An interactive simulator for experiencing isentropic flow, choking, and normal shocks by directly tuning the geometry and back pressure of a converging-diverging nozzle.

Mach Architect#

Studying converging-diverging (C-D) nozzle flow from textbooks alone makes it hard to develop intuition about how the area ratio $A/A^*$ and back-pressure ratio $P_b/P_0$ change the flow regime.

Design the nozzle yourself, then turn the back-pressure dial. Clear four levels and you'll absorb the essentials of 1D compressible nozzle flow.

MACH ARCHITECT1D COMPRESSIBLE FLOW PUZZLE
Level 1 -- Subsonic Acceleration
Objective: Exit Mach M_exit = 0.45 ~ 0.55
IN PROGRESS
NOZZLE CROSS SECTION — DRAG CONTROL POINTS
A*P0 INEXIT
MACH NUMBER M(x)
0.511.522.53M=1
PRESSURE P/P0(x)
0.250.50.751Pb/P0=0.880
BOUNDARY CONDITIONS
Pb/P0 = 0.880
0.04Pb*=0.5281.0
FLOW REGIME
SUBSONIC
M_exit0.4313
M_throat0.4313
A_e/A*1.000
Pb_design0.5282
MACH COLOR MAPM=0M=1M=2M=4-- SHOCK-- THROAT

Controls#

Nozzle geometry: drag the blue control points on the SVG up and down to adjust cross-section
Back pressure: the right-side slider sets $P_b/P_0$
Level select: switch with the LV1–LV4 buttons at the top

Level Walkthrough#

Level 1: Subsonic Acceleration#

The most basic case. In a converging nozzle, subsonic flow accelerates as the area shrinks. The isentropic relation:

\frac{A}{A^*} = \frac{1}{M}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}

Since $M < 1$ , decreasing $A$ increases $M$ . The goal is exit Mach 0.45–0.55.

Level 2: Choking#

When the flow chokes, $M = 1$ at the throat. Lowering back pressure further leaves the upstream mass flow unchanged.

Critical pressure ratio ( $\gamma = 1.4$ ):

\frac{P^*}{P_0} = \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}} = 0.5283

Drop back pressure below this value, or make the throat narrow enough.

Level 3: Supersonic Nozzle Design#

In a C-D nozzle, the flow accelerates supersonically once the area opens up beyond the throat. To reach $M = 2$ :

\frac{A_e}{A^*} = \frac{1}{2}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}\cdot 4\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}} \approx 1.6875

And the back pressure must drop to the design pressure (Pb_design) for fully supersonic flow without shocks.

Level 4: Shock Positioning#

When the back pressure is above design but below critical, a normal shock forms inside the nozzle.

Pre/post-shock relations:

M_2^2 = \frac{M_1^2(\gamma-1)/2 + 1}{\gamma M_1^2 - (\gamma-1)/2}

\frac{P_2}{P_1} = \frac{2\gamma M_1^2 - (\gamma-1)}{\gamma+1}

Adjusting back pressure moves the shock. Higher back pressure pushes the shock toward the throat, lower toward the exit. The target is to land it within $x = 0.65 \sim 0.80$ .

The Simulator's Physics Model#

This simulator rests on these assumptions:

1D quasi-1D flow: only area variation; viscosity/heat transfer ignored
Isentropic flow (except across shocks): $P_0$ , $T_0$ conserved
Calorically perfect gas: $\gamma = 1.4$ (air)
Normal shocks: Rankine–Hugoniot relations
Nozzle geometry: smooth area distribution via Hermite interpolation

The shock position is found by discretely searching for the location that satisfies the exit pressure condition. Upstream of the shock is isentropic supersonic; downstream is isentropic subsonic with stagnation pressure loss applied.

Going Further#

Compressible Multiphase Flow CFD: Introduction — what gets harder when this physics extends to multiphase flow
From the Riemann Problem to Godunov-Type Schemes — the Riemann solver that captures normal shocks numerically
FDM vs FEM vs FVM — three discretization approaches for this 1D problem

Mach Architect: Learn Compressible Flow by Designing Nozzles