Python library · v0.7.0 · MIT · Shanthosh KV

rockmoc

A method-of-characteristics design code for rocket nozzles. You give it a gas, a throat radius, and one expansion target. It marches Mach waves, returns a wall you can send to CAD, and draws the characteristic net the wall came from.

pip install rockmoc PyPI How to use All figures

Mach, throat to exit C+ characteristics C− characteristics Ideal bell · γ = 1.2 · M_e = 3 · R_t = 50 mm

14design calls: bells, a 15° cone, the NASA planar MLN, and the aerospike stack

98unit tests, plus a 50-check offline suite and a 43-case literature campaign

12cited sources: NASA, NACA, Rao, Anderson, Zucrow, Sutton, Sauer

0.03 simport time in 0.7.0, down from 1.6 s, by lazy imports

What this program is

A design code, not a CFD solver

rockmoc builds the inviscid wall of a supersonic nozzle by the method of characteristics. It does not solve the Navier–Stokes equations, and it does not replace a hot-fire test. It is the step before those: a contour whose exit Mach, area ratio, and thrust coefficient you can defend from the mesh.

Inputs

A gas (constant γ, a γ(M) table, or a frozen CEA / Cantera / CoolProp state), throat radius in metres, and exactly one of exit Mach, area ratio, or p_e/p_c. Angles in the API are radians. The geometric throat sits at x = 0, r = R_t.

What comes back

A wall polyline with θ and M, a characteristic net if you ask for it, length, geometric area ratio, vacuum and ambient thrust coefficient, and specific impulse. result.summary(), result.report(), and result.plot() are the three ways to read it.

Where it goes

CSV for a report, a header-less point list for CAD, an R12 DXF, and an OpenFOAM axisymmetric-wedge blockMeshDict. The same call can grow the wall by a displacement thickness before you export it.

Every figure on this page was drawn by the library. Two design points are in use, and the captions say which. Feature figures in images/rockmoc/ are γ = 1.2, M_e = 3, R_t = 50 mm, usually 40 characteristics (website/make_images.py, make_images_extra.py). Dashboard figures in images/rockmoc/ are γ = 1.25, M_e = 3, R_t = 50 mm, 48 characteristics (tests/generate_all_graphs.py), regenerated from the 0.7 solver. Do not read a metric box on a γ = 1.25 plot as the γ = 1.2 table.

One design point

Four bells, same gas, same exit Mach

γ = 1.2, M_e = 3.0, R_t = 50 mm, 40 characteristics. One-dimensional area ratio is ε = A_e/A* = 6.74. Length below is L/R_t. Thrust is the exit-plane integral of momentum and pressure, divided by the one-dimensional vacuum C_F.

Design	L / R_t	C_F,vac / 1-D
MLN	10.45	100.0%
Ideal	11.26	99.5%
TIC, 80% of ideal	9.01	99.3%
Rao, 80% of a 15° cone	4.80	97.0%

At this Mach number Rao’s length convention (80% of the 15° cone of the same ε) is much shorter than a minimum-length contour. The thrust gap is mostly exit divergence, not a mass-flow error. A 15° cone at the same point keeps 96.9% of one-dimensional vacuum C_F.

Overlay of MLN, ideal, TIC and Rao walls, normalized by throat radius, with a wall-angle inset — Walls normalized by throat radius. The inset is wall angle. Rao turns hard and finishes; the ideal bell keeps a long, shallow canceling region.

Library dashboard overlay of MLN, ideal, TIC and Rao walls at gamma 1.25 — Dashboard overlay at γ = 1.25. Rao is still the short bell. The cone is in the bar chart, not in this overlay.

Bar chart comparing length, area ratio and vacuum thrust coefficient across nozzle families at gamma 1.25 — Same comparison from the plotter, at the other design point: γ = 1.25, N = 48, and the 15° cone included. Lengths are MLN 9.76, ideal 10.5, TIC 8.4, Rao 4.22, cone 5.28. Vacuum C_F is about 1.65, 1.65, 1.64, 1.60, 1.57. TIC’s area ratio drops to 5.70 because the cut happens before the ideal exit.

design_mln · axisymmetric

Minimum-length nozzle

A sharp corner at the throat launches a Prandtl–Meyer fan. The corner angle θ_max is solved so the last left-running wave from that corner hits the axis at the design Mach. It is not ν_e/2. That half-angle rule is the planar one, and using it on an axisymmetric nozzle was a real bug.

