Choked Cryogenic Flow and Cavitating-Venturi Validation of OpenFLUME

Validation against experiment: maximum flow of nitrogen and methane through nozzles and a sharp-edged orifice (NASA TP-1484, TM X-71760), and liquid-hydrogen venturi cavitation (NASA CR-2054)

Generated by scripts/choked-flow-validation-report.ts. All model numbers and figures come from live solves of the current solver. Cases, transcription QC and models are defined once in src/validation/chokedFlow.ts and src/validation/cavitationHord.ts; the CI gates are src/validation/__tests__/chokedFlow.test.ts, src/validation/__tests__/cavitationHord.test.ts (scientific tier, npm run test:all) and src/validation/__tests__/experimentalData.test.ts (source hash, fast tier).

Abstract

OpenFLUME represents a flow restriction with one of two algebraic closures: the orifice, whose expansibility factor uses the real-fluid isentropic exponent κ = c_s²ρ/P and chokes at the corresponding critical pressure ratio, and the cavitatingVenturi, which treats the liquid as incompressible down to its vapor pressure and chokes there. This report compares both with 352 measured maximum mass fluxes of nitrogen and methane, 326 through four converging–diverging nozzles (Simoneau & Hendricks, NASA TP-1484, 1979) and 26 through a sharp-edged orifice (Simoneau, NASA TM X-71760, 1975), and checks the venturi closure's throat-pressure assumption against 32 developed liquid-hydrogen cavities (Hord, Anderson & Hall, NASA CR-2054, 1972). The agreement depends on the inlet regime, and the report is organized by it. For subcooled liquid nozzle inlets (P0 ≥ 2 P_v(T0)) the choked-liquid closure predicts the measured flux with a median ratio of 1.045 (range 0.966–1.171), and for gaseous inlets the expansibility closure gives 0.944–1.047. Two regimes are outside what the closures represent. Near saturation the measured flux exceeds the prediction (median 0.879, down to 0.154), the signature of liquid that stays metastable below its vapor pressure; an equilibrium model under-predicts there too. For dense supercritical inlets the expansibility closure, with κ frozen at the inlet state, over-predicts by a median +30 % (up to +57 %), where an isentropic homogeneous-equilibrium (HEM) calculation has a median ratio of 1.029. Through the sharp-edged orifice, the orifice closure with the textbook C_d = 0.61 at the measured back pressure is within −6.1 % to +1.3 % for subcooled inlets. In Hord's venturi the cavity leading edge sits 6–55 % below P_v(T0) because evaporation cools it, so for a choked venturi at these conditions the closure would under-predict the flow by 10–41 %.

Introduction

Feed systems for cryogenic propellants are sized around restrictions that choke: injector orifices, cavitating venturis used as flow regulators, and nozzles and valves that pass flashing liquid. A network code reduces each of these to an algebraic relation between upstream state, downstream pressure and mass flow, and the quality of that relation sets the predicted flow. The NASA Lewis choked-flow program of the 1970s measured exactly that quantity, the maximum mass flux through a restriction as a function of the stagnation state, across subcooled, saturated, near-critical, supercritical and gaseous inlets, and Hord's cavitation program measured the cavity pressures and temperatures that a cavitating-venturi closure assumes. Both are therefore direct tests of the closures, independent of any network.

Experiments and Data

Nozzles (NASA TP-1484)

Simoneau and Hendricks measured the maximum (choked) mass flux of nitrogen through four converging–diverging nozzles (7° and 3.5° conical, elliptical, and two-dimensional) and of methane and hydrogen through the elliptical nozzle, with stagnation temperatures from well below to above the critical temperature and stagnation pressures up to about 9 MPa. Each table row gives T0, P0, the back pressure, the throat-to-stagnation pressure ratio, the measured G = ṁ/A_throat, and properties computed by the authors (stagnation entropy and saturation pressures). The digitized run-summary columns are used here; the axial pressure profiles were not digitized. The hydrogen table is excluded: most of its rows have no transcribed stagnation state.

