Isentropic Nozzle Flow¶

Model compressible flow through a changing-area nozzle by coupling steady one-dimensional mass and energy conservation with reversible adiabatic state relations, exposing subsonic and supersonic branches, sonic choking, and the exact boundaries at which shocks, heat, or friction invalidate the model.

Version: v2 · 2026-09-06 · History
Domain-specific #: 2100
Origin domain: fluid mechanics
Subdomain: compressible flow and gas dynamics
Aliases: Isentropic flow through a nozzle, One-dimensional isentropic nozzle flow

Core Idea¶

Isentropic nozzle flow is the reusable gas-dynamics model that explains how a compressible fluid trades pressure and temperature for directed speed while passing through a duct of changing cross-sectional area. Its identity is not merely “flow in a nozzle” and not merely “an isentropic process.” It couples a particular geometric carrier—a slowly varying nozzle—with steady quasi-one-dimensional conservation of mass and energy, inviscid momentum balance, and reversible adiabatic state change. Within that envelope, entropy and stagnation pressure remain constant and local state variables become functions of Mach number and area ratio.^[1]^[2]

For a calorically perfect ideal gas with constant specific-heat ratio $\gamma$ and gas constant $R$, let $V$ be axial velocity, $a=\sqrt{\gamma RT}$ the local sound speed, and $M=V/a$ the Mach number. Steady continuity requires

\[ \dot m=\rho A V=\text{constant}. \]

Adiabatic energy conservation gives $h_0=h+V^2/2$ and, with constant heat capacity, $T_0=T+V^2/(2c_p)$. Reversibility supplies constant entropy and therefore the familiar stagnation-to-static relations

\[ \frac{T}{T_0}=\left(1+\frac{\gamma-1}{2}M^2\right)^{-1}, \]

\[ \frac{p}{p_0}=\left(1+\frac{\gamma-1}{2}M^2\right)^{-\gamma/(\gamma-1)}, \qquad \frac{\rho}{\rho_0}=\left(1+\frac{\gamma-1}{2}M^2\right)^{-1/(\gamma-1)}. \]

Combining continuity, energy, and the isentropic state relation yields the area–Mach relation

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

where $A^*$ is the area the same isentropic stream would require at $M=1$. NASA Glenn presents these ratios and emphasizes that a given $A/A^*>1$ has both a subsonic and a supersonic solution.^[2] The differential form makes the geometric reversal vivid:

\[ \frac{dA}{A}=(M^2-1)\frac{dV}{V}. \]

A converging passage accelerates subsonic flow but decelerates supersonic flow; a diverging passage does the reverse. Smooth acceleration from subsonic to supersonic speed therefore requires a converging–diverging nozzle whose minimum area is sonic. NASA's nozzle-design account uses this relation to connect geometry, throat choking, and exit Mach number.^[3]

The locked identity is:

steady compressible stream + slowly varying nozzle area + quasi-one-dimensional inviscid balance + adiabatic and reversible change + declared gas model + stagnation state and back-pressure boundary conditions -> coupled area, Mach, pressure, temperature, density, velocity, and mass-flow solution valid only on shock-free isentropic intervals.

Choking is a consequence, not a synonym. At $M=1$, the mass flux reaches its isentropic maximum for fixed $p_0$, $T_0$, $\gamma$, and throat area:

\[ \dot m^*=A^*\frac{p_0}{\sqrt{T_0}} \sqrt{\frac{\gamma}{R}} \left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(2(\gamma-1))}. \]

Lowering downstream pressure cannot then increase mass flow through that fixed sonic section under the same upstream stagnation conditions.^[4] But $A^*$ is a critical-flow reference area, not automatically the physical throat: the geometric throat equals $A^*$ only when it is choked. Likewise, a choked nozzle is not automatically isentropic everywhere. A normal shock can stand downstream of a sonic throat; the flow is adiabatic across the shock but irreversible, entropy rises, and stagnation pressure falls.^[5]