For M_e = 3 and γ = 1.4 the axisymmetric corner is about 11.7°. The old solver reported 25.9°. Planar theory at the same point wants ν_e/2 ≈ 24.9°. The gap is the axisymmetric source term, not a fudge factor.

Downstream of the kernel, a Goursat problem turns the flow onto a uniform axial exit. The wall is the streamline that carries the throat mass flow, so exit area is an output of mass conservation. On this page’s design point the field-integrated vacuum thrust matches the one-dimensional value.

The wall angle rises a few degrees past the corner and then comes back. That overshoot is converged and shock-free. It is listed in the known limits.

Minimum-length nozzle Mach field with C+ and C− characteristics — Fan at the sharp corner, kernel, transition, then a uniform exit wedge. Color is Mach. Lines are the stored C+ and C− connectivity.

MLN wall contour with metrics and wall-angle color — Dashboard contour at γ = 1.25, N = 48. The metric box reads L/R_t = 9.76 and θ_max = 18.4°. The 11.7° corner in the text is the γ = 1.4 case, not this gas.

MLN radius, wall angle and Mach along x — r(x), θ(x), M(x). Exit angle heads toward zero as the mesh is refined. Peak angle is the corner, not the exit.

Mach and static pressure over chamber pressure along the MLN wall — Edge Mach and p/p_c along the wall. These are the isentropic edge values a heat-flux estimate starts from.

design_ideal

Ideal contour

The ideal bell replaces the sharp corner with a circular-arc throat. The march starts on a Sauer transonic initial-value line lying on that arc, then runs the full characteristic lattice out to a uniform axial exit.

Version 0.7 stopped building this shape by stretching an MLN and gluing an arc on the front. The lattice is the design. Exit area again follows the mass-carrying streamline, so geometric ε reflects the discharge of that start line rather than a forced rescale onto the one-dimensional A/A*.

Sauer’s series is the published one (Mendelson, NASA TM X-53084). It is comfortable for throat curvature R_c/R_t ≳ 2. Smaller radii or low exit Mach fall back to a uniform near-sonic start, and that fallback is written into meta["warnings"].

On the shared design point the ideal bell is the longest of the four, L/R_t = 11.26, and keeps 99.5% of one-dimensional vacuum thrust.

Ideal nozzle characteristic field, longer canceling region than the MLN — Ideal MOC field. The canceling region is longer than the MLN because the throat arc swallows part of the turning that a sharp corner does in one fan.

Ideal nozzle wall contour — Ideal wall, metrics box, wall-angle color.

Ideal nozzle radius, angle and Mach profiles — Radius grows more slowly than the MLN. Mach along the wall is monotonic.

Ideal nozzle wall Mach and pressure ratio — Edge Mach and pressure ratio for the long bell.

Wall Mach, static pressure and temperature for the ideal nozzle at 5 MPa and 3300 K — Same ideal wall with chamber state p_c = 5 MPa, T_c = 3300 K: Mach, static pressure, and static temperature. This is the figure you hand a cooling-channel estimate.

design_tic · truncate_fraction

Truncated ideal contour

A TIC is the ideal family cut at a fraction of its length. truncate_fraction is that fraction: 0.80 means the axial station at 80% of the ideal length. The exit angle is then whatever the ideal wall was doing there, so you pay a divergence loss for the mass you saved.

At 80% on this design point, L/R_t drops from 11.26 to 9.01 and vacuum thrust moves from 99.5% to 99.3% of one-dimensional. The interesting part of the trade is shorter than that. The sweep from 50% to 100% is the first figure below.

Flight bells are often this compromise. The code keeps the parent MOC field so you can still see which waves were thrown away with the missing length.

Truncation trade-off: vacuum thrust coefficient versus length for TIC fractions from 50 to 100 percent — Field-integrated vacuum C_F against length as the cut moves from 50% to 100% of the ideal bell. Labels are the truncate fraction.

Characteristic field of a truncated ideal nozzle — TIC characteristic field. The exit station cuts through waves that the full ideal bell would still be canceling.

Truncated ideal contour wall — The shortened wall. Exit slope is visibly nonzero compared with the ideal contour above.

