Reduction to blast equation

Another issue is the estimate of the peak value of detonation decay along the channel axis. This is an important ingredient in the approximate model of detonation failure developed by Schultz (2000). In Fig. 5.6, the normalized time derivative of the axial speed, ˙Da, is plotted as a function of the distance from the corner,xa. In the interval shown here, the behavior of ˙Da is remarkably similar for all cases, and the minimum value scales almost linearly withθCJ. As expected, when the expansion signal arrives at the axis of symmetry, a more sensitive reaction mechanism (larger θCJ) results in a more rapid decoupling of the shock from the reaction zone and therefore in a larger shock decay rate. This decoupling is not immediate, a distance of 4 to 6 half- reaction zones is necessary to reach the peak value of deceleration. Simulations in a non-reacting gas indicate that at least part of this delay is due to the finite thickness associated with the numerical representation of the shock. The similar evolution of acceleration in Fig. 5.6 only occurs at early times, while the subsequent evolution is different. The near-critical and super-critical diffraction cases present positive accelerations that accompany the re-ignition mechanisms, whereas in the sub-critical case Da decreases monotonically.

The three curves in Fig. 5.6 are compared with the estimate from a blast model, discussed in this section, and with the result from Whitham’s theory, presented in the next section.

Very little experimental data are available from the literature; Edwards et al. (1979) report the variation of frontal velocity from a CJ value of 2400 m/s in det-

x

dD

/d

t

H

/

D

Figure 5.6: Shock deceleration as a function of the distance from the corner vertex, parametrized by θCJ.

onation diffraction of oxyacetylene from a rectangular tube. Velocities are obtained from streak photographs at near-critical and sub-critical conditions as a function of distance. An approximate estimate of the peak value ˙Da, normalized byDCJ2 /H, can be extracted from these diagrams by measuring the slope of the two curves. The two values we found, −0.213 (near-critical) and −0.246 (sub-critical), are useful in the sense that they are of the same order of magnitude of the results from the numerical simulations.

In this section, we treat the flow near the channel axis as a cylindrical blast (Ko- robeinikov 1991), of radius r. This approach was followed by Eckett et al. (2000) in the study on critical energy of initiation, and extended by Schultz (2000) to detona- tion diffraction. From the blast similarity relation, we have

dr

dt =δ r

t, (5.2)

1/2) we obtain r =rb r t tb . (5.3)

We take rb to be the shock distance from the (virtual) blast singularity when the expansion signal arrives at the axis of symmetry. The transition from planar wave to cylindrical blast of radius rb is assumed to be instantaneous. Then

tb =

rb 2DCJ

. (5.4)

Since here κ = 1/r by definition, Equation (5.3) allows us to derive a linear relation between detonation speed and front curvature

Da=DCJ

κ κb

, (5.5)

whereκb = 1/rb. Comparison of this result with a typicalDn–κcurve of a sub-critical diffraction, Fig. 5.5, indicates that the blast model can be applied only qualitatively; No portion of the curve from the simulation can be approximated by a straight line passing by the origin. Differentiating Da with respect to time gives

˙ Da =− 1 2 Da t . (5.6)

The estimated maximum deceleration at rb is therefore ˙

Db =−DCJ2 κb. (5.7)

Up this point, the parameterκb is unknown. If we estimate the initial blast radius to be the x distance from the corner vertex to the point of arrival of the corner signal, then rb = 1 κb = H tanα. (5.8) From Equation (5.7), ˙ Db =− D_CJ2 H tanα. (5.9)

Forα = 22.6◦ (from Skews’ construction) we find κbH =− H D_CJ2 ˙ Db = 0.416. (5.10)

With a reasonable estimate of the initial blast radius, the cylindrical blast model provides the correct magnitude of the initial shock decay rate (Fig. 5.6). However, the derivation above ignores the reactivity of the flow, so that it is independent from the reaction mechanism. In addition, the value of κ is roughly one order of magni- tude smaller than the maximum value obtained from numerical simulations. Shock curvature is difficult both to compute consistently (see Appendix A) and to model, since a small differential velocity of the wavefront can result in a large curvature. The curvature peak value in Figs. 5.3 to 5.5 is reached in a transient situation where there is a strong gradient of shock deceleration. This gradient is absent in the cylindrically symmetric blast wave. Thus, the usefulness of the blast decay model is limited to con- firming the scaling of the shock deceleration with D2_CJ/H and to provide a reference for an average value of the shock decay rate.