The Flame as a Wave: A Hyperbolic Framework for Combustion Chemistry

This preprint proposes a hyperbolic mathematical framework for combustion chemistry, departing from the parabolic diffusion foundation that every operational combustion model — EDM, EDC, flamelet, FPV — has inherited from the 1970s. The central argument is that the flame is fundamentally a wave phenomenon: at the quantum level chemistry is wave mechanics, at the molecular level reactions propagate causally at finite speed, and the parabolic diffusion approximation is a computational compromise imposed by 1970s hardware, not a physical necessity.

The proposed governing equation for reaction progress c is hyperbolic:

∂²c/∂t² = c²_wave ∇²c − γ∇⁴c + Ṙ(c, |∇p|)

where c_wave is the reaction wave speed, γ is a composition-space surface tension analogue, and |∇p| is the pressure gradient magnitude — computed by every Navier-Stokes solver at zero additional cost. The pressure gradient replaces mixture fraction as the primary signal: the flame front is where |∇p| is large, and the rate expression f(|∇p|) maps the complete S-curve behaviour — ignition, stable burning, extinction — through pressure alone. Two model constants, p_ref and p_ext, are extracted from a single 1D flamelet calibration solve. Arrhenius chemistry never appears in the CFD solver. The flamelet is a calibration tool, not a lookup table.

The framework eliminates mixture fraction Z, the β-PDF and its collapse under swirling oxidiser streams, the flamelet lookup table, the scalar dissipation rate χ, the k/ε mixing timescale, and the EDC fine structure volume fraction — all computational compromises from the pre-GPU era with no physical necessity.

The physical intuition behind the framework is a droplet striking a water surface: the ripple is the reaction, propagating outward from the impact point, carrying energy, interfering with other ripples. Constructive interference accelerates the flame; destructive interference quenches it. Turbulence-chemistry interaction emerges naturally from wave mechanics, without a separate model.

The flame front is already described exactly as a pressure-driven discontinuity by the Rankine-Hugoniot conditions — no turbulence model, no PDF, no mixture fraction. The parabolic diffusion framework was never a physical necessity. Diffusion is what waves look like when you stop resolving them.

Radiation simplifies significantly: the wave equation provides honest local emission source terms κI_b at every cell every timestep at zero cost. A lightweight ray casting pass on GPU RT cores handles long-range radiative transport. Chemistry on GPU shader cores and radiation on GPU RT cores run simultaneously, with one CPU-GPU handshake per timestep. Ray tracing is the geometric optics limit of wave optics — wave chemistry is the parent framework, ray tracing falls out as a special case.

Two open problems remain: a rigorous definition of c_wave as a function of local flow conditions, and an adaptive wave speed scheme for stiffness across reaction zones of varying timescale. A third problem — the mathematical bridge between |∇p| and Arrhenius kinetics — is closed: the bridge goes through the reference flamelet, which extracts p_ref and p_ext directly from the 1D solution.

Why now: The parabolic assumption was never seriously challenged for deflagrations because the mathematical framework did not exist in tractable form, the computation was prohibitive before GPU hardware, and the parabolic models worked well enough for the flames they were validated against. None of these conditions hold in 2026. GPU hardware makes the computation conceivable. The flames that existing models fail on — oxy-fuel, high-strain, swirl-stabilised — are precisely the flames industrial combustion now needs to simulate. The computational constraints that forced the parabolic assumption are gone. The model built around those constraints has remained.

This is the third framework developed from a 15 kW CH₄/O₂ non-premixed swirl flame on a DESAG co-annular diffusion burner that refused to resolve in the author's master's thesis: Development of an efficient CFD model for high temperature processes, Rohan Anil, TU Bergakademie Freiberg, defended 5 June 2026. The thesis established the experimental reference case and diagnosed the triply constrained nature of the problem.

The first response was the thermochemical ray tracing framework, recasting combustion chemistry as GPU ray traversal through C-H-O composition space with radiation emerging naturally from the ray physics: https://doi.org/10.5281/zenodo.19975933

The second was മഴ (Mazha, meaning "rain" in Malayalam) — a modification to the SST-SAS turbulence model that solves the bootstrap deadlock through Gaussian perturbation seeding triggered by a stagnation sensor, with amplitude scaled as ℘ ~ μ/μ_t. Rain falls uniformly; μ_t decides what survives: https://doi.org/10.5281/zenodo.20555825

A related earlier concept — LBM-guided domain decomposition for RANS/LES simulations, proposing GPU-accelerated LBM as a fast initialisation and domain pruning tool for conventional CFD — was the author's first published open-source concept: https://doi.org/10.5281/zenodo.17771873

All four works share the same philosophy: listen to what the flow already knows. Do not impose structure. Let the physics decide.

This preprint is version 0.1. The full document is currently restricted. The Zenodo record establishes priority of conception as 6 June 2026.