TIC wall radius, angle and Mach — Profiles stop early. θ at the new exit is the truncation penalty.

TIC wall Mach and pressure — Edge thermo on the truncated wall. Exit pressure is higher than the full ideal exit because Mach has not finished rising.

design_rao · design_rao_control · design_rao_toc

Rao thrust-optimised bell

Rao’s 1958 contour maximises thrust at a prescribed length. In rockmoc that length is a fraction of the 15° cone with the same area ratio. length_fraction=0.8 is the usual engineering choice, which is why this bell is 4.80 R_t long here while the MLN is 10.45.

The 0.7 solver builds it as MOC. A control surface satisfies Rao’s conditions (V cos(θ−μ)/cos μ held in the thrust balance, and r ρ V² sin²θ tan μ matched), mass flow closes the exit radius, and the wall is a Goursat solution off that surface. The previous version ignored its own kernel and drew a cubic Hermite through two angles.

Length and exit radius land within about 0.1% of the request, and Mach rises monotonically along the wall. If you ask for a bell shorter than the lattice can support, the nearest feasible length is used and a warning is stored.

design_rao_toc is the length-referenced TOC variant: the length is taken from an ideal MOC bell rather than from the cone. Attachment angle stays above exit angle, which is the TOP/TOC signature (a validation case records θ_n = 26.8° and θ_e = 10.6°).

Rao bell characteristic field from the smooth throat to the mass-flow wall — Rao field: smooth-throat lattice, control surface, mass-flow wall. The exit is short and the turning is concentrated near the throat.

Rao thrust-optimised contour — Rao wall. Compact next to the ideal bell of the same exit Mach.

Rao radius, angle and Mach profiles — θ drops quickly after the attachment region. Exit angle stays finite on purpose.

Rao wall Mach and pressure ratio — Edge Mach and p/p_c on the short bell.

The other geometries

Cone, NASA planar MLN, aerospike

15° cone

design_conical is a straight wall at a constant half-angle, 15° unless you change it. Length is (R_e − R_t) / tan θ_c with R_e from the one-dimensional area ratio. Divergence efficiency is about (1 + cos θ_c)/2. It exists so a comparison has a textbook baseline. On this design point its field-integrated vacuum thrust is 96.9% of one-dimensional, close to the Rao bell and well below the MLN.

Planar NASA MLN

design_planar_mln_nasa follows NASA TM-X-1502 (Vanco and Goldman, 1968): two-dimensional, sharp throat, θ_max = ν_e/2, exit half-height equal to the one-dimensional area ratio. The figure uses M_e = 3, γ = 1.4, where that corner is 24.88°. The paper’s own sample, M_e = 4.05 and γ = 1.36, gives θ_max = 34.96° and y_e/h_t = A/A* = 12.98, with the throat at (0, 1). Planar results did not move in the 0.7 axisymmetric fix.

Annular aerospike

design_spike builds a Lee / Vernacchia plug. The spike angle at the lip satisfies δ = π/2 − ν_e. Truncation, a base-pressure model, and a primary cowl are separate calls: design_spike_truncated, truncated_spike_with_base, design_spike_with_cowl. The figure is p_c = 5 MPa, T_c = 3300 K, γ = 1.2, molar mass 22, lip radius 0.12 m, expansion ratio 18: I_sp = 254 s, C_F = 1.445, exit Mach 3.69. That is a constant-γ isentropic plug, not a reacting plume.

15 degree cone overlaid with MLN, ideal and Rao walls — 15° cone against MLN, ideal, and Rao. The cone is the long straight reference Rao’s length fraction is measured from.

Planar NASA minimum-length nozzle with characteristic nodes, Me 3, gamma 1.4 — Planar MLN, TM-X-1502 rules, M_e = 3, γ = 1.4. Points are the characteristic nodes. Exit height is fixed by A/A*.

Annular aerospike spike contour with shroud lip radius — Spike contour from the Lee / Vernacchia march. The lip radius is marked. Radius falls toward the tip.

Aerospike contour from the library dashboard plotter — The same family from `plot_aerospike`, the figure `result` dashboards use for a plug.