Sharp-edged orifice (NASA TM X-71760)

Simoneau measured the maximum flow of subcooled liquid nitrogen through a 0.318 cm sharp-edged orifice at stagnation temperatures of 95–130 K, with the back pressure, the back (jet) temperature and the authors' discharge coefficient C_d = G/√(2ρ0(P0 − P_b)) tabulated.

Transcription quality control

The CSVs were extracted from scanned tables by OCR with per-cell flags, and some rows carry shifted or mis-read columns. A row is used only if every field the comparison needs is present and unflagged, and if values the paper tabulates redundantly agree with values recomputed from the transcribed fields: the saturation pressure at T0 within 5 % of CoolProp (both datasets); for the orifice also the back temperature within 1.5 K of T_sat(P_b) and the tabulated C_d within 3 % of G/√(2ρ0(P0 − P_b)). These checks use only the transcribed numbers and the equation of state, never a flow model.

Nozzle	Source	Rows	Usable
7° conical nozzle	TP-1484 Table VI	119	108
3.5° conical nozzle	TP-1484 Table VII	46	42
2-D nozzle	TP-1484 Table VIII	91	85
elliptical (N₂) nozzle	TP-1484 Table IX	152	65
elliptical (CH₄) nozzle	TP-1484 Table X	39	26
Sharp-edged orifice	TM X-71760 Table I	109	26

Reason a row was excluded (first failing check)	Rows
T0, P0 or G flagged OCR-uncertain, fused, unparsed or ditto	62
T0, P0, G (or Pb, T_b) blank	59
row flagged by the digitizer (fit failed / fields blank in report)	52
tabulated C_d ≠ G/√(2ρ0(P0 − P_b)) ± 3 %	17
tabulated P_v(T0) disagrees with CoolProp by > 5 %	10
back temperature ≠ T_sat(P_b) ± 1.5 K (column shift)	4

The orifice C_d check makes the orifice comparison at the measured back pressure equivalent to comparing 0.61 Y with the usable rows' discharge coefficients, which range from 0.446 to 0.618; Y is the closure's expansibility factor (below).

Inlet regimes

Results are segmented by the stagnation state, with T_c = 126.19 K and P_c = 3.396 MPa for nitrogen (T_c = 190.56 K for methane):

Regime	Definition	Physics at the restriction
subcooled liquid, P0/Pv ≥ 2	T0 < T_c, P0 ≥ 2 P_v(T0)	liquid accelerates to near P_v, then flashes
near-saturated liquid, P0/Pv < 2	T0 < T_c, P0 < 2 P_v(T0)	flashing begins close to the inlet; non-equilibrium matters most
supercritical, 1 ≤ T0/Tc < 1.5	T_c ≤ T0 < 1.5 T_c	dense fluid whose compressibility changes strongly on expansion
gas, T0/Tc ≥ 1.5	T0 ≥ 1.5 T_c	near-ideal gas

Models

Each data point is one OpenFLUME steady network: a boundary node at (P0, T0), one branch, and a downstream boundary node, with the CoolProp equation of state for the fluid. Nothing is fitted.

• Liquid inlets (T0 < T_c): cavitatingVenturi, choked branch, ṁ = C_d A √(2ρ0(P0 − P_v(T0))). The downstream boundary is set to 0.5 P_v(T0), so the closure is evaluated choked, the state in which the experiments were run.
• Supercritical and gaseous inlets: orifice, ṁ = C_d A Y(r, κ) √(2ρ0 ΔP) with κ = c_s²ρ/P at the inlet state and choking at r* = (2/(κ + 1))^(κ/(κ−1)); the downstream boundary is at 0.05 P0, below r*.
• Sharp-edged orifice data: the orifice closure at the measured back pressure (the model one would build for this restriction), and the cavitatingVenturi closure for comparison.
• Discharge coefficient: C_d = 1 for the nozzles (the papers compare their data with ideal flow at the measured throat area) and C_d = 0.61 for the sharp-edged orifice (the standard high-Reynolds-number value; White, Fluid Mechanics).
• Reference model (not an OpenFLUME closure): isentropic homogeneous equilibrium flow, G* = max over P of ρ(P, s0)√(2(h0 − h(P, s0))), computed directly from CoolProp and multiplied by the same C_d. It separates the error of each closure's idealization from the non-equilibrium physics that no equilibrium model contains. Its implementation is checked against the ideal-gas critical flux of nitrogen at 300 K (gate test, < 0.5 %).

