1 Governing equations and network discretization

This document derives the equations a network solves, starting from the integral conservation laws of gas dynamics and reducing them, under the standing assumptions, to the algebraic jump conditions attached to each element. It settles a distinction that is easy to conflate: steadiness is what removes the time-derivative term and leaves an algebraic balance, whereas the zero-volume limit is a separate modeling choice whose effect is only to discard the volumetric terms while the lumped ones survive. The algebraic balances derived here are more than the mean-flow model: differentiated at the converged operating point, they are the content of the acoustic network, so that the same operator serves both problems — the reuse that is the through-line of the whole method (see perturbation network).

1.1 Standing assumptions

The equations below are posed under the overview assumptions, of which three are load-bearing here:

1. Inviscid dynamics at the component scale. The gas obeys the Euler equations over a component; viscous and turbulent losses enter only through lumped constitutive terms, not a resolved shear field.
2. A steady operating point is sought. The transient term vanishes because \partial/\partial t \equiv 0 at the operating point, independently of any element’s internal volume.
3. Port quantities are section averages. Each port flux is formed from the single representative edge state, exact for a uniform profile and corrected otherwise by a profile-shape factor (see framework).

1.2 Integral conservation laws

We take the gas to be inviscid on the scale of a component, so its dynamics are governed by the Euler equations, written in integral form over a control volume \mathcal{V} with bounding surface \mathcal{S}:

\frac{\partial}{\partial t}\int_{\mathcal{V}} \widetilde{\mathbf{q}}\,\mathrm{d}\mathcal{V} \;+\; \oint_{\mathcal{S}} \mathbf{f}_n \,\mathrm{d}\mathcal{S} \;=\; \dot{\boldsymbol{\Omega}},

where \widetilde{\mathbf{q}} is the vector of conserved densities, \mathbf{f}_n their flux through the boundary along the outward normal, and \dot{\boldsymbol{\Omega}} collects any sources acting within \mathcal{V}. It can be interpreted as the statement that the rate of change of a conserved quantity stored in \mathcal{V}, plus its net efflux through the boundary, equals its rate of production inside. For mass, momentum, and energy the densities and fluxes are given as:

\widetilde{\mathbf{q}} = \begin{bmatrix} \varrho \\ \varrho u \\ \varrho e_t \end{bmatrix}, \qquad \mathbf{f}_n = \begin{bmatrix} \varrho u_n \\ \varrho u\, u_n + p\, n \\ \varrho u_n h_t \end{bmatrix},

where \varrho is the density, u the velocity, p the static pressure, e_t the total energy per unit mass, u_n the velocity component along the outward normal, n the corresponding component of the surface normal, and h_t = h + \tfrac{1}{2}u^2 the total enthalpy — the sum of the static enthalpy h = c_p T and the specific kinetic energy. Intuitively, h_t is the energy currency of a steady stream: it is exactly conserved along an adiabatic flow no matter how the gas accelerates, decelerates, or loses pressure, which is precisely why it, and not the temperature, is the quantity transported across the network (see transport).

The surface integral over a port is a section average of the flux, in the sense of the framework: the boundary term \oint_{\mathcal{S}} \mathbf{f}_n\,\mathrm{d}\mathcal{S} evaluated over a port of area A is A\,\langle \mathbf{f}_n\rangle, and the single-state closure of the final section is exactly the statement that this average is adequately represented by the flux formed from the edge state. Under that closure the stochastic contributions to the port flux, the turbulent departures X'' from the section-averaged profile, are taken to cancel on averaging wherever they are not already carried by the constitutive coefficients of the element relations, so the balances below are those of the averaged profile. The fluctuation that survives them, and on which the acoustic layer of perturbation network is built, is the organized one, X': the coherent, phase-resolved departure that a harmonic excitation of the network produces.

1.3 The steady balance and jump conditions

We seek a steady operating point, so the time-derivative term vanishes and the integral law reduces to a balance of surface fluxes against the source. This reduction requires only steadiness: at the operating point the stored mass, momentum, and energy do not change in time, and the transient term is zero for an element of any internal volume. Under the linear-acoustics assumption of the overview the steady operating point coincides with the temporal mean of the unsteady flow: averaging the exact balances returns the steady equations together with correlation sources \overline{\varrho' u'}, \overline{p' u'}, and their kin, every one of which is of second order in the organized fluctuation and is therefore discarded at the same stroke as the nonlinear acoustics. The two part company once that fluctuation is no longer infinitesimal, since the correlation sources then bias the mean state by an amount growing with the square of the amplitude, most sharply across the elements whose losses are quadratic in the velocity (see limitations).

The correlations of the two fluctuating fields are thus removed from the mean balances by arguments that must not be conflated. The stochastic correlations are modeled: they do not vanish physically, and the loss coefficients and discharge coefficients of elements exist precisely to carry their effect, so the closure is empirical and its accuracy is that of the constitutive model. The organized correlations are neglected: they are genuinely small, uniformly of the order the acoustic layer already discards, so the closure is one of asymptotic ordering and needs no model at all. An overbar therefore denotes an average over both fields, but only the second of the two closures is a matter of order, and only the first is a matter of calibration. For an element P whose boundary consists of its port surfaces (the incident edges e_i) together with solid walls, the balance is given as:

\sum_{e_i} \mathbf{F}_i \;=\; \dot{\boldsymbol{\Omega}}_P, \qquad \mathbf{F}_i \;=\; \sigma_{P,e_i}\,\mathbf{f}_i\, A_i,

where \mathbf{f}_i is the flux evaluated at the representative state of port i, A_i is the port area, \sigma_{P,e_i} \in \{+1,-1\} is the orientation factor (see framework) that aligns each edge’s arbitrary arrow with the outward normal of P, and \dot{\boldsymbol{\Omega}}_P is the net source of P — the integral of the volumetric source density \dot{\boldsymbol{\omega}} over the element volume:

\dot{\boldsymbol{\Omega}}_P \;=\; \int_{\mathcal{V}_P} \dot{\boldsymbol{\omega}}\,\mathrm{d}\mathcal{V}.