What the march actually solves

Characteristics, throat, gas, exit plane

Irrotational, isentropic, perfect-gas (or frozen-table) flow. The equations live in src/rockmoc/moc/equations.py and were checked against the exact conical source flow.

Wave slopes in the meridional plane. μ = arcsin(1/M)(dr/dx)_± = tan(θ ± μ)

Planar, δ = 0. Riemann invariantsalong C+: θ − ν = constant
along C−: θ + ν = constant

Axisymmetric source, δ = 1. This sign pair is the 0.7 fix.along C+: d(θ − ν)/dx = − sinθ · sinμ / (r · cos(θ+μ))
along C−: d(θ + ν)/dx = + sinθ · sinμ / (r · cos(θ−μ))

Interior points are C+ ∩ C−. The axis point forces θ = 0. Near r = 0 the source looks singular; sinθ/r stays finite and the axis unit process handles it. Unit processes use predictor–corrector with averaged angles, in the style of TM-X-1502 and Zucrow & Hoffman.

Local error on the exact conical source flow falls as h³: 7.1×10⁻⁵, 8.9×10⁻⁶, 1.1×10⁻⁶, 1.4×10⁻⁷ as the step goes from 0.08 down to 0.01.

Sauer transonic initial line: Mach and flow angle versus radius for several throat curvatures — Sauer initial-value line for R_c/R_t = 1.5, 2, 3 and 4. Left: Mach across the throat. Right: flow angle. Tighter curvature starts the march further above Mach 1.

Exit-plane Mach and flow angle for MLN, ideal, TIC and Rao — Exit plane taken from the real MOC net, not a synthetic ray. MLN and ideal are nearly uniform and axial. TIC and Rao still carry angle, which is why their thrust sits under the one-dimensional value.

Variable gamma along Mach compared with a constant gamma of 1.22, and the Prandtl-Meyer angle that results — A multi-γ table from γ_c = 1.25, γ_t = 1.22, γ_e = 1.17, interpolated with a monotone cubic. Right: Prandtl–Meyer angle against a constant γ = 1.22. The MOC march calls `gamma_at_mach` at each point.

Inviscid ideal wall and the same wall offset by displacement thickness — Displacement thickness on the ideal wall at p_c = 5 MPa, T_c = 3300 K. The corrected radius is what you export if the CFD mesh should start outside the boundary layer. The model is an integral compressible estimate, in the spirit of NASA TM-X-2343, not a solved boundary-layer ODE.

Gases

Core installs ship PerfectGas and VariableGammaGas. Teaching presets are frozen engineering estimates, not flight thermochemistry. Optional extras replace the chamber γ, R, and T_c and then the MOC still runs frozen.

Preset	γ	R, J/(kg·K)	What the name means
air	1.40	287	Dry air, the teaching default
air_hot	1.33	287	Hot-air stand-in
lox_rp1	1.22	355	LOX/RP-1 near stoichiometric, frozen
lox_rp1_fuel_rich	1.25	370	Fuel-rich LOX/RP-1 estimate
lox_lh2	1.20	550	LOX/LH2 frozen estimate
lox_ch4	1.20	380	Methalox frozen estimate
n2o_htpb	1.25	320	Hybrid-ish teaching estimate
monoprop_h2o2	1.30	380	High-test peroxide decomposition

RocketCEA, Cantera, and CoolProp are extras: pip install "rockmoc[cea]", [cantera], [coolprop]. Tests skip them when the package is absent.

design_production

The call that ships a contour

One function runs the accuracy stack. Pick kind as mln, ideal, tic, or rao. The rest is optional and each piece can be turned off.

Fine MOC design. The chosen family, on a denser characteristic count than a homework sketch.
Expansion table. Perfect gas, or a multi-γ table (γ_c, γ_t, γ_e) such as (1.25, 1.22, 1.18).
Sauer throat metadata. The initial-value line is recorded with the design, not invented after the fact.
Boundary layer. Optional δ* growth of the inviscid wall. prod.bl.corrected is the wall you machine or mesh.
Field thrust. Momentum plus pressure on the exit station of the net, normalised by the field mass flow.
Altitude sweep. Ambient pressure from an ISA-style p_a(h), so C_F and I_sp versus height for that fixed contour.
Mesh certificate and Monte Carlo. Optional. mesh_certificate and monte_carlo_mln quantify grid and γ uncertainty.

