Converging-Diverging Nozzle Area-Ratio Calculator

Inspectable records

Ledger state

The current ledger marks this instrument verified with errata that must be applied at binding time.

Area

Thermodynamics, Fluids & Aero

Tranche

tranche-6

Evidence ID

t6-085

Ledger rows

1

No ledger note is attached.

Review findings

The core area-Mach formula is correct under the stated ideal 1D assumptions, but the instrument specification is incomplete for its claimed back-pressure-based regime output. Define Pb/P0, enforce Ae/A*>=1 and gamma>1, add the pressure and normal-shock algorithms, and distinguish formal branch roots from the actual unchoked operating Mach. The honest-label warning should explicitly state whether normal shocks and shock losses are modeled; otherwise 'shock-in-nozzle' is only a classification, not a physically computed solution.

Formula and scope findings

• The area-Mach relation is standard for quasi-one-dimensional, steady, adiabatic, isentropic flow with constant gamma, using A*=the sonic throat area. It is valid only for A/A* >= 1; this input constraint is not stated.
• The relation has two distinct roots only for A/A* > 1. At A/A*=1 the roots coalesce at M=1, so the claimed dual-root output needs an explicit boundary rule.
• The stated formula alone cannot determine the operating regime from back pressure. Shock-in-nozzle classification additionally requires the isentropic pressure relation, the choked-flow criterion, and normal-shock relations, including shock location or an equivalent pressure threshold algorithm.
• The back-pressure ratio is undefined. It must explicitly mean Pb/P0, where P0 is stagnation pressure upstream of the nozzle.
• For a shock-in-nozzle result, the model must include a normal shock and the associated total-pressure loss. Treating the entire nozzle as isentropic while reporting a shock-in-nozzle regime is inconsistent.
• The exit Mach solutions are independent of back pressure only after the nozzle is choked. For sufficiently high back pressure, the nozzle is unchoked and the supplied A/A* relation does not directly provide the actual exit Mach.

Unit assumptions

• gamma is dimensionless and is the constant ratio of specific heats; test cases use gamma=1.4 for air.
• Area ratio means the geometric exit-area-to-sonic-throat-area ratio Ae/A*.
• Back-pressure ratio means Pb/P0, with Pb static back pressure and P0 upstream stagnation pressure.
• Pressure ratios are dimensionless; no absolute pressure units are required.
• The ideal regime thresholds below assume a converging-diverging nozzle, ideal choking, isentropic regions away from a normal shock, and a normal shock when classified as shock-in-nozzle.

Boundary hardening

BREAK_CLAIMED · SELF_FLAGGED

The specified formula has an undefined gamma=1 input that requires explicit rejection, and the back-pressure ratio convention is not defined well enough to guarantee a unique operating-regime result.

Errata and corrections

Tranche errata

Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

{
  "inputs": {
    "area_ratio_A_over_Astar": 2,
    "back_pressure_ratio_Pb_over_P0": 0.05,
    "gamma": 1.4
  },
  "expected_outputs": {
    "subsonic_M": 0.305904,
    "supersonic_M": 2.1972,
    "exit_supersonic_pressure_ratio_Pe_over_P0": 0.093934,
    "critical_choking_pressure_ratio_Pb_over_P0": 0.528282,
    "shock_at_exit_pressure_ratio_Pb_over_P0": 0.5135,
    "operating_regime": "supersonic"
  },
  "working": "For gamma=1.4, A/A*=(1/M)[(5/6)(1+0.2M^2)]^3. Newton solve on g(M)=-ln(M)+3ln[(5/6)(1+0.2M^2)]-ln(2), with g'(M)=-1/M+1.2M/(1+0.2M^2). Subsonic branch: M0=0.300000; g(M0) is approximately +0.01738, g'(M0)=-2.97970, so M1=0.300000-(+0.01738)/(-2.97970)=0.305834. Second iteration gives M2 approximately 0.305904; subsequent iterations give M=0.305904. Supersonic branch: M0=2.00000; g(M0)=-0.169899, g'(M0)=0.833333, so M1=2.00000-(-0.169899)/(0.833333)=2.20388. Second iteration gives M2 approximately 2.19724; subsequent iterations give M=2.19720. Check: for M=2.19720, M^2=4.82768; (5/6)(1+0.2M^2)=(5/6)(1.965536)=1.637947; 1.637947^3/2.19720 approximately 2.00000. Exit pressure: Pe/P0=(1+0.2M^2)^(-gamma/(gamma-1))=1.965536^(-3.5)=0.093934. Choking limit: (2/(gamma+1))^(gamma/(gamma-1))=(5/6)^3.5=0.528282. Normal-shock pressure multiplier at M=2.19720 is 1+[2gamma/(gamma+1)](M^2-1)=1+(7/6)(3.82768)=5.46563; shock-at-exit Pb/P0=0.093934 times 5.46563 approximately 0.5135. Since Pb/P0=0.05 is below the design exit pressure ratio, the ideal flow is supersonic through the diverging section, with any adjustment shock outside the nozzle."
}

Related workflows

No source-defined multi-instrument workflow membership is recorded.

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json · VERIFIED_WITH_ERRATA
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json