Nozzle operating map

Measured

• shock station walks 0.656 → 0.930 m as p_back/p0 falls 0.90 → 0.60
• exit Mach 2.123 once the shock leaves the duct, against a design 2.197
• analytic references: first critical p/p0 = 0.9372, exit-plane shock at 0.5134
• all four gates PASS

Each row runs a 1-D compressible duct march over 128 cells. The sweep collects them into one table.

Geometry is a converging–diverging nozzle: inlet 2 m², throat 1 m², exit 2 m² over 1 m of length, fed from a 300 kPa / 500 K reservoir.

Output

  p_back/p0   M_exit   shock x [m]      Cf
      0.90    0.325       0.6562    0.242
      0.80    0.368       0.7656    0.277
      0.70    0.420       0.8516    0.316
      0.60    0.493       0.9297    0.365
      0.30    2.123         none    0.857
      0.10    2.123         none    1.254

As back-pressure falls the normal shock moves downstream through 0.656, 0.766, 0.852 and 0.930 m, then leaves the duct. The last two rows are shock-free supersonic exits.

Exit Mach holds at 2.123 across those two rows. Once the nozzle is fully supersonic the exit Mach is set by the area ratio, and further back-pressure reduction alters the plume outside the nozzle rather than the internal flow. The thrust coefficient continues to rise from 0.857 to 1.254 because it depends on the pressure difference.

Comparison with the analytic values

analytic references: first critical p/p0 = 0.9372, exit-plane shock at p/p0 = 0.5134

Both values come from closed-form quasi-1-D relations rather than from the march. The choking gate requires the throat to be choked below the first critical ratio. The shock-position gate requires the computed stations to fall between the throat and the exit plane and to move monotonically downstream.

Measured exit Mach is 2.123 against a design value of 2.197, 3.4% low, consistent with a first-order scheme smearing the expansion.

Concurrency

Six marches run concurrently under the parallel feature, through deep_causality_par. The rows are independent by construction, so the result is order-preserving and identical to the serial run.

Applicability limits

The duct march is first-order and quasi-1-D. Shocks smear over several cells, so each shock station is a cell-resolution estimate.

The gas is perfect at γ = 1.4, with no wall friction, no heat transfer and no real-gas effects.

The sweep covers the choked regimes only. The unchoked window above the first critical ratio is not included, so the map covers the design envelope and not startup transients.

Stated limitation

A first-order quasi-1-D scheme, so shocks smear over a few cells. Perfect gas at γ = 1.4 with no wall friction, heat transfer, or real-gas effects. The unchoked window above the first critical ratio is not swept.