All 378 solves converged.

Results

Nozzles

Ratios are G_model/G_measured; "HEM" is the reference model.

Data	Regime	OpenFLUME closure	n	Median	Range	Mean abs. deviation	HEM median	HEM range
Nozzles (all four)	subcooled liquid, P0/Pv ≥ 2	cavitatingVenturi (choked)	115	1.045	0.966–1.171	4.5 %	1.058	0.969–1.166
Nozzles (all four)	near-saturated liquid, P0/Pv < 2	cavitatingVenturi (choked)	124	0.879	0.154–1.111	17.4 %	0.968	0.558–1.146
Nozzles (all four)	supercritical, 1 ≤ T0/Tc < 1.5	orifice (κ = c²ρ/P, choked)	81	1.304	1.075–1.568	31.8 %	1.029	0.981–1.213
Nozzles (all four)	gas, T0/Tc ≥ 1.5	orifice (κ = c²ρ/P, choked)	6	1.036	0.944–1.047	4.5 %	1.031	0.940–1.042

By nozzle and fluid:

Nozzle	Fluid	Regime	Closure	n	Median	Range	HEM median	HEM range
7° conical nozzle	N₂	subcooled liquid, P0/Pv ≥ 2	cavitatingVenturi (choked)	37	1.028	0.991–1.062	1.051	1.022–1.067
7° conical nozzle	N₂	near-saturated liquid, P0/Pv < 2	cavitatingVenturi (choked)	43	0.877	0.502–1.003	0.962	0.558–1.050
7° conical nozzle	N₂	supercritical, 1 ≤ T0/Tc < 1.5	orifice (κ = c²ρ/P, choked)	26	1.422	1.172–1.548	1.031	1.004–1.053
7° conical nozzle	N₂	gas, T0/Tc ≥ 1.5	orifice (κ = c²ρ/P, choked)	2	1.036	1.034–1.038	1.031	1.029–1.034
3.5° conical nozzle	N₂	subcooled liquid, P0/Pv ≥ 2	cavitatingVenturi (choked)	14	1.040	0.998–1.055	1.059	1.047–1.096
3.5° conical nozzle	N₂	near-saturated liquid, P0/Pv < 2	cavitatingVenturi (choked)	26	0.872	0.311–1.004	0.924	0.583–1.066
3.5° conical nozzle	N₂	gas, T0/Tc ≥ 1.5	orifice (κ = c²ρ/P, choked)	2	1.045	1.043–1.047	1.040	1.039–1.042
elliptical nozzle	N₂	subcooled liquid, P0/Pv ≥ 2	cavitatingVenturi (choked)	14	1.019	0.968–1.079	1.056	1.037–1.069
elliptical nozzle	N₂	near-saturated liquid, P0/Pv < 2	cavitatingVenturi (choked)	15	0.879	0.635–0.961	0.995	0.779–1.036
elliptical nozzle	N₂	supercritical, 1 ≤ T0/Tc < 1.5	orifice (κ = c²ρ/P, choked)	36	1.251	1.075–1.508	1.030	0.981–1.213
2-D nozzle	N₂	subcooled liquid, P0/Pv ≥ 2	cavitatingVenturi (choked)	30	1.074	1.006–1.171	1.080	1.052–1.166
2-D nozzle	N₂	near-saturated liquid, P0/Pv < 2	cavitatingVenturi (choked)	39	0.925	0.154–1.111	0.996	0.663–1.146
2-D nozzle	N₂	supercritical, 1 ≤ T0/Tc < 1.5	orifice (κ = c²ρ/P, choked)	14	1.314	1.197–1.568	1.020	0.999–1.075
2-D nozzle	N₂	gas, T0/Tc ≥ 1.5	orifice (κ = c²ρ/P, choked)	2	0.945	0.944–0.946	0.941	0.940–0.942
elliptical nozzle	CH₄	subcooled liquid, P0/Pv ≥ 2	cavitatingVenturi (choked)	20	1.043	0.966–1.071	1.051	0.969–1.067
elliptical nozzle	CH₄	near-saturated liquid, P0/Pv < 2	cavitatingVenturi (choked)	1	0.929	0.929–0.929	0.946	0.946–0.946
elliptical nozzle	CH₄	supercritical, 1 ≤ T0/Tc < 1.5	orifice (κ = c²ρ/P, choked)	5	1.269	1.240–1.400	1.026	1.011–1.052