Structural Signature¶

Sig role-phrases:

the compressible working fluid — a gas or other compressible medium with a declared equation of state and caloric model, commonly an ideal gas with fixed $R$ and $\gamma$
the slowly varying nozzle geometry — an axial area function $A(x)$ whose cross-sections are represented by section-averaged state variables
the steady quasi-one-dimensional stream — one mass-flow rate and one dominant axial velocity, with time dependence and transverse structure neglected or bounded
the upstream stagnation state — $p_0$, $T_0$, and composition, which set the available specific energy and mass-flux scale
the isentropic validity conditions — adiabatic, reversible, inviscid, shock-free evolution over the interval being solved
the local Mach-state package — $M$, $V$, $T$, $p$, $\rho$, and $a$, coupled rather than chosen independently
the conservation closure — continuity and total-enthalpy conservation joined to the isentropic equation of state
the area–Mach branch choice — the subsonic or supersonic root consistent with inlet conditions, throat status, geometry, and downstream pressure
the sonic critical section — the $M=1$ state at $A^*$, which bounds mass flux and mediates smooth subsonic-to-supersonic passage
the back-pressure regime selector — the downstream condition that determines whether the ideal branch is realized or a shock/non-isentropic adjustment is required
the validity and failure diagnostics — constancy of mass flow, stagnation temperature, entropy, and stagnation pressure within tolerance, plus named residuals for shocks, friction, heat transfer, chemistry, or multidimensionality

Recognition test. A case instantiates Isentropic Nozzle Flow when a changing-area compressible passage is modeled by section-averaged steady variables; the mass and total-enthalpy balances close; entropy and stagnation pressure are constant over the modeled interval; and the selected subsonic, sonic, or supersonic branch satisfies both geometry and boundary conditions. If a shock lies inside the interval, wall friction generates appreciable entropy, heat crosses the wall, chemical or phase change alters the caloric model, or separation makes the one-dimensional area representation inadequate, the simple identity fails there. The model may still apply piecewise on shock-free regions, but it cannot be continued unchanged across the failure.

What It Is Not¶

Not all compressible nozzle flow. The broader class includes viscous, heat-transferring, reacting, separated, unsteady, and shock-containing flows. Isentropic Nozzle Flow is the reversible-adiabatic reference member.
Not merely an adiabatic process. Zero heat transfer does not prohibit viscous dissipation or a shock. Those adiabatic but irreversible processes generate entropy and lose stagnation pressure.
Not incompressible Bernoulli flow. Density, temperature, sound speed, and Mach number are coupled state variables; treating density as constant erases choking and the double-valued area–Mach relation.
Not “a de Laval nozzle.” A de Laval nozzle names a converging–diverging device. Its actual operating state may be wholly subsonic, choked with a shock, overexpanded, underexpanded, or near the ideal isentropic design condition.
Not synonymous with choked flow. Choking fixes a sonic controlling section and caps mass flux under fixed upstream conditions; a choked stream can contain downstream entropy-producing phenomena.
Not synonymous with supersonic flow. An isentropic nozzle can operate wholly subsonically. Conversely, supersonic nozzle flow can include shocks and boundary-layer loss.
Not one algebraic area-ratio lookup. Since $A/A^*>1$ has two Mach roots, branch selection requires inlet information and back-pressure compatibility.^[2]
Not globally valid across a normal shock. Static properties jump, entropy rises, and total pressure decreases; separate upstream and downstream solutions must be joined by shock relations.^[5]
Not automatically accurate because the nozzle is smooth. Boundary layers change effective area and loss, and sufficiently adverse pressure gradients can separate the flow even without a geometric discontinuity.

Scope of Application¶

The abstraction is the first reference model for rocket and jet nozzles, wind-tunnel contractions and test-section feeds, turbine stators, compressor and turbine passages, gas pipelines with strong area change, blowdown orifices, and metering devices. It answers four recurring questions: what Mach number corresponds to a section's area ratio; how static pressure and temperature change as kinetic energy grows; whether a throat is choked; and which measured discrepancy signals a non-isentropic mechanism.

Its cleanest scope is a steady, single-phase gas in a duct whose area changes gradually enough that a cross-sectional average is meaningful. A calorically perfect ideal gas makes the equations above closed-form. A thermally perfect but calorically imperfect gas may remain isentropic, yet variable heat capacity requires a revised property calculation; NACA Report 1135 explicitly distinguishes perfect-gas tables from high-temperature corrections.^[1] Real-gas equations of state can likewise preserve the reversible-adiabatic structure while replacing the simple constant-$\gamma$ ratios.