Production-path characteristic field on a finer mesh — Production MOC field. Finer than the N = 40 sketches above; this is the mesh the package figures use.

Production nozzle contour — Production wall with the dashboard metric box.

Production wall radius, angle and Mach — Full wall state for the package: radius, angle, Mach.

Production wall Mach and pressure — Edge Mach and pressure ratio on the production contour.

Production boundary-layer correction, inviscid wall versus displacement thickness — Inviscid wall and δ*-corrected wall. A validation check requires δ*_e > 0 and corrected radius at least the inviscid radius (one case: 81.2 mm grows to 82.3 mm).

Production altitude performance dashboard — Off-design package: thrust coefficient and specific impulse as ambient pressure falls with altitude.

Trades and uncertainty

Altitude, mixture ratio, area ratio, γ, mesh

Vacuum C_F uses Sutton’s one-dimensional form as the reference. Divergence loss uses the exit angle, not a tax on how curved the wall looked upstream. When a net exists, field_integrated_cf integrates the exit plane.

Thrust coefficient and specific impulse versus altitude for area ratio 6.7 at 5 MPa — Ideal contour, ε = 6.74, p_c = 5 MPa, from sea level to 60 km. C_F climbs as p_a drops and asymptotes at the vacuum value.

MLN performance versus altitude from the dashboard plotter — MLN altitude sweep at the dashboard point, γ = 1.25, ε = 5.83, from `plot_performance_vs_altitude`. Not the same contour as the γ = 1.2 altitude figure in the previous card.

Oxidizer to fuel ratio sweep of gamma, area ratio and performance — O/F from 1.8 to 3.2, LOX/RP-1 style. RocketCEA supplies chamber γ when it is installed. Otherwise the plotter maps mixture ratio onto γ with an engineering correlation and redesigns the bell.

Area ratio sweep of nozzle length and performance — Area-ratio sweep from the dashboard. Larger ε buys vacuum performance and costs length.

Area ratio sweep generated for the site — Site sweep at γ = 1.2, R_t = 50 mm, 16 characteristics. Same question, coarser mesh, so use it for the trend.

Effect of gamma at fixed exit Mach on nozzle length and area ratio — γ sweep at fixed M_e = 3. Lower γ stretches both the Prandtl–Meyer turning and the area ratio, so the bell gets longer.

Gamma sweep dashboard — Dashboard γ sweep. Pair it with the site figure: one is the plotter’s house style, one is the raw trade.

Monte Carlo histograms of length, vacuum thrust coefficient and area ratio — Monte Carlo, 40 samples. γ is drawn from N(1.2, 0.03) and the characteristic count from {12, 16, 20}. Histograms are L/R_t, one-dimensional vacuum C_F, and area ratio.

Grid convergence of length, exit angle and maximum wall angle versus number of characteristics — Richardson study on the ladder N = 12, 24, 48, 96. Observed order is about 1.52 for L/R_t and 1.20 for exit wall angle. Guides of slope 1 and 2 are drawn on the error plot. `analysis.grid_convergence` also reports the grid convergence index.

Bar chart of import time, test-suite time and ideal-design time, version 0.6.1 versus 0.7.0 — The library itself, measured for 0.7.0. Import 1.57 s → 0.03 s. Full test suite 337 s → 40 s, from float kernels in `rockmoc._kernels` (Numba if you install `[fast]`). One ideal design at N = 40: 16.1 s → 0.6 s.

Figure atlas

Every plot the project draws

Click a figure to enlarge it. Filters only hide cards. Nothing here is a stock render: each file is an output of make_images.py, make_images_extra.py, or tests/generate_all_graphs.py.

Ideal nozzle Mach field used as the page hero — Ideal field, borderless. The hero image.

MLN characteristic field — MLN field with fan, kernel, and canceling region.

MLN dashboard MOC field — MLN dashboard field: C+, C−, wall nodes, axis nodes.

Ideal dashboard MOC field — Ideal dashboard field.

TIC dashboard MOC field — TIC dashboard field.

Rao characteristic field — Rao control-surface field.

Production MOC field — Production-path field on the fine mesh.

Family overlay with wall-angle inset — Four walls, throat-normalised, wall-angle inset.

Cone versus contoured bells — 15° cone versus MLN, ideal, and Rao.