Subcooled liquid. With C_d = 1 the choked-liquid closure over-predicts the measured flux by a median +4.5 %, with every point between −3.4 % and +17.1 %, and the HEM reference behaves almost identically (median 1.058). For a liquid well above its vapor pressure the two models coincide: the liquid expands nearly isentropically to about P_v and flashes there. The remaining few-percent over-prediction is of the size expected from discharge coefficients slightly below one; no C_d is fitted here.

Near-saturated liquid. The closure's driving pressure P0 − P_v(T0) vanishes as the inlet approaches saturation, but the measured flux does not. For P_v/P0 ≤ 0.9 the median ratio is 0.894 (0.502–1.111, n = 110); above 0.9 it is 0.498 (0.154–0.697, n = 14), and HEM, which lets the fluid flash in equilibrium, is also low there (median 0.729). The measured flux exceeding both is the signature of liquid that remains metastable below P_v(T0) before flashing. Neither closure represents thermal non-equilibrium. For inlets within 10 % of saturation the choked-liquid closure under-predicts by up to a factor of 6.5 and HEM by up to a factor of 1.7 (Figure 1).

Figure 1. Liquid inlets, all four nozzles: OpenFLUME's choked-liquid closure (C_d = 1) and the HEM reference against measurement, plotted against the inlet saturation ratio P_v(T₀)/P₀. Points right of the dashed line are near-saturated (P₀ < 2P_v).

Supercritical inlets. The expansibility closure over-predicts every point, by a median +30.4 % (+7.5 % to +56.8 %), while HEM's median ratio is 1.029 (0.981–1.213). The closure applies the ideal-gas expansibility law with κ evaluated once, at the inlet. For a dense supercritical fluid κ = c_s²ρ/P changes by a large factor along the expansion, so a constant-κ isentrope is a poor approximation to the real one. The data confirm that this is a limitation of the closure and not of the property data, since the HEM calculation uses the same equation of state along the real isentrope (Figure 2).

Gaseous inlets. For the six room-temperature nitrogen points the closure is within −5.6 % to +4.7 % of measurement, and HEM gives the same values: here κ is nearly constant and the closure reduces to the classical isentropic critical flux.

Figure 2. Supercritical and gaseous inlets, all nozzles: OpenFLUME's expansibility closure (κ frozen at the inlet state, C_d = 1) and the HEM reference against measurement.

Figure 3. All usable nozzle points: OpenFLUME's primary closure against the measured maximum mass flux, by inlet regime (dashed: ±10 %).

Sharp-edged orifice

Data	Regime	OpenFLUME closure	n	Median	Range	Mean abs. deviation	HEM × C_d median	HEM × C_d range
Sharp-edged orifice	subcooled liquid, P0/Pv ≥ 2	orifice at measured Pb	13	0.978	0.939–1.013	2.6 %	0.897	0.802–0.997
Sharp-edged orifice	near-saturated liquid, P0/Pv < 2	orifice at measured Pb	13	0.996	0.960–1.260	6.9 %	0.697	0.412–0.806
Sharp-edged orifice	subcooled liquid, P0/Pv ≥ 2	cavitatingVenturi (choked)	13	0.883	0.772–1.000	10.3 %	0.897	0.802–0.997
Sharp-edged orifice	near-saturated liquid, P0/Pv < 2	cavitatingVenturi (choked)	13	0.632	0.317–0.763	38.9 %	0.697	0.412–0.806

