Rocket Nozzle Flow: Quasi-1D Euler PDE Solver

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

Solver

Marching speed med

Live Solver Readout

PDE exit Mach M_e

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Isentropic M_e (algebra)

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Solver vs. formula

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Mass-flow conservation

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Iterations / residual

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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

This solves the same real gas-dynamics problem as the 3D nozzle sim, but with a completely different computational method: instead of algebraically inverting the isentropic area–Mach relation, it time-marches the actual unsteady quasi-1D Euler equations on a 181-point grid until the flow relaxes to its steady state, exactly like a real one-dimensional CFD solver:

∂(ρ')/∂t' = −ρ'∂V'/∂x' − ρ'V'∂(lnA')/∂x' − V'∂ρ'/∂x'
∂V'/∂t' = −V'∂V'/∂x' − (1/γ)[∂T'/∂x' + (T'/ρ')∂ρ'/∂x']
∂T'/∂t' = −V'∂T'/∂x' − (γ−1)T'[∂V'/∂x' + V'∂(lnA')/∂x']

(primed = nondimensionalized by chamber stagnation density/temperature/sound speed). Each pseudo-time step uses MacCormack's predictor–corrector scheme — forward differences to predict, backward differences to correct, averaged — the classic explicit finite-difference method for this equation set, plus a small pressure-switched artificial-viscosity term for numerical stability. It starts from a crude, deliberately-wrong initial guess and visibly relaxes toward the true profile as iterations accumulate.

Two independent checks confirm the PDE solver actually reached the correct physics: its converged exit Mach number is compared live against the closed-form isentropic area–Mach relation (solved separately by bisection, exactly as the 3D sim does), and mass flow ρ'V'A' — which must be constant along a steady nozzle — is checked for how flat it stays across the grid. Both should agree to within a fraction of a percent once the residual has dropped to machine precision.

Expansion ratio / throat radius — rebuild the nozzle contour and reset the solver to a cold-start guess.
Chamber pressure / altitude — set the physical stagnation pressure and ambient pressure used to convert the solver's nondimensional output into real thrust, Isp and exit pressure.
Marching speed — how many solver iterations run per animation frame; higher settles to steady state faster but the intermediate (non-physical) transient states flash by quicker too.

Subsonic (M<1) Supersonic (M>1) Mach-angle characteristics beyond exit