The scope should be declared interval by interval. A nozzle with an internal shock is not discarded wholesale: one may solve an upstream isentropic branch, apply the appropriate shock jump, then solve a downstream branch using its reduced stagnation pressure. A nozzle with small wall losses may be compared against the ideal solution using discharge, thrust, or nozzle-efficiency corrections. But those are extensions around the reference model, not silent changes to its identity.

Clarity¶

The model turns a crowded physical story into a short dependency chain:

stagnation state + gas properties + area distribution + branch/back pressure -> Mach distribution -> static state and velocity -> mass flow and performance.

This chain prevents three common category errors. First, static and stagnation quantities are not interchangeable: the static state changes as energy moves into directed velocity, while the ideal stagnation state remains fixed. Second, adiabatic and isentropic are not interchangeable: reversibility must be added to zero heat transfer. Third, critical area and geometric throat are not interchangeable unless choking has actually occurred.

The most useful reporting discipline is to state the assumptions before the answer: steady or transient, one-dimensional or resolved, ideal or real gas, constant or variable $\gamma$, shock-free or piecewise, and measured back pressure. A Mach number without its area branch is incomplete; an area ratio without the associated $A^*$ and boundary regime is ambiguous.

Manages Complexity¶

The full compressible Navier–Stokes problem contains three-dimensional velocity fields, viscosity, thermal conduction, boundary layers, turbulence, shocks, chemistry, and wall geometry. Isentropic Nozzle Flow manages that complexity by keeping the couplings that dominate ideal nozzle acceleration—mass conservation, energy conversion, compressibility, geometry, and sonic information propagation—while suppressing mechanisms whose effects can be tested as residuals.

The reduction is especially powerful because it converts a spatial field problem into algebraic relations indexed by $A/A^*$. Geometry becomes a state selector, and the sonic throat becomes a control boundary. Rather than guessing whether a converging section accelerates a fluid, the sign of $M^2-1$ decides the response. Rather than treating choking as an empirical surprise, the mass-flow function shows an extremum at $M=1$.^[4]

Complexity returns in an organized way. A failure of mass-flow constancy suggests leakage, accumulation, or measurement error. A drop in stagnation pressure with nearly constant stagnation temperature suggests adiabatic irreversibility such as shock or friction. A change in stagnation temperature points to heat transfer, shaft work, or chemical energy exchange. A mismatch concentrated near walls suggests boundary-layer displacement or separation. The ideal model thus acts as a diagnostic baseline, not only a calculator.

Abstract Reasoning¶

The abstraction supports reasoning about branches, extrema, invariants, and admissibility. The area–Mach equation is not single-valued: for every $A/A^*>1$, one root lies below one and another above one. A numerical root is therefore not a physical answer until the regime chooses the branch. The throat is a topological bottleneck in the solution family: a smooth transition through $M=1$ requires $dA=0$, so the sonic point occupies an extremal area.

The invariants separate what geometry can change from what ideality preserves. Static pressure, temperature, density, and velocity vary strongly, while $\dot m$, $T_0$, $p_0$, and entropy remain fixed along one isentropic stream. A shock keeps mass, momentum, and total enthalpy balanced but breaks entropy and stagnation-pressure invariance. This selective invariant loss is a powerful classifier.

Back pressure supplies an admissibility constraint rather than appearing directly in the local area–Mach formula. The same nozzle geometry can support different operating regimes because downstream compatibility selects among subsonic flow, a choked stream with an internal shock, or an isentropic supersonic branch. The abstraction therefore teaches why local equations and global boundary conditions must be solved together.

Knowledge Transfer¶

Once learned, the structure transfers across propulsion and gas-dynamic devices. In a rocket nozzle, chamber stagnation state and throat area set the choked mass flow, while exit area ratio selects an ideal exit Mach and pressure. In a supersonic wind tunnel, the same area–Mach relation designs the nozzle contour for a target test-section Mach number. In a turbine stator, a pressure drop is interpreted as conversion to directed kinetic energy, with loss measured against the isentropic reference.

The diagnostic logic also transfers to model comparison. “Hold total enthalpy, inspect entropy generation” is useful for shocks, diffusers, and turbomachinery. “Find the critical section where a flux function is extremized” reappears in compressible orifice flow and in other hyperbolic conservation systems. What does not transfer freely is the domain accent: Mach number, stagnation state, gas caloric properties, sonic choking, and nozzle back pressure have specific fluid-mechanical meanings.