With C_d = 0.61 at the measured back pressure the orifice closure is within −6.1 % to +1.3 % for subcooled inlets and −4.0 % to +26.0 % for near-saturated ones; the largest over-prediction is the point closest to saturation. Part of this agreement is coincidental. At 95 and 110 K the expansibility factor is 0.954–0.993 and the jet behaves as an incompressible jet discharging at the back pressure, with flashing outside the orifice. At 119–124 K the tabulated discharge coefficient falls toward the critical point, and so does Y (0.850–0.926, because κ = c_s²ρ/P falls); the closure tracks the data through a gas-dynamic factor, not through a model of flashing, and that correspondence should not be extrapolated. Both choked models are wrong for a sharp-edged orifice: choking at P_v gives a median 0.883 (subcooled) and 0.632 (near-saturated), and HEM × C_d gives 0.897 and 0.697 (Figure 4). A sharp-edged orifice passing subcooled cryogen should be modeled with the orifice closure at the real downstream pressure, not with a choked closure.

Figure 4. Sharp-edged orifice, liquid nitrogen: the orifice closure at the measured back pressure, the choked-at-P_v closure and HEM, all with C_d = 0.61, against measurement.

Part B: Cavitating Venturi (NASA CR-2054)

Data

Hord, Anderson and Hall photographed developed (super)cavities on the quarter-round contour of a transparent venturi of 1.905 cm throat diameter in liquid para-hydrogen, and tabulated (Table A-1b) for each of 32 runs the inlet state (P0, T0), the vapor pressure at T0 (as the "saturation head" h* and the pressure p*), and the wall pressures and temperatures at seven ports along the cavity on both walls.

What is compared, and what is not

The cavitatingVenturi closure assumes that a choked throat sits at P_v(T0). Hord's measurements test that assumption directly:

1. OpenFLUME's P_v(T0) (CoolProp ParaHydrogen) against the tabulated p*;
2. the measured cavity pressure at the leading edge, p_c (port 2, nearest the throat, mean of the two walls), and the minimum over all ports, against P_v(T0);
3. the mass flux the closure predicts (an OpenFLUME solve with C_d = 1, downstream at 0.5 P_v) against the flux of a choked throat at the measured p_c, G_c = √(2ρ0(P0 − p_c)). Both carry the same C_d and area, so the ratio needs no venturi geometry.

The measured mass flow is not compared. The table gives a velocity "at the test-section inlet", but the digitized record does not establish which area it refers to: with the throat diameter and the listed 3.175 cm inlet diameter, the throat velocity head would exceed P0 − P_v. The hydrofoil and ogive cavitation sets of the same program (CR-2156, CR-2242) are external flows over bodies with no network-component counterpart and are out of scope.

Results

All 32 closure solves converged. OpenFLUME's vapor pressure agrees with the tabulated value to within 0.91 % (median 0.08 %), so the equation of state is not a factor. The measured leading-edge cavity pressure lies 6–55 % below P_v(T0), and the minimum cavity pressure 7–68 % below. At the leading edge the cavity is close to saturation at its own measured temperature (p/P_sat(T_port) = 0.888–1.170), so the depression is thermal: evaporation into the cavity cools the liquid around it below T0. Further downstream the ports read up to 2.613 × P_sat(T_port) as pressure recovers along the cavity.

For a choked venturi at these conditions the closure's throat pressure is therefore too high, and its predicted flux is 0.594–0.900 times the flux through a throat at the measured p_c: an under-prediction of 10–41 %. The effect is largest for the warmer runs (T0 ≈ 22.4–23 K), where P0 − P_v is a small fraction of P0 (Figure 5). The closure has no thermodynamic-suppression term; for liquid hydrogen near its normal boiling point a user calibrating a venturi should expect its C_d to absorb this effect.

