The approximate 'all-shock' reactive Riemann solver of Beccantini and Studer (2010) solves the reactive Riemann problem for thermally perfect gases with a single infinitely fast irreversible reaction and a prescribed fundamental flame speed, using an entropy-fixed Roe-type flux that spans weak and Chapman–Jouguet deflagrations and Chapman–Jouguet and strong detonations with continuous regime transitions; it powers the Reactive Discrete Equations Method (RDEM) combustion model of the industrial EUROPLEXUS code, validated at 10–100 m scales. Two robustness gaps of this Roe-type solver remain open in the published record: the solver degrades for slow deflagrations at low Mach number, where Turkel-type preconditioning has not been adapted to the reactive jump structure, and it is prone to carbuncle-type multidimensional shock instabilities on strong shocks; the authors' own 2024 tangential-artificial-viscosity cure addresses single-component and multi-material non-reactive flows and has not been extended to reactive flows. The problem is to extend this solver to both regimes — low-Mach accuracy and carbuncle-free strong multi-dimensional shocks — while preserving all-regime coverage, regime-transition continuity, and consistency of the reactive jump conditions and flame-speed closure.

Background

In the numerical simulation of premixed gaseous combustion for industrial safety (hydrogen risk in nuclear containments, vented explosions, Li-ion battery thermal-runaway events), a common model is the reactive Euler equations: the compressible inviscid balance laws of mass, momentum and energy, closed by a progress variable tracking a single irreversible, infinitely fast chemical reaction, so that every gas state is either unburnt or fully burnt. Interface-based finite-volume methods need, at each cell face between a burnt and an unburnt state, the solution of the reactive Riemann problem — the initial-value problem with piecewise-constant left and right data in which the classical gas-dynamic waves are supplemented by a reactive discontinuity (a combustion wave). Across that wave the Rankine–Hugoniot jump conditions — the conservation statements linking the states on the two sides — include the chemical energy release qq. When the gases are thermally perfect, i.e. ideal-gas mixtures whose specific heats (and hence enthalpies) depend on temperature, as is the case for high-temperature combustion products, these jump conditions lose their closed-form calorically-perfect solutions and must be solved numerically. The physically admissible combustion waves form four regimes: weak deflagration, Chapman–Jouguet (CJ) deflagration, CJ detonation and strong detonation, where the Chapman–Jouguet states are the sonic limiting states at which the wave speed relative to the burnt gas equals the local sound speed. A deflagration is not determined by conservation alone; a burning-rate closure is required, and Beccantini and Studer (2010) close it by prescribing the fundamental flame speed K0K_0, the flame speed relative to the gas ahead of it.

Beccantini and Studer (2010) solved this reactive Riemann problem with a field-by-field iterative algorithm, verified that transitions between the four regimes vary continuously with K0K_0 and with the piston velocity driving the flow, and — replacing every genuinely nonlinear rarefaction wave by an entropy-fixed shock (the 'all-shock' approximation, with an entropy fix so that inadmissible discontinuities are corrected) — constructed an approximate reactive Riemann solver of Roe type, i.e. a flux-difference-splitting scheme that builds the numerical flux from a linearization of the flux jump across the interface. Implemented inside the Reactive Discrete Equations Method (RDEM), a conservative interface-capturing finite-volume framework for reactive mixtures, this solver computes one- and two-dimensional deflagration and detonation flows robustly and now underpins the combustion model of the EUROPLEXUS fast-transient industrial code, validated against large-scale experiments.

Two robustness gaps of this solver line remain open in the published record. First, low-Mach behavior: like all Roe-type schemes, the solver's numerical dissipation scales with the acoustic speed, so for slow deflagrations — laminar hydrogen–air flame speeds are of order 0.10.1–0.30.3 m/s against acoustic speeds near 10001000 m/s — the acoustic time-step restriction and excessive dissipation degrade accuracy. The standard remedy, Roe–Turkel preconditioning (multiplying the scheme's dissipation matrix by a Turkel-type preconditioner that rescales the acoustic eigenvalues down to convective magnitudes at low Mach number), has not been adapted to the reactive jump structure and its four-regime selection. Second, shock stability: Roe-type solvers are prone to the carbuncle phenomenon, a grid-aligned odd–even decoupling instability that corrupts strong multi-dimensional shocks on structured meshes. Beccantini, Galon, Lelong and Baj (2024) proposed a tangential artificial viscosity (an extra viscosity term in the momentum component tangential to the cell face, applied near shocks) that cures the carbuncle for single-component and multi-material non-reactive flows; its extension to the reactive Riemann solver remains open. Whether such cures can be transferred to the reactive all-shock solver without disturbing the fundamental-flame-speed closure, the regime selection, and the regime-transition continuity is the open question.

Problem Statement

Extend the approximate 'all-shock' reactive Riemann solver of Beccantini and Studer (2010) — the Roe-type flux solver for the reactive Euler equations with thermally perfect gases and prescribed fundamental flame speed K0K_0, spanning the weak-deflagration, Chapman–Jouguet-deflagration, Chapman–Jouguet-detonation and strong-detonation regimes within the Reactive Discrete Equations Method (RDEM) finite-volume framework — to the two regimes in which it is currently unreliable, while preserving its established properties (all-regime coverage, continuous regime transitions with respect to K0K_0 and the piston velocity, and convergence to exact one-dimensional reactive Riemann solutions):

(i) Low-Mach-number regime: equip the solver so that slow-deflagration computations at low Mach number are free of accuracy loss driven by acoustic stiffness, the flame and interface structure is not over-diffused as the Mach number tends to zero, and the reactive Rankine–Hugoniot jump conditions, the fundamental-flame-speed closure and the four-regime selection remain consistent under the modification; on a fixed slow-deflagration configuration computed at two or more successively lower Mach numbers on a common sequence of refined meshes, the computed flame speed and interface thickness must converge under refinement at the rate of the unmodified solver, and the change caused by lowering the Mach number must not exceed the discretization error on the finest mesh.

(ii) Strong multi-dimensional shocks: cure the carbuncle-type multidimensional shock instability of the Roe-type reactive solver on structured grids without corrupting the combustion wave, its flame-speed closure or the regime-transition continuity, and demonstrate the cure on a strong-shock multi-dimensional reactive configuration (detonation diffraction or shock–flame interaction), on a structured mesh on which the unmodified Roe-type reactive flux develops the grid-aligned odd–even decoupling of the shock front, by showing that the cure suppresses that decoupling there without altering the combustion wave away from the shock.

Production-scale integration and validation of the solver serves as the application context rather than the target: the RDEM combustion model built on this solver has been validated in the EUROPLEXUS code against large-scale experiments covering all major premixed flame combustion regimes (Velikorodny et al. 2015).

References

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