1 Overview
Nefes is a network solver for compressible, reacting flows and their thermoacoustics. It represents a fluid system as a directed graph — components joined at their ports — and solves for the steady mean flow through that graph, then for the linear acoustic behaviour of small perturbations around that flow, without ever resolving the full three-dimensional field. The purpose of this document is to state what the method computes, the assumptions under which it holds, and the four modeling decisions that shape everything downstream, and to serve as the map into the rest of the documentation.
1.1 What the method computes
A flow network is the compressible-flow analogue of an electrical circuit: components such as nozzles, orifices, junctions, and plenums are joined at ports, and gas flows through them driven by pressure differences much as current is driven by voltage differences. Given the network’s connectivity, its element parameters, and its boundary data, a mean-flow solve answers the questions an engineer asks of such a system — how the flow splits between branches, what pressure, temperature, and Mach number each component sees, how much flow a supply delivers — in milliseconds rather than the hours a full computational-fluid-dynamics simulation would take.
Unlike an incompressible or electrical network, a compressible gas network carries three coupled quantities along every connection — mass flow, pressure, and energy — and the relations between them are nonlinear. This admits genuinely compressible behaviour: the density varies along a path, hot and cold streams mix, and the mass flow saturates once the narrowest section reaches the speed of sound, the phenomenon known as choking. On the reacting side, an element may add heat or undergo chemical equilibrium, so the network also resolves flame temperature rise, dilution, and the transport of each injected stream’s composition.
The distinguishing aim of the project, however, lies one step beyond the mean flow. Because every steady element relation is instantaneous and algebraic, its linearization about the converged operating point is the corresponding acoustic jump condition, so the same assembled operator that produced the mean flow, differentiated, becomes the frequency-domain acoustic network. On that acoustic network the tool computes the forced response and scattering between chosen stations, the natural modes and their growth rates (stability), the response of a compact flame as a dynamic source, and — inverting the forced problem — the identification of an unknown element’s dynamic response from a measured network response. It should be emphasized that this unity is structural rather than incidental: the mean flow and its acoustics are two evaluations of one operator, so no second model can drift from the first.
What the method does not compute is the spatial field inside a component. An element is treated as a lumped relation between the states at its ports; the interior flow, the boundary layers, and the turbulence are represented through constitutive models, not resolved.
1.2 Standing assumptions and scope
The following assumptions hold throughout the mean-flow formulation, and each is revisited where a later document depends on it:
- Ideal gas. Each stream obeys p = \varrho R T with R = R_u/W, the gas being a mixture of ideal gases; this is a good approximation for air and combustion products away from high pressures and the critical region, and it holds under both thermodynamic closures. The closures differ in what accompanies it: a non-reacting stream is calorically perfect, with a constant W and constant specific heats, so that h = c_p T, c^2 = \gamma R T, and p/\varrho^\gamma is an entropy invariant; a chemical-equilibrium stream retains only the thermal equation of state, since dissociation makes W, c_p, and the sound speed functions of the state (see thermochemistry).
- Inviscid on the scale of a component. The gas is governed by the Euler equations at the component scale; viscous and turbulent losses enter only through lumped constitutive terms in the element relations, not through a resolved shear field.
- Clean cross-section ports. The flow crosses each port perpendicularly, so a single signed normal velocity carries all of the kinetic energy through it; this constrains the ports, not the interior flow, which may be fully multidimensional.
- Subsonic regime. The present version is restricted to subsonic mean flows, whether flowing or quiescent; the throat may reach M = 1 and choke at an outlet, but supersonic branches and shocks are deferred (see limitations).
- Linear acoustics. The acoustic layer superposes on the mean flow fluctuations small enough that any quantity of second order or higher in the perturbation is negligible, so the response is governed by the first-order (linearized) operator; amplitudes and nonlinear acoustic effects are outside the present scope. This assumption also underwrites the mean flow, and not the acoustics alone: it is what allows the steady operating point the solver converges to be read as the temporal mean of the unsteady flow, the two differing only by correlation sources that are themselves of second order (see governing equations, limitations).
