1 Scope and limitations
Stating the boundary of a method’s validity is part of stating the method, and this document collects, in one place, the scope boundaries and approximations noted in passing throughout the theory. Each limitation below is either a scope boundary, a regime the present version does not attempt, or a bounded approximation, a model that is accurate to a stated order and has a designed path to exactness. None is a hidden assumption: the intent here is to draw the honest edge of the tool so that a result is trusted exactly as far as it should be.
1.1 Mean-flow scope
The mean-flow model is restricted to subsonic operation, flowing or quiescent, up to a sonic throat. Choking to M = 1 at a narrowest section is fully in scope and emerges from the element rows (see choking); what is deferred is supersonic flow inside the domain and the structures that accompany it. A normal shock standing in a diverging passage, and a declared supersonic exit, require an internal shock-position degree of freedom (the dual of the self-vacating row a supersonic inlet would carry), which is designed but not built. It should be noted that an emergent supersonic converging–diverging nozzle is not impossible in principle; it is deferred because it needs that additional unknown, not because the formulation forbids it, and the same enrichment is what finite-frequency acoustics of shocked nozzles will require.
A second, milder bound is that the sudden-contraction and Borda-expansion pressure heads use the incompressible reduction of the momentum balance, accurate to \mathcal{O}(M^2) (see elements). A dedicated contraction element that resolves the vena-contracta state, and so stays exact at higher Mach number, is planned but not present.
1.2 Reacting-flow scope
The reacting closure is equilibrium-based: the frozen, equilibrium, and marker-gated closures of thermochemistry are built, but finite-rate chemistry is not. Where equilibrium is enforced, the model assumes the local mixture has reached chemical equilibrium; for low-Mach devices that is often adequate for the major heat-release chemistry, but formation rates differ widely across species. Nitric oxides (NOx), for example, are kinetically slow, and equilibrium estimates of their levels may not be appropriate even when the bulk flame temperature is well captured. The frozen-to-burnt switch is realized through the marker gate rather than a residence-time balance, and that gate approach assumes that the flame is thin compared with the network elements. Each lumped element also assumes perfect mixing of its incoming streams into one outlet state (see transport), which may or may not represent the geometry of interest.
The equilibrium closure, including its condensed-phase (soot) products, is intended for oxidiser-bearing mixtures: any blend from lean through stoichiometric and deep into the rich, sooting regime is in scope and converges reliably. The one composition it does not solve reliably is a mixture with essentially no oxidiser at all, a pure fuel driven to equilibrium as in a cracking or pyrolysis passage. There, almost every carbon atom condenses to soot and the remaining gas collapses toward a single species, and that limit leaves the equilibrium system degenerate. Such a state is reachable (an equilibrium region fed only by fuel), so it is a genuine boundary rather than an abstract one; when a network reaches it the solve reports a clear non-convergence rather than a silent or misleading result. A related subtlety is that the reacting closure equilibrates at constant enthalpy (adiabatically), which is the combustion setting; the isothermal equilibrium a heated cracking duct would need, holding the passage at a set temperature, is not expressed as a network element. Pure-fuel pyrolysis is therefore outside the present reacting scope; the accompanying note in the repository records the mechanism and the path to lifting it.
1.3 Acoustic-model scope
Before the specific bounds, the premise beneath all of them: the acoustic layer is linear, and its perturbation is an infinitesimal one. The consequence worth recording here is one the mean flow inherits. The state the solver converges is the steady operating point, and it coincides with the temporal mean of the unsteady flow only to the order the linearization retains: averaging the exact balances leaves the correlation sources \overline{\varrho' u'}, \overline{p' u'}, and their kin, which are quadratic in the fluctuation and are therefore dropped (see governing equations). At finite amplitude those sources bias the observed mean away from the steady solution, by an amount growing with the square of the amplitude and concentrating in the elements whose losses are quadratic in the velocity. What the tool therefore predicts is whether a mode grows, not the amplitude at which its growth arrests, nor the mean state a saturated oscillation settles into; a measurement taken on a strongly oscillating rig is not the state this mean flow computes. The forward-compatible path is the amplitude-dependent describing function: retaining the correlation sources and evaluating the dynamic sources of dynamic sources at a prescribed amplitude turns the same operator into a saturation condition, and the machinery to carry it is already the shape of the source block.
The compositional (indirect) noise coupling R_\xi (Magri et al. 2016) is retained everywhere the acoustic linearization is inherited from the mean-flow kernel, but it is dropped by the hand-written analytic terminal closures for a choked-nozzle or constant-mass-flow outlet, which carry the entropy coupling R_s but no composition column; the solver raises a warning precisely when a reacting flow meets such a closure, so the gap is surfaced rather than silent (see dynamic sources).
The automatic de-normalization of a flame’s mean heat release is exact for the perfect-gas heat-release flame, which carries its power as a parameter, but for every other flame it uses the sensible-enthalpy rise \dot m\,\overline{c}_p\,\Delta T and is accurate only to \mathcal{O}(M^2); supplying the mean heat release explicitly removes the approximation (see dynamic sources).
1.4 What is exact
Against these bounded approximations it is worth restating plainly what carries no approximation, so the edge is not mistaken for the whole. The Jacobian is exact to machine precision, obtained by complex-step differentiation of analytic residuals rather than by finite differences (see complex-step). The mean-flow mass balance is exactly linear and globally conservative, and the energy transport is exact up to the quadratically small smoothing of the upwind weights (see transport). The isentropic and choking rows reproduce the classical compressible-flow relations exactly in the subsonic regime, to the order of the fixed complementarity smoothing (see choking). The characteristic maps that turn the base Jacobian into the acoustic operator are the exact linearization of the state definitions, invertible at every physical state (see characteristics).
The honest summary is therefore that the subsonic, equilibrium-reacting mean flow and the acoustics linearized about it form a complete and internally exact core, ringed by a small set of named, bounded approximations — each with a designed path to removal — and two firm scope boundaries: supersonic internal flow, and the finite-amplitude response beyond the linear one. The validation track collects the evidence that the core performs as claimed.