Examples¶

Mapped example 1 — designed air nozzle. Consider calorically perfect air with $\gamma=1.4$, $R=287\ \mathrm{J\,kg^{-1}K^{-1}}$, upstream stagnation pressure $p_0=500\ \mathrm{kPa}$, stagnation temperature $T_0=300\ \mathrm{K}$, and a choked throat of area $A^*=1.00\ \mathrm{cm^2}$. Choose the supersonic exit branch $M_e=2.00$.

Mapped back:

working fluid: ideal air with declared $\gamma$ and $R$
geometry: choked throat plus an exit area to be found
stagnation state: $500\ \mathrm{kPa}$ and $300\ \mathrm{K}$
branch: supersonic downstream of the throat
area–Mach result: $A_e/A^*=1.6875$, so $A_e=1.6875\ \mathrm{cm^2}$
state result: $T_e/T_0=0.55556$, $p_e/p_0=0.12780$, hence $T_e\approx166.7\ \mathrm{K}$ and $p_e\approx63.9\ \mathrm{kPa}$
velocity result: $V_e=M_e\sqrt{\gamma RT_e}\approx517.6\ \mathrm{m/s}$
critical-flow result: $\dot m^*\approx0.1167\ \mathrm{kg/s}$
diagnostic: these values are jointly admissible only if the downstream pressure and contour sustain the shock-free supersonic branch; the identical exit area ratio also has a subsonic root, so area alone does not prove $M_e=2$.

Mapped example 2 — NASA converging–diverging verification nozzle. NASA's CDV verification case uses inlet, throat, and exit areas of $2.5$, \$1.0$, and $1.5\ \mathrm{in^2}$. Three exit-to-total pressure ratios deliberately produce different regimes: $0.89$ for subsonic isentropic flow, $0.75$ for a supersonic solution containing a normal shock, and $0.16$ for supersonic isentropic flow.^[6]

Mapped back:

same geometry: fixed across all three cases
regime selector: back pressure changes, not the nozzle contour
isentropic diagnostic: the $0.89$ and $0.16$ cases can be compared against shock-free analytic solutions and constant total pressure
failure boundary: the $0.75$ case cannot be represented by one global isentropic branch because the normal shock creates entropy and a total-pressure loss
knowledge gained: geometry constrains candidate solutions, while the downstream boundary determines which solution is physically realized.

Worked intervention — locating a discrepancy. Suppose measured $T_0$ is constant from inlet to exit but measured $p_0$ falls abruptly at one axial station. Do not “correct” the area ratio until the loss is classified. An abrupt loss paired with a supersonic-to-subsonic jump indicates a shock; a gradual loss suggests distributed friction; loss concentrated where adverse pressure gradient is strong suggests separation. The intervention is to segment the nozzle at the detected loss, preserve the upstream isentropic solution, apply the appropriate non-isentropic relation, and restart downstream with the new stagnation pressure.

Structural Tensions¶

Geometry determines state ↔ boundary conditions select state. Area ratio narrows the possible Mach numbers but does not choose between subsonic and supersonic roots. Diagnostic: two valid roots at the same $A/A^*$ disappear only after inlet and back-pressure information is supplied.
Acceleration ↔ pressure recovery. A nozzle converts enthalpy and pressure into velocity, while a diffuser reverses that exchange; which area trend accelerates the stream flips at $M=1$. Diagnostic: the sign relation $dA/A=(M^2-1)dV/V$ changes across the sonic state.
Smooth ideal evolution ↔ discontinuous shock adjustment. Both can satisfy conservation, but only the smooth reversible path preserves entropy and stagnation pressure. Diagnostic: an abrupt static-state jump and total-pressure loss marks the shock boundary.
Maximum throughput ↔ downstream insensitivity. Choking makes mass flow insensitive to further back-pressure reduction at fixed upstream state, yet downstream structure can still change. Diagnostic: $\dot m$ plateaus while shocks move or the external jet pattern changes.
Universal nondimensional form ↔ gas-property dependence. Mach and area ratios organize many gases, but $\gamma$, real-gas behavior, and variable heat capacity alter quantitative results. Diagnostic: the same geometry produces shifted state ratios when the caloric model changes.
One-dimensional economy ↔ multidimensional reality. Section averages enable a compact solution while boundary layers, shocks, and contour curvature create transverse structure. Diagnostic: wall-sensitive measurements or flow separation depart systematically from the cross-sectional prediction.
Adiabaticity ↔ isentropy. No heat transfer is necessary for the ideal nozzle but not sufficient for reversibility. Diagnostic: constant $T_0$ accompanied by falling $p_0$ reveals adiabatic loss rather than an isentropic stream.
Autonomy ↔ reduction to generic flow and conservation. Flow and Conservation Laws provide the broad skeleton, yet they do not entail the Mach-branch reversal, sonic critical area, stagnation-state ratios, or back-pressure regime map. Diagnostic: if replacing gas, nozzle, Mach, and sonic semantics with generic transported quantities preserves the predictions, the node has been reduced too far; if those semantics are required to choose and test the solution, the domain abstraction remains autonomous.

