ strongly	Mach disk / separation risk	Ex 2
3	$p_{e} = p_{b}$ (perfect)	pressure term vanishes, max thrust	Ex 3
4	$p_{e} > p_{b}$ (under-expanded)	expansion fans, jet bulges	Ex 4
5	Degenerate $A_{e} / A^{*} = 1$	throat = exit, $M_{e} = 1$ (sonic)	Ex 5
6	Limiting $p_{b} \to 0$ (vacuum)	always under-expanded	Ex 6
7	Same nozzle, changing $p_{b}$ (climb)	regime flips mid-flight	Ex 7
8	Thrust sign across regimes	$(p_{e} - p_{b}) A_{e}$ can be $+$ or $-$	Ex 8
9	Exam twist: solve for design altitude	invert the logic	Ex 9

We fix $γ = 1.4$ throughout unless stated, so $\frac{γ - 1}{2} = 0.2$ and $\frac{γ}{γ - 1} = 3.5$ .

Figure — Over - under expanded nozzle flows

Worked example Ex 1 — Mild over-expansion

[Cell 1] $p_{0} = 10$ bar, $A_{e} / A^{*} = 4.0$ , $p_{b} = 1$ bar.

Forecast: A fat area ratio (4×) means the gas expanded a lot, so $p_{e}$ is probably small — guess over-expanded.

Find $M_{e}$ from Tool 1. Area ratio $4.0$ ⇒ supersonic root $M_{e} \approx 2.94$ . Why this step? Geometry alone fixes the exit Mach; the throat is choked so we only need the shape.
Find $p_{e}$ from Tool 2. $\frac{p _{0}}{p _{e}} = (1 + 0.2 \cdot 2.9 4^{2})^{3.5} = (2.729)^{3.5} \approx 34.0$ . Why? This converts the Mach number into the pressure it corresponds to.
Compute $p_{e}$ . $p_{e} = 10/34.0 \approx 0.294$ bar. Why? Just rearrange.
Verdict. $p_{e} = 0.294 < p_{b} = 1.0$ ⇒ $PR = 0.294 < 1$ ⇒ over-expanded. Oblique shocks at the lip. See Oblique Shocks.

Verify: Pressure ratio $p_{0} / p_{e} = 10/0.294 = 34.0$ , matching step 2. Units: bar/bar dimensionless ✓. Forecast correct ✓.

Worked example Ex 2 — Strong over-expansion, separation risk

[Cell 2] $p_{0} = 10$ bar, $A_{e} / A^{*} = 8.0$ , $p_{b} = 1$ bar.

Forecast: Even bigger area ratio ⇒ even lower $p_{e}$ ⇒ strongly over-expanded. The pressure gap is huge, so watch for a Mach disk or the shock creeping inside.

$M_{e}$ from Tool 1. $A_{e} / A^{*} = 8.0$ ⇒ $M_{e} \approx 3.65$ . Why? Doubling the area ratio pushes the supersonic root higher.
$p_{e}$ from Tool 2. $p_{0} / p_{e} = (1 + 0.2 \cdot 3.6 5^{2})^{3.5} = (3.665)^{3.5} \approx 92.5$ , so $p_{e} = 10/92.5 \approx 0.108$ bar. Why? Same isentropic conversion.
Verdict + severity. $PR = 0.108/1 = 0.108$ . Extremely below 1: strongly over-expanded. Why care? A large required pressure rise ( $p_{b} / p_{e} \approx 9.3$ ) can exceed what an attached oblique shock provides, so a normal shock / Mach disk forms and the flow may separate inside the diverging wall. See Normal Shock & Mach Disk and Flow Separation in Nozzles.

Verify: $p_{b} / p_{e} = 1/0.108 \approx 9.26$ — a near-order-of-magnitude jump, consistent with "strong." Compare to Ex 1 where $p_{b} / p_{e} \approx 3.4$ (milder) ✓.

Worked example Ex 5 — Degenerate nozzle, throat = exit

[Cell 5] $p_{0} = 10$ bar, $A_{e} / A^{*} = 1.0$ , $p_{b} = 1$ bar.

Forecast: Area ratio 1 means the exit is the throat — no diverging section. The throat sits at $M = 1$ when choked, so guess $M_{e} = 1$ , sonic (not supersonic).

$M_{e}$ from Tool 1. At $A_{e} / A^{*} = 1$ , the only root is $M_{e} = 1$ . Why? Tool 1 equals 1 exactly at $M = 1$ (the throat is the minimum of the area function). See Choked Flow & the Throat (M=1).
$p_{e}$ from Tool 2. $p_{0} / p_{e} = (1 + 0.2 \cdot 1^{2})^{3.5} = (1.2)^{3.5} \approx 1.893$ , so $p_{e} = 10/1.893 \approx 5.28$ bar. Why? Sonic exit has a fixed pressure ratio of about 1.89.
Verdict. $p_{e} = 5.28 > p_{b} = 1$ ⇒ under-expanded — but note the flow is only sonic, so the external adjustment is subsonic-to-transonic in character, not a classic supersonic fan. The over/under machinery still labels it by pressure.

Verify: $(1.2)^{3.5} \approx 1.893$ ; $10/1.893 \approx 5.28$ bar ✓. Critical pressure ratio for $γ = 1.4$ is the textbook $0.528$ (i.e. $p_{e} / p_{0} = 0.528$ ) ✓.

Worked example Ex 9 — Exam twist: design for a target altitude

[Cell 9] "You need $p_{e} = 0.30$ bar (to be perfect at a chosen altitude). $p_{0} = 10$ bar, $γ = 1.4$ . What $M_{e}$ and roughly what $A_{e} / A^{*}$ ?"

Forecast: We run the tools backwards: start from a pressure, get $M_{e}$ , then get area ratio.

Invert Tool 2 for $M_{e}$ . $p_{0} / p_{e} = 10/0.30 = 33.33 = (1 + 0.2 M_{e}^{2})^{3.5}$ . So $1 + 0.2 M_{e}^{2} = 33.3 3^{1/3.5} = 33.3 3^{0.2857} \approx 2.717$ , giving $M_{e}^{2} = (2.717 - 1) /0.2 = 8.585$ , $M_{e} \approx 2.930$ . Why? Take the $1/3.5$ power to undo the exponent, then solve the linear-in- $M^{2}$ relation.
Tool 1 forward for area ratio. Plug $M_{e} = 2.93$ : $A_{e} / A^{*} \approx 3.97 \approx 4.0$ . Why? This tells the designer how big to make the exit relative to the throat.
Verdict at design. By construction $p_{e} = p_{b} = 0.30$ bar there ⇒ perfect.

Verify: $33.3 3^{1/3.5} \approx 2.717$ ; $M_{e} \approx 2.93$ ; forward Tool 1 gives $A_{e} / A^{*} \approx 3.97$ — consistent with the $A_{e} / A^{*} = 4$ nozzle used throughout ✓.

Recall Quick self-test

A nozzle gives $p_{e} = 0.30$ bar. It is fired at $p_{b} = 0.30$ bar. Regime? ::: Perfectly expanded ( $PR = 1$ ), no waves, max thrust for that $V_{e}$ . Same nozzle, $p_{b} = 0.05$ bar. Regime and wave type? ::: Under-expanded ( $PR = 6 > 1$ ); Prandtl–Meyer expansion fans, jet bulges out. Same nozzle, $p_{b} = 2$ bar. Regime and wave type? ::: Over-expanded ( $PR = 0.15 < 1$ ); oblique shocks, possibly Mach disk / separation. Firing into vacuum — over or under? ::: Always under-expanded, $PR \to \infty$ .