Nomenclature
This page is the single source of notation for the Nefes documentation. Every other document links here rather than redefining a symbol, so that a symbol carries one meaning throughout and the sign, orientation, and frequency conventions are stated once. Where a symbol is genuinely reused in two roles, the two are listed together and the disambiguating context is named.
Conventions
The notation follows a small set of rules, applied uniformly:
- Mean (base) states carry an overbar, \overline{X}, and denote the converged steady state about which the acoustics are linearized (for example the mean density \overline{\varrho} and sound speed \overline{c}). A subscript zero is not used for the mean, as numeric subscripts denote port indices.
- Fluctuations carry a prime, X' = X - \overline{X}, and are the small unsteady departures from the mean. They are the organized (coherent, phase-resolved) fluctuations that the acoustic layer resolves; where the stochastic turbulent departure must be named alongside them it carries a double prime, X'', and it is closed by the constitutive models rather than resolved (see governing equations).
- Complex amplitudes carry a hat, \widehat{X}, and are the frequency-domain amplitude of a time-harmonic fluctuation, X'(t) = \Re\{\widehat{X}\,e^{\mathrm{i}\omega t}\}.
- Section (area) averages carry angle brackets, \langle X\rangle \equiv \tfrac{1}{A}\int_A X\,\mathrm{d}A, and denote the average of a field over a port cross-section; the edge quantities are averages of this kind (see framework).
- Vectors and matrices are set in bold, \mathbf{X}; scalars are set in plain italic, X.
- Frequency is reported and prescribed as the ordinary frequency f in hertz; the angular frequency \omega = 2\pi f appears only inside derivations, and the crossing between the two is stated where it occurs.
- Edge orientation. Every edge carries an arbitrary reference arrow fixed at build time; it defines the positive sense of all signed edge quantities and makes no claim about the flow direction, which the solver discovers (a negative \dot m_e is flow against the arrow).
- The imaginary unit is \mathrm{i}, reserved for it; a complex-step derivative uses the same unit with a real step h_{\text{cs}} (see complex-step).
Roman symbols
| symbol | meaning |
|---|---|
| \varrho,\ u,\ p,\ T | density, signed normal velocity, static pressure, static temperature |
| h,\ h_t | static / total specific enthalpy, h_t = h + \tfrac{1}{2}u^2 |
| p_t,\ T_t | total (stagnation) pressure / temperature |
| c,\ M | speed of sound, signed Mach number M = u/c |
| s | entropy, in the invariant form p/\varrho^{\gamma} |
| \dot m,\ m | mass flow rate (signed along the edge arrow), mass flux density m = \dot m / A |
| A_e | edge (port) cross-sectional area |
| R,\ c_p,\ c_v | specific gas constant, specific heats at constant pressure / volume |
| H | static enthalpy implied by a trial density, H = h_t - m^2 /(2\varrho^2) |
| \dot Q | lumped heat-release rate of a flame element |
| Z_i | conserved mixture fraction of feed stream i (a transported scalar) |
| \mathbf{Y}_{\text{el}} | elemental mass-fraction vector, expanded from the mixture fractions (\mathbf Y_{\text{el}} = \sum_i Z_i\,\mathbf Y_{\text{el}}^{(i)}) before any equilibrium call |
| b | burnt-marker scalar (transported, gates the reacting closure); g(b) its smooth gate |
| \mathbf{x},\ \mathbf{R},\ \mathbf{J} | unknown vector, residual vector, Jacobian \mathbf J = \partial\mathbf R/\partial\mathbf x |
| \mathbf{x}_e | edge state vector, \mathbf x_e = (\dot m_e,\ p_e,\ h_{t,e}) (plus any transported scalars) |
| E | number of edges in the network |
| f,\ g,\ h | characteristic wave amplitudes (downstream-acoustic, upstream-acoustic, entropy) |
| \mathbf{w} | characteristic amplitude vector, \mathbf w = (f,\ g,\ h)^{\top} |
| L | duct length; L_{\text{eff}} its acoustic effective length (with end corrections) |
| q | dynamic head of a stream, q = \tfrac{1}{2}\varrho u^2 (a signed, smoothed form is used in loss residuals) |
| K_L | loss coefficient of a concentrated loss element, referenced to a dynamic head |
| C_c | vena-contracta contraction coefficient of a sudden contraction (C_c = 1 is loss-free) |
| f_0 | resonant frequency [Hz] |
| \mathbf{A}(\omega) | perturbation system matrix, \mathbf A = \overline{\mathbf J} + \mathrm{i}\omega\mathbf M + \mathbf P(\omega) + \mathbf S(\omega) |
| \overline{\mathbf{J}} | converged mean-flow (base) Jacobian; the algebraic block and the zero-frequency operator, \mathbf A(0) = \overline{\mathbf J} |
| \mathbf{M} | storage block: the finite-volume time-derivative terms (compliance and inertance), entering through \mathrm{i}\omega |
| \mathbf{P}(\omega) | propagation block: the lossless-duct phase relations e^{-\mathrm{i}\omega\tau} |
| \mathbf{S}(\omega) | source block: the prescribed unsteady feedback (a flame’s heat-release response) |
| \widehat{\mathbf{b}} | forcing amplitude of the perturbation system (zero for the stability problem) |
| \mathbf{L}_e | per-edge change of basis from solution variables to characteristics, \mathbf w = \mathbf L_e\,\widehat{\mathbf x}_e |
| \mathbf{T}(f) | 2-port transfer matrix between two stations, \mathbf v_{\text{down}} = \mathbf T\,\mathbf v_{\text{up}} |
| \boldsymbol{\mathcal{S}}(f) | 2-port scattering matrix, mapping incoming to outgoing waves, \mathbf w_{\text{out}} = \boldsymbol{\mathcal S}\,\mathbf w_{\text{in}} |