Figure 5. Hord's liquid-hydrogen venturi: the closure's throat pressure P_v(T₀) (OpenFLUME and tabulated) against the measured cavity pressure at the leading edge and the minimum over all ports.

Trust Levels

Comparison	Evidence class	Trust level	What may be quoted
Nozzles, subcooled liquid, cavitatingVenturi	Validation against experiment	Validated within the stated range	Median 1.045, range 0.966–1.171 at C_d = 1
Nozzles, gas, orifice	Validation against experiment	Validated within the stated range	0.944–1.047 at C_d = 1 (six points)
Nozzles, near-saturated liquid	Validation against experiment	Documented disagreement	Under-prediction, unbounded as P0 → P_v
Nozzles, supercritical, orifice	Validation against experiment	Documented disagreement	Over-prediction, median +30 %
Sharp-edged orifice, orifice at P_b	Validation against experiment	Validated within the stated range	Subcooled −6.1 % to +1.3 % at C_d = 0.61
Hord venturi, throat-pressure assumption	Validation against experiment (closure assumption)	Documented disagreement	Cavity 6–55 % below P_v(T0)
HEM reference	Reference model, not OpenFLUME	—	Context only

Conclusions

OpenFLUME's two restriction closures predict choked cryogenic flow well in the regimes they are built for and badly outside them, and the boundary is sharp enough to state. The choked-liquid cavitatingVenturi closure with C_d = 1 matches nozzle data for liquid nitrogen and methane subcooled to at least twice the vapor pressure within a few percent, and the orifice closure matches gaseous nitrogen and, at the measured back pressure with C_d = 0.61, subcooled liquid nitrogen through a sharp-edged orifice. Three findings bound their use: neither closure represents thermal non-equilibrium, so near-saturated liquid inlets are under-predicted; the expansibility closure's frozen κ over-predicts dense supercritical flow by up to +57 %, where an equilibrium calculation along the real isentrope does not; and the venturi closure's throat pressure ignores the thermal suppression of cavities, which Hord's hydrogen data put at 6–55 % of P_v. These are limitations of the closures, documented here with the data that expose them; no closure was adjusted.

References

1. Simoneau, R. J., and Hendricks, R. C., "Two-Phase Choked Flow of Cryogenic Fluids in Converging-Diverging Nozzles," NASA TP-1484, 1979.
2. Simoneau, R. J., "Maximum Two-Phase Flow Rates of Subcooled Nitrogen Through a Sharp-Edged Orifice," NASA TM X-71760, 1975 (also Advances in Cryogenic Engineering 21, 1975, pp. 299–306).
3. Hord, J., Anderson, L. M., and Hall, W. J., "Cavitation in Liquid Cryogens. Volume I: Venturi," NASA CR-2054, 1972.
4. Hord, J., "Cavitation in Liquid Cryogens. Volume II: Hydrofoil," NASA CR-2156, 1973; "Volume III: Ogives," NASA CR-2242, 1973 (out of scope).
5. White, F. M., Fluid Mechanics, 7th ed., McGraw-Hill, 2011 (orifice discharge coefficients).
6. Bell, I. H., Wronski, J., Quoilin, S., and Lemort, V., "Pure and Pseudo-pure Fluid Thermophysical Property Evaluation and the Open-Source Thermophysical Property Library CoolProp," Industrial & Engineering Chemistry Research 53(6), 2014, pp. 2498–2508.

Nomenclature

Symbol	Meaning
A	branch flow area (throat or bore)
C_d	discharge coefficient
c_s	speed of sound
G	mass flux ṁ/A
G*	HEM critical mass flux
h, h0	specific enthalpy; stagnation value
ṁ	mass flow rate
P0, T0	stagnation pressure, temperature
P_b	back pressure
P_c, T_c	critical pressure, temperature
p_c	measured cavity pressure at the leading edge (port 2)
P_v	vapor (saturation) pressure
r, r*	pressure ratio P_down/P_up; critical value
s0	stagnation entropy
Y	expansibility factor
κ	isentropic exponent c_s²ρ/P
ρ0	stagnation density