Boundary layer wall offset — δ* offset on the ideal wall.

Production boundary layer figure — Production boundary-layer figure.

Production wall profiles — Production r, θ, M.

MLN Mach and pressure — MLN edge Mach and p/p_c.

Ideal Mach and pressure — Ideal edge Mach and p/p_c.

TIC Mach and pressure — TIC edge Mach and p/p_c.

Rao Mach and pressure — Rao edge Mach and p/p_c.

Production Mach and pressure — Production edge Mach and p/p_c.

Wall Mach, pressure and temperature — Ideal wall Mach, pressure, and temperature at 5 MPa and 3300 K.

Exit plane Mach and angle — Exit-plane Mach and flow angle, four families.

Sauer throat initial line — Sauer throat line versus R_c/R_t.

Variable gamma tables — γ(M) and ν(M), variable versus constant.

TIC truncation trade-off — TIC length versus vacuum thrust.

Altitude performance — Altitude sweep, ε = 6.74, 5 MPa.

MLN altitude performance — MLN altitude dashboard.

Mixture ratio sweep — Oxidiser-to-fuel sweep.

Area ratio sweep dashboard — Area-ratio sweep, dashboard.

Area ratio sweep — Area-ratio sweep, site figure.

Gamma sweep — γ sweep at fixed exit Mach.

Monte Carlo histograms — Monte Carlo on γ and mesh density.

Solver speed, 0.6.1 versus 0.7.0 — Import, tests, and one design: 0.6.1 versus 0.7.0.

Aerospike spike — Lee / Vernacchia spike, ε = 18.

OpenFOAM wedge blockMesh preview — OpenFOAM wedge preview: one block, axis collapsed, wall grading 0.12. Every second axial line is drawn. Checked with blockMesh and checkMesh on OpenFOAM v2412.

Version 0.7

What changed, and what it was checked against

The axisymmetric numbers moved because the physics was wrong, not because a mesh default changed. Planar NASA results are the same as before.

C− sign

Compatibility on C− had the wrong sign, and the invariants θ±ν were swapped in every unit process. The axis point was being reached along an upstream C+. The source-flow test above is the check.

MLN corner

θ_max is solved so the last corner wave hits the axis at M_e. For M_e = 3, γ = 1.4 that is 11.7°, not the old 25.9° blend of a partial source correction and a 0.52·ν_e patch. Exit area is no longer rescaled to hide the miss.

Ideal lattice

The ideal contour is a characteristic lattice from a Sauer line on the circular-arc throat. It is not an MLN with an arc glued on and a 1.55× stretch.

Rao MOC

Rao TOC uses the control-surface ODE, a mass-matched exit, and a Goursat wall. It hits the requested length and exit radius to about 0.1%.

Exit thrust

C_F integrates momentum and pressure on the exit plane of the net and divides by the field mass flow. The old “exit ray” was synthetic. An MLN reproduces the one-dimensional vacuum value.

Sauer line

The throat initial-value line is Sauer as published. A made-up Hall factor was removed. Hall’s higher-order series is still not implemented.

Literature campaign

The deep run is 43 cases, 170 metrics, 12 sources, median error about 0%. Thermodynamic table errors sit under about 0.02%. The largest intentional disagreement is axisymmetric θ_max versus planar ν_e/2, which is about 4% by construction and is not a failure. Re-run with py -3.12 tests/run_deep_validation.py. A separate offline suite, run_all_validation(), records 50 of 50 checks.

Anchor	What is checked
Anderson, Modern Compressible Flow	ν(M), A/A, p_e/p_c at γ = 1.4. M = 2: ν = 26.380°. M = 3: ν = 49.757°, A/A = 4.2346.
NACA Report 1135 (Ames, 1953)	Compressible-flow equations and tables behind those digits.
NASA TM-X-1502, Vanco & Goldman, 1968	Planar sharp-throat MLN. Sample M_e = 4.05, γ = 1.36: θ_max = ν_e/2 = 34.96°, y_e = A/A*.
NACA RM E8J12, Shames & Seashore, 1948	Two-dimensional sharp-edge design data, five cases from M_e = 2 to 4.
NASA TM-X-2343, Goldman & Vanco, 1971	MLN plus boundary-layer displacement. The δ* module follows that spirit.
Rao, 1958, Jet Propulsion	Optimum-thrust contour. Length versus the 15° cone, and θ_n > θ_e.
Lee, 1963 / Vernacchia spike notes	Annular plug. δ = π/2 − ν_e, tip radius toward zero, positive vacuum thrust.
Sutton & Biblarz	Ideal one-dimensional thrust coefficient.
Zucrow & Hoffman	Axisymmetric unit processes. The source-term signs above.
Huzel & Huang	15° conical nozzle as the industry baseline.
Sauer / Kliegel, via Mendelson TM X-53084	Transonic throat initial-value line. M > 1 on the line, θ = 0 on the axis.
Shapiro	Prandtl–Meyer function and ν_max as M → ∞.