The symbols, decorations, and sign conventions used here and in every other document are collected once in the nomenclature; the reader is referred there rather than to a redefinition in place.
1.3 The four modeling decisions
The framework rests on four decisions which mainly arise from the requirements:
- Robust mean flow solution in arbitrarily complex topologies,
- Compatibility of the mean flow and acoustic networks.
Together they are what let the solver discover the flow — its directions, its choke points, its backflow — from a cold start, rather than requiring to start from a particular initial guess of the flow field.
- All state lives on the connections, none in the components. Conventional flow network solvers employ a segragated storage scheme for solution variables, e.g. pressure at nodes and mass flow rate at edges. In Nefes, edges carry the flow state; elements own only the physical relations between the states of the edges that meet at them (see framework). An element is a control volume on which the governing balances are applied; it carries no state of its own, because the state lives on the edges and no volume-associated state is needed. This also allows Nefes to follow the standard acoustic network convention: e.g. a transfer matrix relates the acoustic state variables across an element, which can be directly realized as all states live on the edges of the network.
- Each edge carries the triple (\dot m,\ p,\ h_t) — the mass flow rate, the static pressure, and the total enthalpy (see state and recovery). This choice has the property that every other flow quantity — density, velocity, temperature, Mach number, stagnation state — is recovered from the triple uniquely and smoothly, no matter how fast or in which direction the gas flows, a property that a more familiar triple such as (p_t, T_t) does not share.
- A fixed bookkeeping rule makes the system square. Every element contributes exactly as many equations as it has ports, and every edge contributes exactly one transport equation (see equation structure and transport). The count is therefore square independently of which way the gas flows, which is precisely what allows flow directions to be discovered rather than prescribed.
- The formulation is built around smoothness. Every residual is written without any branch,
abs,min, ormaxon the flow state, so it is smooth and complex-analytic (see well-posedness and the smoothness contract). Newton’s method, suitably damped, then converges even from an exactly quiescent start, and the Jacobian is obtained exactly — to machine precision, free of subtractive cancellation — by complex-step differentiation (Lyness and Moler 1967; Squire and Trapp 1998; Martins et al. 2003) (see complex-step).
The fourth decision has a consequence that reaches past the mean flow, which is the foundational feature of the Nefes acoustic network. Because the Jacobian is assembled exactly at convergence, it doubles as the algebraic content of the acoustic problem: restoring the finite-volume storage terms dropped at steady state, the lossless-duct wave phases, and any unsteady flame source turns that same Jacobian \overline{\mathbf{J}} into the frequency-domain operator \mathbf{A}(\omega) = \overline{\mathbf{J}} + \mathrm{i}\omega\mathbf{M} + \mathbf{P}(\omega) + \mathbf{S}(\omega), which reduces to the steady operator \mathbf{A}(0) = \overline{\mathbf{J}} at \omega = 0 (see perturbation network). This is the structural payoff of the design, and it is the reason a single tool serves both the steady and the acoustic question.
1.4 How the documentation is organized
The documentation is arranged into three tracks over a shared foundation of nomenclature and references.
- Theory — the physics and mathematics, from the graph model and governing balances through transport, elements, characteristics, and choking, then the reacting closures, the perturbation network, the dynamic flame sources, the forward acoustic analyses, and the inverse identification analysis. Each theory document opens with its own assumptions ledger and either cites or derives every claim it makes.
- Design philosophy — why the code is shaped the way it is: the smoothness-over-branching principle, the complex-step derivative engine, the regularized-primitive library and its per-kernel safety contract, the assembly of the mean-flow and perturbation operators, the damped-Newton solver with its artificial-resistance continuation, and the reproducibility discipline.
- Validation — the evidence: a master map from every physical claim to the analytic or literature case that checks it, the internal consistency verifications (complex-step against finite difference, edge-direction-flip invariance, thermochemistry against an independent oracle), and the named literature benchmarks.