Rocket Nozzle Expansion — Mach Diamonds

Nozzle Geometry

Expansion ratio ε = A_e/A_t 25

Throat radius R_t 0.15 m

Chamber & Environment

Chamber pressure P_c 10.0 MPa

Altitude (ambient pressure) 0 km

Live Readout

Exit Mach M_e

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Exhaust velocity

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Specific impulse

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Thrust

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Exit / ambient pressure

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How it works

Hot combustion gas is choked to Mach 1 at the throat, then a diverging bell/cone accelerates it to supersonic speed. For a calorically-perfect gas (ratio of specific heats γ), the local Mach number M is fixed purely by the local area ratio A/A* through the area–Mach relation:

A/A* = (1/M) · [ (2/(γ+1)) · (1 + (γ-1)/2 · M²) ] ^ ((γ+1)/(2(γ-1)))

This sim solves that equation numerically along the nozzle contour (bisection on the sub/supersonic branches) to animate real flow acceleration, then applies the isentropic relations at the exit plane:

Te/Tc = 1 / (1 + (γ-1)/2 · Me²)
Pe/Pc = (Te/Tc) ^ (γ/(γ-1))
Ve    = Me · √(γ·R·Te)          (exhaust velocity)
Isp   = Ve / g0                  (specific impulse)
ṁ     = Pc·At·√(γ/(R·Tc)) · (2/(γ+1))^((γ+1)/(2(γ-1)))   (choked mass flow)
F     = ṁ·Ve + (Pe − Pa)·Ae      (thrust equation)

Whether the plume looks under-expanded, perfectly-expanded or over-expanded depends only on how P_e (fixed by geometry) compares with ambient P_a (set by altitude):

P_e > P_a (under-expanded) — the jet keeps expanding past the nozzle lip, over-shoots ambient pressure, then gets compressed back by the surrounding air; the repeated expansion/compression sets up the classic bright "shock diamond" (Mach diamond) pattern seen on real rocket and jet exhaust.
P_e ≈ P_a (perfectly expanded) — the plume stays a smooth cylinder; this is the design condition an engineer picks ε for, at one specific altitude.
P_e < P_a (over-expanded) — the jet is squeezed inward by the higher outside pressure before re-expanding, and at extreme mismatch the flow can separate from the nozzle wall before reaching the exit.

Because a real engine flies from sea level to vacuum, a single fixed nozzle is only perfectly expanded at one altitude — this is exactly why first-stage and vacuum-optimized upper-stage engines use very different expansion ratios ε. The diamond spacing/amplitude drawn here is a physically-motivated stylization (shock-cell spacing scales with exit diameter and √(Me²−1)), not a full CFD shock solution.

Chamber / throat (hottest) Supersonic exhaust plume

Altitude 0 km
P_a = 101.3 kPa

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