Structural–Framed Character¶

Vocabulary travels (0.75). Flow, conservation, branch, invariant, and critical-point language travels, while Mach number, stagnation state, sonic choking, and nozzle back pressure remain gas-dynamic. Evaluative weight (0.0). Membership is a descriptive model claim, not a value judgment. Institutional origin (0.0). Standards and laboratories may measure the regime, but no institution creates its physical validity. Human-practice bound (0.0). Once the carrier and boundary conditions are fixed, the relations do not depend on organized practice. Import versus recognize (0.25). Analogies to other bottlenecked flows may preserve a thin skeleton, but literal recognition requires the nozzle and compressible-gas obligations.

Carrier. The structure is carried by a compressible fluid moving through a changing-area passage.
Invariant. On each ideal interval, mass flow, entropy, stagnation temperature, and stagnation pressure remain constant while static state and velocity change.
Transformation. Nozzle area and boundary conditions redistribute stagnation enthalpy between thermal state and directed kinetic energy, selecting a subsonic or supersonic branch.
Boundary. Shocks, friction, heat transfer, reaction, phase change, strong unsteadiness, and multidimensional separation bound the simple model.
Diagnostic consequence. Area–Mach compatibility plus the invariant ledger makes the abstraction falsifiable: wrong mass flow, entropy growth, total-pressure loss, or branch inconsistency identifies a missing mechanism.

Its character is strongly structural but irreducibly domain-framed. The reusable organization—carrier, conservation, branch, critical point, invariant, and failure boundary—is structural. Yet the meanings of Mach number, stagnation state, sonic information propagation, throat area, gas calorics, and back pressure cannot be removed without changing what is predicted and how failure is diagnosed.

Structural Core vs. Domain Accent¶

Part 1 — candidate structural core. A transported quantity passes through a variable-capacity channel; conservation couples local rate and state; a nondimensional regime variable reverses the response to geometry; and a critical point maximizes flux and separates solution branches.

Part 2 — indispensable domain accent. The carrier is a compressible fluid, capacity is nozzle area, the regime variable is Mach number, the critical point is sonic choking, the conserved energetic quantity is stagnation enthalpy, and reversible adiabatic gas relations connect pressure, density, temperature, and speed. Back pressure determines whether a shock-free branch is globally admissible.

Part 3 — substitution test. Replacing the gas with money in a network, data in a link, or people in a corridor may preserve generic flow and bottleneck metaphors, but it does not preserve $a=\sqrt{\gamma RT}$, the area–Mach double root, stagnation ratios, or shock entropy production. Those are not decorative examples; they do the inferential work. The candidate therefore fails the prime bar as expected and survives as a domain-specific abstraction.

Flow. The node strictly instantiates structured transport: a material stream has direction, rate, channel, driving pressure difference, and continuity. This is the strongest genus-level parent.
Conservation Laws. Constant mass flow and total enthalpy are constitutive, while shock analysis demonstrates that different invariant subsets can survive different processes.
Constraint. Nozzle area, stagnation state, and back pressure constrain admissible Mach branches, but Constraint alone does not generate the gas-dynamic relations.
Transformation. The nozzle produces a rule-governed conversion between thermal/pressure state and directed kinetic energy while ideal invariants remain fixed.
Entropy (Thermodynamic Sense). Entropy constancy distinguishes the ideal stream; entropy increase marks shock and viscous failure.
Reversibility and Irreversibility. Thermodynamic reversibility is essential locally, though the current prime's decision-restoration framing is broader and is not proposed as a direct parent.
Adiabatic Process. This staged domain neighbor supplies the zero-heat-transfer condition but allows irreversible entropy production. It is a semantic supertype candidate only after canonical implementation; staging-only targets cannot serve as live parents in this draft's DAC proposal.
Turbulence. Turbulence and boundary-layer loss are important validity boundaries, not constituents required by the ideal identity.