| \mathbf{A}_0(\omega) | known (passive) perturbation operator, with the unknown element set to its reference |
| n | interaction index of the n–\tau flame response |
| R | acoustic reflection coefficient at a termination, R = \widehat g/\widehat f (context distinguishes it from the gas constant) |
| Z | acoustic impedance at a termination (context distinguishes it from a mixture fraction) |
| \operatorname{cond}(\cdot) | per-frequency condition number of the identification system (the identifiability diagnostic) |
Greek symbols
| symbol | meaning |
|---|---|
| \gamma | ratio of specific heats, \gamma = c_p/c_v |
| \Gamma | caloric constant, \Gamma = c_p/R = \gamma/(\gamma-1) (\approx 3.5 for air) |
| \sigma_{P,e} | orientation factor of edge e at element P: +1 if P is the tail, -1 if the head |
| \beta_\psi | profile-shape factor of a convected scalar \psi (\beta_\psi = 1 for a uniform profile) |
| \varepsilon | smoothing width of a regularized primitive (mass-flow units) |
| \theta,\ w,\ \xi | smooth upwind / donor / boundary-regime blending weights |
| \kappa | artificial-resistance (stabilization) coefficient |
| \kappa_s | dimensionless artificial-resistance continuation schedule |
| r_{\text{art}} | artificial-resistance scale |
| \varphi_\varepsilon | smoothed complementarity residual (Fischer–Burmeister), selecting the subsonic and choked regimes of a single row |
| \lambda | Levenberg–Marquardt damping parameter |
| \tau_+,\ \tau_-,\ \tau_u | duct transit times of the downstream-acoustic, upstream-acoustic, and convected paths |
| \tau | flame time lag of the n–\tau response |
| \mathcal{F}(f) | dynamic-source transfer function; for the n–\tau flame, \mathcal F = n\,e^{-\mathrm{i}\omega\tau} |
| \omega | angular frequency, \omega = 2\pi f (derivations only); a mode’s complex frequency is \omega = \omega_r + \mathrm{i}\omega_i |
| \sigma | modal growth rate, \sigma = -\omega_i = -\Im(\omega); a mode is unstable when \sigma > 0 (convention e^{+\mathrm{i}\omega t}) |
An important remark on two reused letters: the operator source block \mathbf{S}(\omega) and the frequency-domain scattering matrix \boldsymbol{\mathcal{S}}(f) are distinct objects and are typeset differently for that reason; likewise R and Z denote a reflection coefficient and an impedance in the acoustic context and the gas constant and a mixture fraction in the mean-flow context, and no document uses a given letter in both roles at once. A related remark on numeric subscripts: they denote port indices (p_0, p_{t,1} for ports 0 and 1), never the mean state, which is why the mean/base state is written with an overbar (\overline{\varrho}, \overline{c}) rather than a subscript zero — the two would otherwise collide on the element residuals.
Decorations, sub- and superscripts
| notation | meaning |
|---|---|
| \overline{X} | converged mean / base value of X |
| X' | fluctuation of X about the mean |
| \widehat{X} | complex (frequency-domain) amplitude of X' |
| \langle X\rangle | section (area) average of X over a port |
| \dot X | a rate (per unit time) |
| X_t | a total (stagnation) quantity |
| X_e,\ X_P | a quantity carried on edge e / owned by element P |
| \dot m^{\text{out}}_{P,e},\ \dot m^{\text{in}}_{P,e} | mass flow leaving / entering element P through edge e, \dot m^{\text{out}}_{P,e} = \sigma_{P,e}\dot m_e |
Terms
| term | meaning |
|---|---|
| characteristic variables | the wave amplitudes (f, g, h) that diagonalize the linearized 1-D Euler system |
| choking | mass-flow saturation when the narrowest section reaches M = 1 |
| complex-step derivative | an exact derivative from a single imaginary-perturbed evaluation, free of subtractive cancellation |
| edge | a port cross-section shared by two elements (or an element and the exterior); owns the state vector |
| element | a network component (graph node): a control volume on which the governing equations are applied; it owns equations, not state (state lives on the edges) |
| flame transfer function (FTF) | the linear frequency response of a flame’s heat release to a reference fluctuation |
| identification (de-embedding) | recovering an unknown element’s dynamic response from a measured network response, given a model of the rest |
| Jacobian | the matrix of sensitivities \partial R_i/\partial x_j |
| jump condition | an algebraic relation between the states on the two sides of a compact element (one taken in the zero-volume limit) |
| residual | how far an equation is from being satisfied at the current guess; all zeros means solved |
| scattering matrix | a frequency-domain 2-port relating the incoming waves at two stations to the outgoing ones |
| seeding | tagging one unknown with an imaginary perturbation, x \leftarrow x + \mathrm{i}h_{\text{cs}} |
| stamp | a local contribution an element writes into the assembled (mean-flow or perturbation) operator |
| storage | the finite-volume compliance and inertance restored to an element under an unsteady perturbation |
| terminal | a single-port boundary element and its incident edge at which the perturbation network imposes an acoustic boundary law on the waves |
| transfer matrix | a frequency-domain 2-port relating the flow variables at two stations along their arrows |
| transport (edge) equation | the donor/upwind relation that carries total enthalpy (and any scalar) along an edge |
| well-posed | having exactly as many independent conditions as unknowns — solvable and unambiguous |