Influence of discrete sources on detonation propagation in a Burgers equation analog system

An analog to the equations of compressible flow that is based on the inviscid Burgers equation is utilized to investigate the effect of spatial discreteness of energy release on the propagation of a detonation wave. While the traditional Chapman-Jouguet (CJ) treatment of a detonation wave assumes that the energy release of the medium is homogeneous through space, the system examined here consists of sources represented by $\delta$ functions embedded in an otherwise inert medium. The sources are triggered by the passage of the leading shock wave following a delay that is either of fixed period or randomly generated. The solution for wave propagation through a large array $({10}^3–{10}^4)$ of sources in one dimension can be constructed without the use of a finite difference approximation by tracking the interaction of sawtooth-profiled waves for which an analytic solution is available. A detonation-like wave results from the interaction of the shock and rarefaction waves generated by the sources. The measurement of the average velocity of the leading shock front for systems of both regular, fixed-period and randomized sources is found to be in close agreement with the velocity of the equivalent CJ detonation in a uniform medium, wherein the sources have been spatially homogenized. This result may have implications for the applicability of the CJ criterion to detonations in highly heterogeneous media (e.g., polycrystalline, solid explosives) and unstable detonations with a transient and multidimensional structure (e.g., gaseous detonation waves).

Figure 1

Conservation of $E$ in the control volume.

Figure 2

Sample solution for spatially homogeneous source energy release.

Figure 3

Schematic illustration of the method of characteristics in an $x\text{−}t$ diagram. The right-running characteristics [Eq. (16)] are plotted as dashed, arrowed blue lines; the interface characteristics as thick, solid, arrowed blue lines; the characteristics of particle paths with zero velocity [Eq. (14)] as thin, solid, arrowed red lines; the trajectories of shocks as thick, solid curves; and the locus of sources releasing energy as black, solid circles that become red, open squares upon being shocked and red, solid squares upon releasing their energy.

Figure 4

Illustration of the implementation of source activation: (a) source during delay period after being shocked, (b) source energy release inserted as a triangular profile to approximate $\delta$ function, and (c) shortly after being activated, and (d) the second shock now caught up and merged with the leading shock.

Figure 5

Sample solutions of $\rho$ for randomly spaced energy sources with random period delay. Circles are unshocked sources, empty squares shocked sources during delay period, and solid squares sources with energy released.

Figure 6

Velocity history of a system where sources were equally spaced with an average energy density of $q=0.5$ and (a) a fixed delay of $\tau =0.5t_{\mathrm{c}}$ over a short distance in linear scale, and (b) fixed delays of $\tau =0.5t_{\mathrm{c}}$ and $\tau =0.1t_{\mathrm{c}}$ over a long distance in logarithmic scale.

Figure 7

$x\text{−}t$ diagrams of wave processes, (a) in a laboratory-fixed reference frame and (b) in a wave-attached frame moving at the CJ velocity, of a system where sources were equally spaced with an average energy density of $q=0.5$ and a fixed delay of $\tau =0.5t_{\mathrm{c}}$. $x\text{−}t$ diagrams of wave processes, (c) in a laboratory-fixed reference frame, and (d) in a wave-attached frame moving at the CJ velocity, of a system where sources were randomly spaced with an average energy density of $q=0.5$ and random delay in the range $0.1$ to $1t_{\mathrm{c}}$. The interface characteristics are plotted as blue lines; the trajectories of shocks as black curves; and the locus of sources releasing energy as black, solid circles that become red, open squares upon being shocked and red, solid squares upon releasing their energy.

Figure 8

(a) Velocity history of a system where sources were randomly spaced with an average energy density of $q=0.5$ and random time delays over the range $0.1$ to $1t_{\mathrm{c}}$ over a short distance in linear scale. (b) The average wave velocity for a set of 20 realizations of a system of randomly spaced sources with an average energy density of $q=0.5$ and random time delays over the range $0.1$ to $1t_{\mathrm{c}}$ over a long distance in logarithmic scale. (c) The average wave velocity for two sets of 20 realizations of systems of randomly spaced sources with an average energy density of $q=0.5$ and random time delays over the ranges $0.01$ to $0.1t_{\mathrm{c}}$ and 1 to $10t_{\mathrm{c}}$ over a long distance in logarithmic scale. The thick dashed lines in (b) and (c) are the ensemble averaged velocities as a function of $x/L$.

Figure 9

Schematic representation of the control volume enclosing a nonreactive shock wave.

Figure 10

The analog of the Chapman-Jouguet condition shown in $\rho \text{−}f\left(\rho \right)$ diagram.

Figure 11

Illustration of (a) a shock wave located between two linear profiles of $\rho$ upstream and downstream of the shock and (b) the trajectory of the shock wave solved using the method of characteristics in an $x\text{−}t$ diagram.