Two separate decay timescales of a detonation wave modeled by the Burgers equation and their relation to its chaotic dynamics

This study uses a simplified detonation model to investigate the behavior of detonations with galloping-like pulsations. The reactive Burgers equation is used for the hydrodynamic equation, coupled to a pulsed source whereby all the shocked reactants are simultaneously consumed at fixed time intervals. The model mimics the short periodic amplifications of the shock front followed by relatively lengthy decays seen in galloping detonations. Numerical simulations reveal a sawtooth evolution of the front velocity with a period-averaged detonation speed equal to the Chapman-Jouguet velocity. The detonation velocity exhibits two distinct groups of decay timescales, punctuated by reaction pulses. At each pulse, a rarefaction wave is created at the reaction front's last position. A characteristic investigation reveals that characteristics originating from the head of this rarefaction take 1.57 periods to reach and attenuate the detonation front, while characteristics at the tail take an additional period. The leading characteristics are amplified twice, by passing through the reaction fronts of subsequent pulses, before arriving at the shock front, while the trailing characteristics are amplified three times. This leads to the two distinct groups of timescales seen in the detonation front speed.

Figure 1

Phase diagrams of period-2 detonations (three left subfigures) and a chaotic detonation (right) in three models: (a) the Euler equations [4], (b) an asymptotic model of the Euler equations [2], and (c) an evolution equation derived from Fickett's detonation analog [9]; two groups of decay rates are highlighted in red and blue.

Figure 2

Effect of resolution on profiles and error at $t=99.5$; (a) profiles of $u$ (solid) and $\lambda$ (dashed), and (b) error vs resolution; the dashed line shows ${(1/\mathrm{\Delta }x)}^{-1/2}$.

Figure 3

Detonation speed evolution.

Figure 4

Phase diagrams of ten superimposed cycles (black, $90\le t\le 100$); (a) $D$ over the cycle, (b) $\dot{D}$ and $D$, and (c) timescale; dashed lines show fits; $x_{\mathrm{s},n}$ is the shock position at the $n=90$ pulse.

Figure 5

Profiles of $u$ (solid, left axis) and the reaction front (at the shaded/unshaded interface) in the laboratory frame of reference during the initial cycles; expansion fan head (circle) and expansion fan tail (square).

Figure 6

Profiles of $u$ (solid, left axis) and the reaction front (at the shaded/unshaded interface) over one regular cycle in a frame of reference moving at $D_{\mathrm{CJ}}$; expansion fan heads from current (solid circle) and previous (dashed circle) cycles, and expansion fan tail (square); $n=99$.

Figure 7

Characteristics diagram (black) and reaction front position (red dotted line, reacted where shaded) over three periods of the regular pulsating behavior; blue and green characteristics delimit characteristics amplified twice from those amplified three times ($n=97$).