A numerical study of the dynamics of detonation initiated by cavity collapse

Abstract

The dynamics of detonation initiated by the shock-induced collapse of a gas-filled ellipsoidal cavity embedded in a condensed-phase explosive is examined computationally. Attention is focused on the details of the evolutionary process following the collapse. The reaction rate is assumed to be pressure-dependent, switching on when the pressure exceeds an ignition threshold. The strength of the incident shock is taken to be such that the reaction would not be initiated without the interaction of the shock with the cavity. The system is modeled as a multi-material mixture, and a high-resolution, Godunov-type scheme is employed to solve the governing equations numerically. The computations are carried out in parallel, and adaptive mesh refinement is used to obtain accurate and well-resolved solutions. It is found that collapse of the cavity produces a detonation provided that the cavity is large enough, or the rate of reaction strong enough; otherwise any reaction initiated by the collapse fizzles out. Details of how the detonation is established are found to depend strongly upon the shape of the cavity.

Appendix

We consider the structure of a planar, Chapman–Jouguet detonation in the solid. Assuming that the detonation is propagating at speed D into a stationary state characterized by the suffix 1, conservation of mass, momentum, and energy provide the conditions:

$$\begin{aligned} \rho (D-u)= & {} \rho _1 D, \end{aligned}$$

(9)

$$\begin{aligned} p+\rho (D-u)^2= & {} p_1+\rho _1D^2, \end{aligned}$$

(10)

$$\begin{aligned} h+\frac{1}{2}(D-u)^2= & {} h_1+\frac{1}{2}D^2. \end{aligned}$$

(11)

Here, h is the enthalpy given by

$$\begin{aligned} h = e+\frac{p}{\rho } = \frac{(1+w_1)p+w_2}{\rho }-\lambda q_\mathrm{s}, \end{aligned}$$

(12)

for the equation of state in (5) with $q=q_\mathrm{s}$ for the solid. We note that

$$\begin{aligned} h_1 = \frac{\bigl (1+w^{(u)}_1\bigr )p_1+w^{(u)}_2}{\rho _1}, \end{aligned}$$

(13)

since $\lambda =0$ and $\mathbf {w}=\mathbf {w}^{(u)}$ at state 1. Now, (12) and (13) yield

$$\begin{aligned} h-h_1= & {} \frac{(1+w_1)p+w_2}{\rho }\nonumber \\&- \frac{\bigl (1+w^{(u)}_1\bigr )p_1+w^{(u)}_2}{\rho _1}-\lambda q_\mathrm{s}. \end{aligned}$$

(14)

Elimination of u from (9) and (10) provides the Rayleigh line

$$\begin{aligned} p-p_1 = J^2\left( \frac{1}{\rho _1}-\frac{1}{\rho } \right) , \end{aligned}$$

(15)

where $J=\rho _1D$ is the mass flux through the steady detonation wave. Elimination of u between (9) and (11) leads to

$$\begin{aligned} h-h_1 = \frac{1}{2} J^2 \left( \frac{1}{\rho _1^2}-\frac{1}{\rho ^2} \right) , \end{aligned}$$

and then, the use of (15) to eliminate J yields

$$\begin{aligned} h-h_1 = \frac{1}{2}(p-p_1)\left( \frac{1}{\rho _1}+\frac{1}{\rho } \right) . \end{aligned}$$

(16)

By virtue of (14), (16) transforms into

$$\begin{aligned} \frac{1}{2}(p-p_1)\left( \frac{1}{\rho _1}+\frac{1}{\rho } \right)= & {} \frac{(1+w_1)p+w_2}{\rho }\nonumber \\&-\frac{\bigl (1+w^{(u)}_1\bigr )p_1+w^{(u)}_2}{\rho _1}-\lambda q_\mathrm{s},\nonumber \\ \end{aligned}$$

(17)

which specifies a family of Hugoniot curves, parameterized by $\lambda$. To find the intersection of the Rayleigh line and a Hugoniot curve for any choice of $\lambda$, we eliminate $\rho$ between (15) and (17) to get

$$\begin{aligned}&\frac{1}{2}(p-p_1)\left( \frac{2}{\rho _1} - \frac{p-p_1}{J^2} \right) \nonumber \\&=\bigl [(1+w_1)p+w_2\bigr ]\left( \frac{1}{\rho _1} - \frac{p-p_1}{J^2} \right) \nonumber \\&\quad -\frac{\bigl (1+w^{(u)}_1\bigr )p_1+w^{(u)}_2}{\rho _1}-\lambda q_\mathrm{s}. \end{aligned}$$

(18)

On setting $P=p-p_1$, (18) can be rearranged into the quadratic

$$\begin{aligned}&P^2 - \frac{2\rho _1w_1}{2w_1+1} \left[ D^2 - \frac{(1+w_1)p_1+w_2}{\rho _1w_1} \right] P\nonumber \\&\quad + \frac{2D^2\rho _1^2}{2w_1+1}\left[ \lambda q_\mathrm{s} - \frac{ \bigl (w_1-w^{(u)}_1\bigr ) p_1 + \bigl (w_2-w^{(u)}_2\bigr ) }{\rho _1} \right] \nonumber \\&\quad =0. \end{aligned}$$

(19)

Here we have replaced J in favor of $D=J/\rho _1.$ For a specified D the appropriate solution of this quadratic yields the pressure increment $P= p-p_1$ as a function of $\lambda .$ With P known, the original conservation conditions determine the other state variables, $\rho$ and u.

