When the chemical energy of an explosive medium is concentrated into discrete, sheet-like sources separated by inert gas, direct simulations of the one-dimensional Euler equations (Mi, Timofeev & Higgins, J. Fluid Mech. 817 (2017) 306–338) show that the resulting quasi-periodic ensemble of interacting blast waves propagates at average speeds up to 15% above the Chapman–Jouguet (CJ) value, with a time-averaged structure whose effective sonic surface sits in a non-equilibrium state — interpretable as a weak detonation. The authors supported these observations only with a heuristic model that patches together the planar Taylor–Sedov point-blast solution and Sakurai's energy-partition result, and posed as an open question whether a rational asymptotic solution of this discrete-source detonation problem, derived rigorously from the governing Euler equations in the joint limit of point-like sources (discreteness parameter Γ→0\Gamma \to 0) and zero release delay (τ→0\tau \to 0), can be found. This problem asks for exactly that: an asymptotic derivation from the conservation laws yielding closed-form predictions of the existence and structure of the quasi-periodic limiting propagation, of the super-CJ average speed (its dependence on the ratio of specific heats and its insensitivity to the released energy and delay), and of the averaged wave structure at the effective sonic surface — or a rigorous demonstration that no such limiting solution exists.

Background

A detonation is a combustion wave in which a shock compresses and heats the reactants, triggering exothermic reaction that in turn supports the shock. Classical theory assumes a homogeneous energetic medium and a steady, one-dimensional wave: the ZND (Zel'dovich–von Neumann–Döring) structure, in which a lead shock is followed by a reaction zone terminating where the flow becomes sonic relative to the wave (the sonic surface). The self-sustained wave of minimum speed compatible with conservation of mass, momentum, and energy is the Chapman–Jouguet (CJ) detonation; the CJ speed is conventionally treated as a fixed upper bound for a given energy release, and measured gaseous detonation speeds typically agree with equilibrium CJ predictions to within about 1%. A weak detonation is a solution branch in which the wave speed exceeds the CJ value and the flow reaches sonic conditions before completing its transition — a branch normally ruled out for homogeneous media with irreversible exothermic kinetics, and previously encountered mainly as 'pathological detonations' enabled by special thermodynamics. The sonic surface is the locus where the flow speed relative to the wave equals the local sound speed; it bounds the region that can communicate with the shock front, so its state controls the propagation speed. Mi, Timofeev and Higgins (J. Fluid Mech. 817 (2017) 306–338) studied a deliberately simplified model that isolates the effect of spatially discrete energy release: a calorically perfect gas in which a prescribed energy release per unit mass QQ is concentrated into sheet-like sources of width WW spaced a distance LL apart (discreteness parameter Γ=W/L\Gamma = W/L; Γ→0\Gamma \to 0 means point-like, delta-function-like sources), each source releasing its energy a prescribed delay time after the shock arrives (non-dimensional delay τ\tau; τ→0\tau \to 0 means release at the shock front). The governing equations are the one-dimensional Euler equations for the inert gas with this prescribed, discrete energy deposition — there is no reaction-progress variable or kinetic rate. Numerically, as the front passes through hundreds of sources the dynamics become quasi-periodic and the average wave speed converges: as the sources become highly discrete, speeds reach as much as 15% above CJ (for ratio of specific heats γ=1.1\gamma = 1.1), the deviation grows as γ→1\gamma \to 1, and it is insensitive to the total energy release QQ, the delay τ\tau, and whether sources are fixed in the laboratory frame or convected with the flow. Temporal averaging of the simulations onto a steady one-dimensional projection shows a structure consistent with ZND — except that the effective sonic surface does not correspond to an equilibrium state, so the super-CJ waves act as weak detonations. The authors' Appendix constructed an analytic model for the Γ→0\Gamma \to 0, τ→0\tau \to 0 limit by patching known results: the planar Taylor–Sedov self-similar point-blast solution, Sakurai's energy-partition factor η\eta for blast energy split between forward- and backward-running waves at the release point, a recursive carried-over particle velocity, and hypersonic-similarity analogies. This construction reproduces qualitative trends (increasing deviation from CJ with decreasing γ\gamma) but over-predicts the plateau speed, and it is explicitly ad hoc rather than derived from the conservation laws. Classical detonation asymptotics has successfully exploited other singular limits — high or zero activation energy, high overdrive, small heat release, and the Newtonian limit γ→1\gamma \to 1 — but the limit of spatially discrete, delta-function energy release has no such rational asymptotic treatment. Whether one can be constructed, and what it would predict, is the open question the paper 'left to the detonation theory community'.

Problem Statement

For the one-dimensional discrete-source detonation model of Mi, Timofeev and Higgins (2017) — a calorically perfect gas governed by the Euler equations, with a prescribed energy release per unit mass QQ concentrated into sheet-like sources of width WW and spacing LL (discreteness Γ=W/L\Gamma = W/L), each source releasing its energy a non-dimensional delay τ\tau after the shock front arrives — construct, or rigorously rule out, a rational asymptotic solution in the joint limit of point-like sources (Γ→0\Gamma \to 0) and zero delay (τ→0\tau \to 0), derived directly from the governing conservation laws rather than by patching known solutions. A satisfactory solution must be asymptotically systematic (with stated scalings, matching conditions, and controlled error), and must yield closed-form predictions for: (i) the existence and structure of the quasi-periodic limiting propagation; (ii) the average wave speed relative to the CJ speed of the equivalent homogeneous medium as a function of the ratio of specific heats γ\gamma, including its observed insensitivity to the total energy release QQ and to the delay τ\tau; and (iii) the time-averaged wave structure, in particular the non-equilibrium state at the effective sonic surface that makes the super-CJ waves weak detonations. A refutation of the negative branch must rule out the whole class — all of (i)–(iii) together with the stated systematicity requirement — and not merely a single conjunct.

References

  1. XiaoCheng Mi, Evgeny V. Timofeev, Andrew J. Higgins (2017), 'Effect of spatial discretization of energy on detonation wave propagation', Journal of Fluid Mechanics 817, 306–338. DOI: 10.1017/jfm.2017.81; https://doi.org/10.1017/jfm.2017.81 (preprint arXiv:1608.07665, https://arxiv.org/abs/1608.07665)
  2. XiaoCheng Mi, Andrew J. Higgins (2015), 'Influence of discrete sources on detonation propagation in a Burgers equation analog system', Physical Review E 91, 053014. DOI: 10.1103/physreve.91.053014; https://doi.org/10.1103/physreve.91.053014 (preprint arXiv:1502.00980, https://arxiv.org/abs/1502.00980)
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