The orientation factor plays the role of the outward-normal sign of a finite-volume scheme, so an edge shared by two elements enters their balances with opposite signs and the shared face conserves each quantity by construction.

No flux crosses the solid walls, so they contribute to none of the balances through convection. They enter one balance only — the momentum balance — through the pressure force they exert on the gas, and it is precisely this wall-pressure term that distinguishes one element from another. The mass and energy balances take the same universal form for every interior element, while the momentum/pressure relation encodes the geometry: it is what separates a nozzle from an orifice from a dump diffuser, each exerting a different wall force. The catalogue of these relations is the subject of elements.

1.4 The zero-volume limit and the fate of sources

The second simplification is the zero-volume limit, a modeling convenience applied to an element rather than a necessity — the steady balance above already holds for a finite volume. Its effect is to discard whatever scales with the internal volume; the question worth settling is what the net source \dot{\boldsymbol{\Omega}}_P = \int_{\mathcal{V}_P}\dot{\boldsymbol{\omega}}\,\mathrm{d}\mathcal{V} leaves behind as \mathcal{V}_P \to 0. The answer depends on how the source density behaves in the limit, and it is cleanest to split it into a regular part, which stays bounded, and a concentrated part, whose magnitude grows as the volume shrinks so that its integral does not:

\dot{\boldsymbol{\Omega}}_P = \underbrace{\int_{\mathcal{V}_P}\dot{\boldsymbol{\omega}}_{\text{reg}}\,\mathrm{d}\mathcal{V}}_{\;=\,\mathcal{O}(\mathcal{V}_P)\,\to\,0\;} \;+\; \underbrace{\int_{\mathcal{V}_P}\dot{\boldsymbol{\omega}}_{\text{conc}}\,\mathrm{d}\mathcal{V}}_{\;\to\,\dot{\boldsymbol{\Omega}}_P^{\,\text{lumped}}\;},

where \dot{\boldsymbol{\omega}}_{\text{reg}} is a bounded density and \dot{\boldsymbol{\omega}}_{\text{conc}} is a density that concentrates onto the vanishing element. A bounded density integrates to a quantity of order \mathcal{V}_P, which vanishes with the volume, exactly as the transient term did. A concentrated density is, in the limit, a Dirac source — formally \dot{\boldsymbol{\omega}}_{\text{conc}} \to \dot{\boldsymbol{\Omega}}_P^{\,\text{lumped}}\,\delta(\mathbf{x} - \mathbf{x}_P) — whose integral is the finite rate \dot{\boldsymbol{\Omega}}_P^{\,\text{lumped}} the element imposes regardless of its size. The net source that survives the limit is therefore exactly this lumped rate, and it is the term that appears in the jump condition:

\dot{\boldsymbol{\Omega}}_P^{\,\text{lumped}} \;=\; \lim_{\mathcal{V}_P \to 0}\int_{\mathcal{V}_P}\dot{\boldsymbol{\omega}}\,\mathrm{d}\mathcal{V}.

Consequently the framework fully supports in-element sources, provided they are lumped: a heat-release element retains its full power \dot{Q}_P in the energy balance, a fan or pump adds a finite shaft work or momentum source, and a reactor adds a finite species production rate. A compact flame is thus a legitimate jump condition carrying a finite source, and mechanically such a source enters through the element’s donor term without altering the per-edge transport equations or the equation count (see transport).

A less obvious but equally important point is that the volume an element discards is not dynamically idle. At steady state its accumulation rate is zero, but its capacity to store mass and energy reappears the moment the flow is unsteady: the transient term, restored under a harmonic perturbation, becomes the element’s acoustic storage (see perturbation network). Taking an element to the zero-volume limit therefore has a consequence beyond dropping volumetric sources — it removes that element’s storage contribution to the acoustic problem, which is why the limit is applied selectively: a component whose compliance or inertia shapes the acoustic response is kept at finite volume, while one whose internal volume is dynamically negligible, such as a thin flame, is idealized to zero volume.

1.5 Edge-state closure

The steady balance references the flux \mathbf{f}_i at each port, which is a section average over that port surface (see framework). We model this average by evaluating the flux formulas at the single representative state carried on the edge — the same closure a finite-volume method makes at a coarser scale, in which one cell-face state stands for the average over the face. For a convected quantity \psi this replaces the true port flux A\,\langle m\,\psi\rangle by \dot m\,\psi_e, and the leading correction is the profile-shape factor:

A\,\langle m\,\psi\rangle \;=\; \beta_\psi\,\dot m\,\psi_e, \qquad \beta_\psi = \frac{\langle m\,\psi\rangle}{\langle m\rangle\,\psi_e},

where \beta_\psi = 1 for a uniform profile. The present work takes \beta_\psi = 1 throughout, so the port flux is a function of the edge’s own state alone; the element balances then couple only to the states of their incident edges, and the network’s sparsity mirrors its connectivity. A less obvious benefit of writing the correction explicitly is that it names the route to a wider validity: supplying a measured or modeled \beta_\psi per element — a developed-profile kinetic-energy coefficient, say — restores the exact section flux without touching the equation structure, so the single-state closure can be extended to non-uniform profiles as an element-level refinement rather than a change of framework.

The choice of the variables that carry each edge’s state, on which the solver’s robustness turns, is state and recovery.