Diffraction of weakly unstable detonation through an obstacle with different sizes and shapes

Detonation diffraction has been extensively studied because it is a very fundamental problem and has broad practical applications. In this study, two-dimensional simulations considering detailed chemistry and transport are conducted to investigate the weakly unstable detonation diffracting through an obstacle. The emphasis is placed on assessing the effects of obstacle size and shape on detonation diffraction. For a semicircular obstacle, the subcritical, critical, and supercritical regimes are identified by increasing the obstacle radius. The competition between the energy release rate and expansion rate is analyzed; it interprets the regime distribution. Increasing the radius of a semicircular obstacle can reduce the curvature and thereby the expansion rate. The energy release rate dominates over the loss due to expansion during the diffraction process with a large obstacle radius. For triangular and rectangular obstacles, the supercritical regime without detonation quenching is not observed in the present simulations. Furthermore, the critical regime for transverse detonation formation is investigated and the critical obstacle sizes are compared for different obstacle geometries. It is found that the semicircular obstacle has the smallest critical size while the rectangular obstacle has the largest critical size. Unlike the critical obstacle size, the wall reflection distance of transverse detonation is independent of the obstacle geometry and it varies in the range of 10–15 times the detonation cell width. The characteristics of transverse detonation are compared for two weakly unstable mixtures, hydrogen/air without and with nitrogen dilution. A pair of transverse detonations induced by local explosion is observed in hydrogen/air with nitrogen dilution, which is quite different from the single transverse detonation in hydrogen/air. The comparison of the three regimes’ distributions for these two mixtures shows that a larger obstacle size is required for transverse detonation formation in hydrogen/air with nitrogen dilution. However, the critical obstacle size normalized by the corresponding detonation induction length becomes closer for two mixtures. This indicates that the formation of transverse detonation is nearly independent of nitrogen dilution for weakly unstable mixtures.

Figure 1

Schematic of detonation diffracting through a semicircular obstacle.

Figure 2

(a) Density contour and (b) distribution of six mesh levels for detonation propagating through a semicircular obstacle and corresponding formation of Mach reflection.

Figure 3

Schlieren contour for $0.22{\mathrm{H}}_2+0.11{\mathrm{O}}_2+0.67\mathrm{Ar}$ at $P_0=100\mathrm{kPa}$ and $T_0=295\mathrm{K}$. (a) experimental result from [13] and (b) present simulation result. Red circles represent the positions of local explosion, E.

Figure 4

Schematic of three regimes: (a) subcritical regime, (b) critical regime, and (c) supercritical regime during detonation diffracting through a semicircular obstacle (in gray). The blue lines represent the shock wave and the red lines denote the reaction zone.

Figure 5

Evolution of temperature contours during detonation diffracting through a semicircular obstacle with $r/l_i=12.5$. RR: regular reflection; MR: Mach reflection. The black isolines correspond to temperature equal to 2400 K.

Figure 6

Evolution of density gradient contours during detonation diffracting through a semicircular obstacle with $r/l_i=12.5$. RR: regular reflection; MR: Mach reflection.

Figure 7

Evolution of temperature contours during detonation diffracting through a semicircular obstacle with $r/l_i=50$. MR: Mach reflection. The black isolines correspond to temperature equal to 2400 K.

Figure 8

Density gradient distributions during detonation diffracting through a semicircular obstacle with $r/l_i=50$. The time sequence for lines 1–7 is 1: 6 μs; 2: 7.5 μs; 3: 9 μs; 4: 10.5 μs; 5: 12 μs; 6: 13.5 μs; 7: 15 μs.

Figure 9

Enlarged temperature, pressure, and density gradient contours at $t=12\mu \mathrm{s}$ for detonation diffracting through a semicircular obstacle with $r/l_i=50$.

Figure 10

Change of normalized pressure, $P/P_0$, and temperature, $T/T_0$, after the leading shock with normalized shock position, $x/l_i$, along the lines $a\text{–}c$ in Fig. 7 for a semicircular obstacle with $r/l_i=50$. The gray zones correspond to the shock-reaction front decoupling.

Figure 11

Evolution of temperature contours for a semicircular obstacle with $r/l_i=100$. RR: regular reflection; MR: Mach reflection.

Figure 12

Enlarged temperature contours for a semicircular obstacle with $r/l_i=100$.

Figure 13

Density gradient distributions for a semicircular obstacle with $r/l_i=100$ at different simulation times.

Figure 14

Normalized pressure of leading shock, $P/P_{\mathrm{vN}}$, near the obstacle surface with its normalized horizontal position, $x/r$, for a semicircular obstacle with different obstacle radii.

Figure 15

Evolution of temperature contours for $r/l_i=100$ with different obstacle shapes.

Figure 16

Change of the normalized wall reflection distance from the obstacle apex to the reflection point of transverse detonation, $l_w/\lambda$, with normalized obstacle size, $r/l_i$, for different obstacle shapes.

Figure 17

Temperature, pressure, and density gradient contours of transverse detonation formation in mixture 2 for a semicircular obstacle with $r/l_i=25$ and $l_i=1.254\mathrm{mm}$.

Figure 18

Evolution of density gradient contours for a semicircular obstacle with $r/l_i=25$ and $l_i=1.254\mathrm{mm}$. T1–T3 represent triple points, TW2 represents transverse wave of T2, P1 and P2 represent pressure waves, E represents local explosion, and TD1 and TD2 represent transverse detonations.

Figure 19

Distributions of three regimes for mixtures 1 and 2 with induction lengths $l_{i,1}$ and $l_{i,2}$, respectively. Different meshes with and without diffusion term are considered. (a) No normalization, (b) with normalization by induction length.

Figure 20

Numerical soot foils (a), (c), (e) and pressure contours (b), (d), (f) for ZND (a), (b) and cellular (c), (d), (e), (f) detonations diffracting through a semicircular obstacle with different obstacle radii.

Figure 21

The distributions of the density gradient contours for four types of mesh. The semicircular obstacle with $r/l_i=50$ (i.e., critical regime) was used.

Figure 22

Temperature contours for detonation diffracting through a semicircular obstacle with diffusion terms (left column) and without diffusion terms (right column). Different obstacle radii of $r/l_i=12.5$ (top row), 50 (middle row), and 100 (bottom row) are considered.

Figure 23

(a), (b) Density gradient contour considering reflective and wall boundary condition on an obstacle surface, respectively; (c), (d) flow speed contour considering reflective and wall boundary condition on obstacle surface, respectively, for detonation propagating through a semicircular obstacle with $r/l_i=50$.