In the weakly nonlinear asymptotic theory of self-sustained gaseous detonations derived by Faria, Kasimov and Rosales (J. Fluid Mech. 784, 2015), the reactive Euler equations reduce, at small heat release, to a forced version of the unsteady transonic small-disturbance (Zabolotskaya–Khokhlov) equation coupled to a rate equation. Numerical solutions of this reduced system develop cellular fronts whose transverse shocks intersect the lead shock in configurations resembling the triple points of classical cellular detonations. However, algebraic arguments of Tabak–Rosales (1994) and Hunter–Brio (2000) for the corresponding non-reacting asymptotics show that three shocks cannot meet at a point with smooth flow in between — the classical von Neumann paradox. The open problem, stated explicitly in the source paper, is to determine the actual local wave structure at these triple-point-like intersections in the forced detonation model — genuine multi-shock discontinuities, sharp but continuous waves, or Guderley-type singularities modified by heat release.

Background

A detonation is a combustion wave in which a strong leading shock heats the gas and thereby triggers exothermic chemical reactions that, in turn, sustain the shock. The classical one-dimensional, steady description is the ZND model (after Zel'dovich, von Neumann and Döring), in which a planar lead shock is followed by a reaction zone governed by the reactive Euler equations with a heat-release law; for a given heat release qq the self-sustained wave of minimum speed is the Chapman–Jouguet (CJ) detonation. In experiments, gaseous detonations are almost never planar and steady: the lead front breaks into cells produced by transverse shock waves that sweep along it and intersect it at triple points — points where three shocks (incident, transverse and Mach stem) meet, as in Mach reflection of a shock from a wall. The regularity and cellular dynamics of gaseous detonations are surveyed in standard monographs on the detonation phenomenon.

When a shock is weak, the classical triple-shock picture itself runs into a fundamental difficulty known as the von Neumann paradox. Von Neumann's two-shock/three-shock reflection theory, based on the Rankine–Hugoniot jump conditions (the conservation relations across a discontinuity) for the full Euler equations, predicts that below a critical shock strength regular reflection should be impossible; yet experiments and computations display apparent triple-point (Mach) configurations in exactly this regime. For weak shock reflection at near-grazing incidence, the natural asymptotic limit is governed by the unsteady transonic small-disturbance equation, written in terms of a velocity potential ϕ\phi as ϕxτ+(12ϕx2)x+ϕyy=0\phi_{x\tau}+\left(\tfrac{1}{2}\phi_x^2\right)_x+\phi_{yy}=0, where xx is the streamwise, yy the transverse coordinate and τ\tau a retarded time; in nonlinear acoustics the same equation is called the Zabolotskaya–Khokhlov equation. Tabak and Rosales (Phys. Fluids 6:1874–1892, 1994) showed that in this asymptotic framework the 'triple-point paradox' arises: a simple algebraic argument of shock-polar type (the shock polar is the locus of post-shock flow states attainable across an oblique shock, parametrized by flow deflection) rules out configurations in which three shocks meet at a point with smooth flow between them, and they proposed a resolution involving an unusual, non-classical singularity. Hunter and Brio (J. Fluid Mech. 410:235–261, 2000) analyzed weak shock reflection with these asymptotic equations and confirmed the non-existence of classical triple-shock solutions while computing numerical configurations of von Neumann reflection — the limiting weak-reflection pattern in which the reflected shock degenerates into a continuous compression merging with an expansion fan, with no slipstream. The companion Euler computations of Zakharian, Brio, Hunter and Webb (J. Fluid Mech. 422:193–205, 2000) resolved Guderley's decades-old proposal: behind the apparent triple point there is an extremely small supersonic patch, an expansion fan and a tiny Mach stem (Guderley reflection), consistent with the asymptotic picture; Skews and Ashworth (J. Fluid Mech. 542:105–114, 2005) and later work on the triple-point singularity further examined the physical nature of this resolution. Thus, on the non-reacting side, the paradox is by now largely understood as the replacement of the forbidden triple point by a Guderley-type singular structure.

Detonations bring this same fluid-dynamic regime together with chemistry. Building on earlier weakly nonlinear detonation asymptotics of Rosales and Majda (SIAM J. Appl. Math. 43:1086–1118, 1983), Faria, Kasimov and Rosales (J. Fluid Mech. 784:163–198, 2015; arXiv:1407.8466) derived, from the compressible reactive Navier–Stokes equations at small heat release qq, large activation energy θ\theta and γ−1\gamma-1 small, the two-dimensional asymptotic model

uτ+u ux+vy=−12λx,vx=uy,λx=−k (1−λ) exp⁡ ⁣[θ(q u+q λ)],u_\tau + u\,u_x + v_y = -\tfrac{1}{2}\lambda_x, \qquad v_x = u_y, \qquad \lambda_x = -k\,(1-\lambda)\,\exp\!\left[\theta\left(\sqrt{q}\,u + q\,\lambda\right)\right],

