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Theoretical approach to one-dimensional detonation instability

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Abstract

Detonation instability is a fundamental problem for understanding the micro-behavior of a detonation front. With the theoretical approach of shock dynamics, detonation instability can be mathematically described as a second-order ordinary difference equation. A one-dimensional detonation wave can be modelled as a type of oscillators. There are two different physical mechanisms controlling the behaviors of a detonation. If the shock Mach number is smaller than the equilibrium Mach number, the fluid will reach the sonic speed before the end of the chemical reaction. Then, thermal chock occurs, and the leading shock becomes stronger. If the shock Mach number is larger than the equilibrium Mach number, the fluid will be subsonic at the end of the chemical reaction. Then, the downstream rarefaction waves propagate upstream, and weaken the leading shock. The above two mechanisms are the basic recovery forces toward the equilibrium state for detonation sustenance and propagation. The detonation oscillator concept is helpful for understanding the oscillating and periodic behaviors of detonation waves. The shock dynamics theory of detonation instability gives a description of the feedback regime of the chemical reaction, which causes variations of the leading shock of the detonation.

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Abbreviations

a :

sonic speed

E a :

chemical activation energy

k :

chemical rate factor

L CJ :

reaction zone length in the Chapman-Jouguet (CJ) state

M :

moving Mach number of leading shock

M e :

equilibrium Mach number of leading shock

p :

pressure

Q :

chemical heat

R :

gas constant

S :

chemical reaction source term

t :

time

u :

velocity

x :

distance from shock position to its equilibrium state

ρ :

density

γ :

specific heat ratio

λ :

chemical reaction progress

μ :

function of M and γ

ω :

chemical reaction rate

ν :

reaction order

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Authors and Affiliations

  1. State Key Laboratory of High Temperature Gas Dynamics, Chinese Academy of Sciences, Beijing, 100190, China

    Chun Wang, Gaoxiang Xiang & Zonglin Jiang

Authors
  1. Chun Wang
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  2. Gaoxiang Xiang
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  3. Zonglin Jiang
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Corresponding author

Correspondence to Chun Wang.

Additional information

Project supported by the National Natural Science Foundation of China (No. 90916028) and the Innovation Program of the State Key Laboratory of High Temperature Gas Dynamics of Institute of Mechanics, Chinese Academy of Sciences

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Wang, C., Xiang, G. & Jiang, Z. Theoretical approach to one-dimensional detonation instability. Appl. Math. Mech.-Engl. Ed. 37, 1231–1238 (2016). https://doi.org/10.1007/s10483-016-2124-6

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  • DOI: https://doi.org/10.1007/s10483-016-2124-6

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