Anderson / NACA digits at γ = 1.4, as stored by the validation suite. “Solver” is rockmoc’s perfect-gas kernel.
M	ν reference	ν solver	A/A* reference	A/A* solver
1.5	11.905°	11.90521°	1.1762	1.17617
2	26.380°	26.37976°	1.6875	1.68750
3	49.757°	49.75735°	4.2346	4.23457
4	65.785°	65.78482°	10.7188	10.71875
5	76.920°	76.92022°	25.000	25.00000

From a checkout to a wall file

Install, design, export

Python 3.10 or newer. Core dependencies are NumPy and SciPy. Plots need rockmoc[plot]. Interactive Plotly fields need [viz]. Numba kernels need [fast].

from rockmoc import DesignSpec, PerfectGas, design_ideal
from rockmoc.export import export_dxf, export_blockmesh

gas  = PerfectGas(gamma=1.2, R=350.0)
spec = DesignSpec(gas, throat_radius=0.05, exit_mach=3.0, n_characteristics=60)
r    = design_ideal(spec, store_net=True)

print(r.summary())
r2 = r.with_boundary_layer(pc=5e6, Tc=3300)
export_dxf(r2.contour, "nozzle.dxf", units="mm")
export_blockmesh(r2.contour, "case/system/blockMeshDict")
r.plot(show=False, save_dir="figures")   # needs rockmoc[plot]

from rockmoc import (
    DesignSpec, PerfectGas, design_mln, design_tic, design_rao,
    design_conical, design_planar_mln_nasa, design_spike,
    design_production, compare_nozzle_designs,
)

spec = DesignSpec(PerfectGas(1.25, R=350), throat_radius=0.05, exit_mach=3.0)
mln  = design_mln(spec, store_net=True)
tic  = design_tic(DesignSpec(PerfectGas(1.25), throat_radius=0.05,
                             exit_mach=3.0, truncate_fraction=0.80))
rao  = design_rao(spec, length_fraction=0.80)
cone = design_conical(spec, half_angle_deg=15)

# NASA TM-X-1502 sample: Me = 4.05, gamma = 1.36
planar = design_planar_mln_nasa(4.05, 1.36, dnu_deg=0.1, throat_half_height=0.05)

spike = design_spike(Pc=5e6, Tc=3200, gamma=1.25, molar_mass=22,
                     Re=0.15, expansion_ratio=25, n_points=5000)

prod = design_production(spec, kind="mln", apply_bl=True,
                         multi_gamma=(1.25, 1.22, 1.18), pc=5e6, Tc=3200)
print(compare_nozzle_designs(spec).to_markdown())

# CLI needs no extra dependency beyond the install
rockmoc design --type ideal --Me 3 --gamma 1.2 --Rt 0.05 --export csv,dxf,openfoam --units mm -o out
rockmoc design --type rao --area-ratio 25 --length 0.8 --export cad -o out
rockmoc run nozzle.toml
rockmoc --version

# nozzle.toml
[gas]
gamma = 1.2
R = 350.0

[design]
type = "rao"
throat_radius = 0.05
exit_mach = 3.5
length_fraction = 0.85

[export]
formats = ["csv", "cad", "dxf", "openfoam"]
units = "mm"

from rockmoc import DesignSpec, design_mln, compare_nozzle_designs, explainer

spec = DesignSpec.student(Me=3.0, gas_name="lox_rp1", Rt_mm=50)
mln  = design_mln(spec)
print(mln.report(length_unit="mm"))          # millimetres and degrees
mln.lab_report("lab.md", student_name="Alex", course="AE")
print(explainer("mln"))
print(compare_nozzle_designs(spec).to_markdown())

Export formats

CSV keeps x, r, θ, and M. CAD points are a header-less x y z list for a curve-through-points feature in SolidWorks or CATIA. DXF is R12 so older CAD importers accept it. OpenFOAM is an axisymmetric wedge blockMeshDict.