(written for the perturbation velocity components u,vu,v, reaction progress λ\lambda, rate constant kk, streamwise coordinate xx, transverse coordinate yy and retarded time τ\tau; the first equation is a forced Burgers-type balance for uu, the second imposes irrotationality). Without the reaction term the system reduces exactly to the Zabolotskaya–Khokhlov equation above. Numerical solutions of this forced model reproduce pulsating and period-doubling instabilities in one dimension and self-organized cellular fronts in two dimensions, with transverse shocks whose intersections with the lead front closely resemble detonation triple points. Yet, precisely because the unforced asymptotics forbid classical triple points, the observed triple-point-like structures cannot be what they appear to be, and the algebraic no-go arguments of the non-reacting theory carry over to the model. The source paper states plainly that it makes no attempt to address this problem and that whether the structures are true discontinuities, sharp waves, or singularities remains open. This is a reacting, forced, and non-self-similar analogue of the weak-reflection von Neumann paradox: unlike the steady wedge-reflection geometry in which Guderley reflection was identified, the detonation triple-point-like structures arise inside unsteady, cellular, chaotic dynamics, and the heat-release forcing acts inside the very balance that would otherwise host the singularity.

Problem Statement

For the weakly nonlinear asymptotic detonation model of Faria–Kasimov–Rosales (J. Fluid Mech. 784, 2015), namely

uτ+u ux+vy=−12 λx,vx=uy,λx=−k (1−λ) exp⁡ ⁣[θ(q u+q λ)],u_\tau + u\,u_x + v_y = -\tfrac{1}{2}\,\lambda_x, \qquad v_x = u_y, \qquad \lambda_x = -k\,(1-\lambda)\,\exp\!\left[\theta\left(\sqrt{q}\,u + q\,\lambda\right)\right],

whose solutions in the unstable regime develop cellular fronts with triple-point-like intersections between transverse shocks and the lead shock, determine the actual local wave structure at these intersections. Specifically, taking as established — as the source argues via a simple algebraic argument of the Tabak–Rosales (1994) and Hunter–Brio (2000) type — that the asymptotic equations do not admit classical triple points where three shocks separated by smooth flows meet, establish which configurations are possible or excluded for the forced system, and identify what replaces the forbidden configuration (a sharp but continuous compression, a non-classical singularity, a Guderley-type supersonic patch with an expansion fan, or another structure). This is the von Neumann paradox posed by the source paper, in its natural generality: no restriction on parameters, geometry, or answer form is intended, and the question is the reacting analogue of the Guderley/von-Neumann-reflection resolution achieved for non-reacting weak shock reflection.

References

  1. L. M. Faria, A. R. Kasimov, R. R. Rosales, "Theory of weakly nonlinear self-sustained detonations", Journal of Fluid Mechanics 784 (2015) 163–198. DOI 10.1017/jfm.2015.577; arXiv:1407.8466. https://doi.org/10.1017/jfm.2015.577
  2. E. G. Tabak, R. R. Rosales, "Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection", Physics of Fluids 6 (1994) 1874–1892. DOI 10.1063/1.868246. https://doi.org/10.1063/1.868246
  3. J. K. Hunter, M. Brio, "Weak shock reflection", Journal of Fluid Mechanics 410 (2000) 235–261. DOI 10.1017/S0022112099008010. https://doi.org/10.1017/S0022112099008010
  4. A. R. Zakharian, M. Brio, J. K. Hunter, G. M. Webb, "The von Neumann paradox in weak shock reflection", Journal of Fluid Mechanics 422 (2000) 193–205. DOI 10.1017/S0022112000001609. https://doi.org/10.1017/S0022112000001609
  5. B. W. Skews, J. T. Ashworth, "The physical nature of weak shock wave reflection", Journal of Fluid Mechanics 542 (2005) 105–114. DOI 10.1017/S0022112005006543. https://doi.org/10.1017/S0022112005006543
  6. R. R. Rosales, A. Majda, "Weakly nonlinear detonation waves", SIAM Journal on Applied Mathematics 43 (1983) 1086–1118. DOI 10.1137/0143071. https://doi.org/10.1137/0143071
  7. L. M. Faria, A. R. Kasimov, R. R. Rosales, "Study of a model equation in detonation theory: multidimensional effects", SIAM Journal on Applied Mathematics 76 (3) (2016) 887–909. DOI 10.1137/15M1039663; arXiv:1512.07503. https://doi.org/10.1137/15M1039663
  8. A. Sakurai, S. Kobayashi, "Triple-Point Singularity and the Neumann Paradox of Mach Reflection", in: K. Kontis (ed.), Shock Wave Interactions, Springer (2018) 203–212. DOI 10.1007/978-3-319-73180-3_16. https://doi.org/10.1007/978-3-319-73180-3_16