# A shock dynamics approach for analyzing two-dimensional numerical detonation cells in gases

Hiroaki Watanabe, Ashwin Chinnayya, Pierre Vidal

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Hiroaki Watanabe, Ashwin Chinnayya, Pierre Vidal. A shock dynamics approach for analyzing two-dimensional numerical detonation cells in gases. 30th International Colloquium on the Dynamics of Explosions and Reactive Systems (ICDERS 2025), Jul 2025, Ottawa, Canada. hal-05410530

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Submitted on 11 Dec 2025

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# A shock dynamics approach for analyzing two-dimensional numerical detonation cells in gases

Hiroaki Watanabe, Ashwin Chinnayya, Pierre Vidal

Institut Pprime UPR 3346 CNRS, ISAE-ENSMA, Université de Poitiers
1, avenue Clément Ader, BP 40109, 86961 Futuroscope-Chasseneuil, France

The reaction zone of a detonation in gases exhibits a three-dimensional, unsteady cellular structure. Thus, the leading shock of self-sustained detonations consists of Mach stems –the cells– with longitudinal shocks that cyclically decelerate from velocities greater than to lesser than the Chapman–Jouguet value $D_{CJ}$, with continuously decreasing total curvature. Relationships between the normal shock velocity $D_n$, the normal acceleration $\dot{D}_n$, and the total curvature $\kappa$ have been commonly used to analyze the dynamics of these shocks and the physics of the flow that sustains them.

For example, Jackson et al. [1, 2] have analyzed the cell dynamics with $D_n$-$\kappa$ and $D_n$-$\dot{D}_n$-$\kappa$ relationships for the detonation in a straight channel. Based on a $D_n$-$\kappa$ relationship evaluated from the shock front of a cell, they identified three features [1]: reactive Mach stem, inert blast wave, and the transition between the two. With a $D_n$-$\dot{D}_n$-$\kappa$ relationship evaluated along the normal to the detonation cell front, they observed a surface in the $D_n$-$\dot{D}_n$-$\kappa$ space during a cell cycle [2]. However, their analysis addressed the specific case of a very weakly unstable mixture with a one-step chemistry, a high specific heat ratio $\gamma \sim 1.6$, and a single cell in the channel width. By contrast, the present work considers $D_n$-$\dot{D}_n$-$\kappa$ relationships evaluated from two-dimensional simulations, with detailed chemistry and several cells in the channel width [3, 4].

In the main, for the regular hydrogen-based mixture $2H_2 + O_2 + 7Ar$, the same three features as in [1, 2] are observed. For the irregular $2H_2 + O_2$ mixture, forward jetting and compression waves from the delayed combustion of unburnt pockets can significantly modify $\kappa$ and $\dot{D}_n$. Further analysis then confirms the shock-ignition criterion expressing boundedness of the induction time [5–7], i.e., a rate of heat production greater than that of adiabatic cooling from volumetric expansion, that shock-flame decoupling in cells depends more on flow unsteadiness ($\sim \dot{D}_n$) than on flow divergence ($\sim \kappa$).

## Acknowledgment

This material is based upon work supported by the Air Force Office of Scientific Research under award number FA8655-25-1-7053.

## References

[1] Jackson SI, Short M. (2013) The influence of the cellular instability on lead shock evolution in weakly unstable detonation. Combust. Flame 160: 2260-2274.

[2] Jackson SI, Chiquete C, Short M. (2019) An intrinsic velocity-curvature-acceleration relationship for weakly unstable gaseous detonations. Proc. Combust. Inst. 37: 3601-3607.

[3] Watanabe H, Matsuo A, Chinnayya A, Itouyama N, Kawasaki A, Matsuoka K, Kasahara J. (2023) Lagrangian dispersion and averaging behind a two-dimensional gaseous detonation front. J. Fluid Mech. 968: A28.

[4] Watanabe H, Matsuo A, Chinnayya A, Itouyama N, Matsuoka K, Kasahara J. (2025) Lagrangian characterization of induction and reaction timescales in a cellular gaseous detonation. Phys. Fluids. 36: 026101.

[5] Vidal P., Khasainov BA (1999) A necessary condition for adiabatic explosion behind a shock of arbitrary dynamics, C. R. Acad. Sci., Paris, Série IIB, 327(1), 95-102.

[6] Vidal P. (2009) Critical slow dynamics of detonation in a gas with non-uniform initial temperature and composition: a large-activation-energy analysis. Int. J. Spray Combust. Dynamics. 1: 435-471.

[7] Cheevers K, Radulescu M. (2022) Ignition behind decaying shock waves: detonation cells. arXiv, 2211.05216v1.