Propagation of gaseous detonation waves in a spatially inhomogeneous reactive medium

Detonation propagation in a compressible medium wherein the energy release has been made spatially inhomogeneous is examined via numerical simulation. The inhomogeneity is introduced via step functions in the reaction progress variable, with the local value of energy release correspondingly increased so as to maintain the same average energy density in the medium and thus a constant Chapman-Jouguet (CJ) detonation velocity. A one-step Arrhenius rate governs the rate of energy release in the reactive zones. The resulting dynamics of a detonation propagating in such systems with one-dimensional layers and two-dimensional squares are simulated using a Godunov-type finite-volume scheme. The resulting wave dynamics are analyzed by computing the average wave velocity and one-dimensional averaged wave structure. In the case of sufficiently inhomogeneous media wherein the spacing between reactive zones is greater than the inherent reaction zone length, average wave speeds significantly greater than the corresponding CJ speed of the homogenized medium are obtained. If the shock transit time between reactive zones is less than the reaction time scale, then the classical CJ detonation velocity is recovered. The spatiotemporal averaged structure of the waves in these systems is analyzed via a Favre-averaging technique, with terms associated with the thermal and mechanical fluctuations being explicitly computed. The analysis of the averaged wave structure identifies the super-CJ detonations as weak detonations owing to the existence of mechanical nonequilibrium at the effective sonic point embedded within the wave structure. The correspondence of the super-CJ behavior identified in this study with real detonation phenomena that may be observed in experiments is discussed.

Figure 1

Conceptual illustrations of a reactive system with (a) energy sources (red dots) homogeneously embedded within an inert medium (blue dots), (b) energy sources collected in spatially discretized layers (or sheets), and (c) energy sources collected in square-based prisms separated by inert regions.

Figure 2

Schematic showing the initiation method and implementation of spatially discrete reactive (a) layer and (b) squares.

Figure 3

Time evolution [from (a) to (c)] of the pressure (top row) and reaction progress variable (bottom row) profiles for a one-dimensional simulation with reactive layers and the following parameters: $\mathrm{\Gamma }=0.04$ and $L=10$.

Figure 4

Sample two-dimensional contours of the pressure (left column) and reaction progress variable (right column) for the case with reactive layers at (a) early ($t=30.2$) and (b) later ($t=140.5$) times with $\mathrm{\Gamma }=0.04$ and $L=10$ and (c) for the case with reactive squares, $\mathrm{\Gamma }=0.04$, and $L=25$.

Figure 5

History of instantaneous wave propagation velocity normalized by CJ velocity ($V/V_{\mathrm{CJ}}$) as a function of the leading shock position for the cases with (a) one- and (b) two-dimensional reactive layers ($\mathrm{\Gamma }=0.04$ and $L=10$) and (c) reactive squares ($\mathrm{\Gamma }=0.04$ and $L=25$).

Figure 6

For cases with one-dimensional reactive layers (blue circles), two-dimensional reactive layers (green diamonds), and two-dimensional reactive squares (red squares), average wave propagation velocity normalized by CJ velocity as (a) a function of $\mathrm{\Gamma }$ with $L=10$ and (closed symbols) (b) a function of $L$ with $\mathrm{\Gamma }=0.04$ (open symbols).

Figure 7

Spatial profiles of (a) averaged pressure plotted in the wave-attached reference frame and (b) the terms of thermicity in the master equation [Eqs. (A13) and (A14)] plotted in the vicinity of the averaged sonic point for the case with one-dimensional reactive layers, $\mathrm{\Gamma }=0.04$, and $L=10$. Also shown are the spatial profiles of (c) averaged pressure plotted in the wave-attached reference frame and (d) the terms of thermicity in the master equation plotted in the vicinity of the averaged sonic point for the case with two-dimensional reactive layers, $\mathrm{\Gamma }=0.04$, and $L=10$.

Figure 8

Contours of reactive progress variable $Z$ plotted in $x^{\prime }-t$ diagrams for the cases with one-dimensional reactive layers with $\mathrm{\Gamma }=0.04$ and (a) $L=10$, (b) $L=1$, (c) $L=50$, and (d) two-dimensional reactive layers with $L=10$.

Figure 9

For cases with one-dimensional reactive layers (blue circles), two-dimensional reactive layers (green diamonds), and two-dimensional reactive squares (red squares), average wave propagation velocity normalized by CJ velocity as a function of ${\tau }_{\mathrm{c}}$. The open symbols are for the cases with $\mathrm{\Gamma }=0.04$ and various $L$; the closed symbols are for the cases with $L=10$ and various $\mathrm{\Gamma }$.