Relationships to Other Abstractions¶

Current abstraction Isentropic Nozzle Flow Domain-specific

Parents (2) — more general patterns this builds on

Isentropic Nozzle Flow is a kind of Flow Prime

Flow. The node strictly instantiates structured transport: a material stream has direction, rate, channel, driving pressure difference, and continuity.
Isentropic Nozzle Flow presupposes Conservation Laws Prime

Conservation Laws. Constant mass flow and total enthalpy are constitutive, while shock analysis demonstrates that different invariant subsets can survive different processes.

Hierarchy paths (2) — routes to 2 parentless roots

Isentropic Nozzle Flow → Flow

Show alternative path (1)

Neighborhood in Abstraction Space¶

Isentropic Nozzle Flow sits in a sparse region of the domain-specific corpus (90^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Thermoacoustics — 0.80
Divergence Zone — 0.79
Janzen–Rayleigh Expansion — 0.79
Downwelling — 0.78
Wave method — 0.78

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

Compressible nozzle flow. Tell: if shocks, friction, heat, reaction, or separation are allowed without segmenting or correcting the model, the broader class is intended.
Adiabatic nozzle flow. Tell: if entropy may rise while wall heat transfer is zero, the process is adiabatic but not isentropic.
Choked flow. Tell: if the claim concerns a sonic controlling section and maximum mass flux without requiring reversible conditions everywhere, it is choking rather than the full isentropic-nozzle identity.
De Laval nozzle. Tell: if the referent is the converging–diverging hardware or contour regardless of operating regime, it is the device, not its ideal flow state.
Normal-shock nozzle flow. Tell: if a supersonic stream jumps to subsonic speed with entropy increase and stagnation-pressure loss, the shock must be modeled explicitly.^[5]
Fanno flow. Tell: if an adiabatic constant-area duct changes state through wall friction, the controlling idealization is Fanno rather than isentropic nozzle flow.
Rayleigh flow. Tell: if heat transfer in a constant-area duct drives the Mach-state change, the controlling idealization is Rayleigh flow.
Incompressible Bernoulli flow. Tell: if density is treated as constant and sonic choking or area-branch multiplicity cannot appear, the incompressible approximation is being used.
Isentropic efficiency. Tell: if the ideal isentropic endpoint is only a comparator for an actual lossy turbine, compressor, or nozzle, the subject is a performance ratio, not an assertion that the actual path is isentropic.
Overexpanded or underexpanded jet. Tell: if exit pressure mismatches ambient and the principal adjustment occurs outside the nozzle through expansion fans or shocks, the external jet regime is focal.

References¶

[1] Ames Research Staff. (1953). Equations, Tables, and Charts for Compressible Flow. NACA Report 1135. NASA Technical Reports Server. https://ntrs.nasa.gov/citations/19930091059 registry ↩a ↩b

[2] NASA Glenn Research Center. (2021). “Isentropic Flow Equations.” Beginner's Guide to Aeronautics. https://www.grc.nasa.gov/www/BGH/isentrop.html registry ↩a ↩b ↩c

[3] NASA Glenn Research Center. (2024). “Nozzle Design.” Beginner's Guide to Aeronautics. https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/nozzle-design/ registry ↩

[4] NASA Glenn Research Center. (2021). “Mass Flow Choking.” Beginner's Guide to Propulsion. https://www.grc.nasa.gov/www/k-12/BGP/mflchk.html registry ↩a ↩b

[5] NASA Glenn Research Center. (2021). “Normal Shock Verification.” NPARC Alliance Validation Archive. https://www.grc.nasa.gov/www/wind/valid/normal/normal.html registry ↩a ↩b ↩c

[6] NASA Glenn Research Center. (2021). “Converging-Diverging Verification (CDV) Nozzle.” NPARC Alliance Validation Archive. https://www.grc.nasa.gov/www/wind/valid/cdv/cdv.html registry ↩