Plots can stay on screen or go to a directory: result.plot(show=False, save_dir="out"). Pass a light theme when the figure has to survive a printed report. SVG is the default vector output.

Where the code sits

thermo/ gases and expansion tables. moc/ unit processes, planar NASA, axisymmetric kernel, transition, ideal, Rao. designers/ the public builders. aerospike/ the plug. analysis/ boundary layer, throat, atmosphere, uncertainty, comparison. performance/ one-dimensional and field thrust. plot/ the dark figures. student/ reports, presets, explainers. validate/ the literature suite.

Preview of the axisymmetric wedge mesh written as blockMeshDict — What `export_blockmesh` writes: one wedge, collapsed axis, spline wall, grading toward the wall. The DXF path is an R12 polyline with wall and axis layers, checked with ezdxf.

Public calls

Call	Returns
design_mln / design_ideal / design_tic	Axisymmetric wall, metrics, optional net
design_rao / design_rao_control / design_rao_toc	Thrust-optimised or length-referenced bell
design_conical	Straight wall at a fixed half-angle
design_planar_mln_nasa	TM-X-1502 planar MLN
design_spike, design_spike_truncated, design_spike_with_cowl, design_primary_cowl, truncated_spike_with_base	Plug, truncated plug, base pressure, cowl
design_production	Fine mesh, table, throat, optional BL, field C_F, altitude
compare_nozzle_designs	Family table, including the cone. Markdown, LaTeX, or CSV
field_integrated_cf, integrated_performance, off_design_altitude_sweep	Exit-plane thrust, one-dimensional metrics, C_F(h)
correct_wall_bl, sauer_throat_ivl	δ* wall, transonic start line
monte_carlo_mln, mesh_certificate, sensitivity_one_at_a_time	Uncertainty and a mesh pass/fail
mesh_convergence_planar, mesh_convergence_axisym, grid_convergence	Mesh studies and Richardson order
example_gas, explainer, KNOWN_LIMITS, run_all_validation	Teaching presets, short explainers, the check suite

For a lab course

Reports

result.report(length_unit="mm") prints millimetres and degrees. lab_report writes a Markdown skeleton with a checklist. explainer("rao") is a short “why this shape” note.

Errors that name the fix

Passing both an exit Mach and an area ratio raises DesignError and shows a working line. Only one expansion target is legal, because the other two are determined by γ.

Notebooks and problems

Four tutorials under examples/notebooks: a first MLN, a family comparison, altitude performance, and the NASA planar sample. Worked problems are in docs/student_problems.md. Symbols are in docs/glossary.md.

Read this before you quote a number

What the contour is allowed to mean

In scope

Homework, a first sizing, a family trade at fixed γ, a mesh study, a wall to drop into CAD or a CFD mesher, and a lab figure with the characteristic net still visible. Axisymmetric bells, the planar NASA MLN, and the constant-γ aerospike each state their assumptions in the result metadata.

Known numerical limits

Hall’s higher-order throat series is not in the code. Sauer is the start, and it is the right start for R_c/R_t ≳ 2. Below that, the ideal solver says it used a uniform near-sonic line. A very short Rao request can fall off the feasible lattice; you get the nearest length plus a warning. The axisymmetric MLN wall angle bumps up a few degrees after the corner before it turns back. A planar ideal can show about a 1° first-order bump after the arc attaches.

Out of scope

Shocks, separation, side loads, film cooling, regenerative-channel conjugate heat transfer, structural thickness, multi-phase or reacting flow, and anything you would sign as flight hardware. Chemistry backends freeze a γ and an R. They do not integrate a reacting nozzle flow. The boundary layer is an integral estimate.

The order of work the docs recommend is: converge the mesh, export the wall, then CFD, thermal, and structural analysis. KNOWN_LIMITS is importable so a generated report can print this list instead of forgetting it.

Source PyPI Full guide Changelog Validation notes Author