1.1 The CJ state

At the CJ point the two roots of the quadratic in (19) are equal when $\lambda$ is set to 1. We note that $\mathbf{w}=\mathbf{w}^{(r)}$ at $\lambda =1.$ Therefore (19) reduces to

$$\begin{aligned}&P^2 - \frac{2\rho _1w^{(r)}_1}{2w^{(r)}_1+1} \left[ D^2 - \frac{\bigl (1+w^{(r)}_1\bigr )p_1+w^{(r)}_2}{\rho _1w^{(r)}_1} \right] P\nonumber \\&\quad + \frac{2D^2\rho _1^2}{2w^{(r)}_1+1}\left[ q_\mathrm{s} - \frac{\varDelta w_1 p_1 + \varDelta w_2 }{\rho _1} \right] =0, \end{aligned}$$

(20)

where $\varDelta w_j=w^{(r)}_j-w^{(u)}_j$, $j=1$ or 2. The equal-root condition

$$\begin{aligned}&\left[ D^2- \frac{\bigl (1+w^{(r)}_1\bigr )p_1+w^{(r)}_2}{\rho _1w^{(r)}_1} \right] ^2 \nonumber \\&\quad =\frac{2(2w^{(r)}_1+1)}{\bigl (w^{(r)}_1\bigr )^2}D^2 \left[ q_\mathrm{s} - \frac{ \varDelta w_1 p_1+ \varDelta w_2 }{\rho _1}\right] \end{aligned}$$

(21)

determines the CJ speed $D_{\mathrm{CJ}}.$ We define

$$\begin{aligned} \alpha= & {} \frac{\bigl (1+w^{(r)}_1\bigr )p_1+w^{(r)}_2}{\rho _1w^{(r)}_1}, \\ \beta= & {} \frac{2w^{(r)}_1+1}{\bigl (w^{(r)}_1\bigr )^2} \left[ q_\mathrm{s}-\frac{\varDelta w_1 p_1+\varDelta w_2}{\rho _1} \right] , \end{aligned}$$

so that (21) condenses into

$$\begin{aligned} (D^2-\alpha )^2=2\beta D^2, \end{aligned}$$

and yields the root $D_{\mathrm{CJ}}$ as

$$\begin{aligned} D_{\mathrm{CJ}}^2 = (\alpha +\beta ) + \sqrt{(\alpha +\beta )^2-\alpha ^2}. \end{aligned}$$

(22)

Once $D_{\mathrm{CJ}}$ is known, (20) determines $P_{\mathrm{CJ}}$ to be

$$\begin{aligned} P_{\mathrm{CJ}} = \frac{\rho _1w^{(r)}_1}{2w^{(r)}_1+1} \left[ D_{\mathrm{CJ}}^2 - \frac{\bigl (1+w^{(r)}_1\bigr )p_1+w^{(r)}_2}{\rho _1w^{(r)}_1} \right] . \end{aligned}$$

Then,

$$\begin{aligned} p_{\mathrm{CJ}}= & {} p_1 + P_{\mathrm{CJ}}, \qquad \frac{1}{\rho _{\mathrm{CJ}}} = \frac{1}{\rho _1} - \frac{P_{\mathrm{CJ}}}{\rho _1^2D_{\mathrm{CJ}}^2},\\&u_{\mathrm{CJ}} = D_{\mathrm{CJ}}\left( 1- \frac{\rho _1}{\rho _{\mathrm{CJ}}} \right) , \end{aligned}$$

and the CJ state is completely determined.

Table 2 Steady detonation states

We are interested, in particular, at the CJ detonation propagating in the region ${\tilde{\varOmega }}_\mathrm{s}$ behind the incident shock; see Fig. 1. The density and pressure in that region are given in Table 1. The von-Neumann and CJ states associated with the detonation are listed in Table 2.

1.2 Reaction-wave structure

The structure of the reaction zone for wave speed $D_{\mathrm{CJ}}$ is determined by the traveling-wave version of the rate law,

$$\begin{aligned} (D_{\mathrm{CJ}}-u)\lambda _x = - {\mathscr {R}}, \end{aligned}$$

(23)

where $\mathscr {R}$ is the pressure-dependent reaction rate defined in (4). We shall assume that the pressure at the end of the reaction zone exceeds the pressure threshold $p_{\mathrm{ign}}$ so that the reaction is not extinguished prior to complete consumption of the reactant. Integration of (23) requires p and u to be known functions of $\lambda .$ The expression for $p(\lambda )$ comes from solving the quadratic (19) with $D=D_{\mathrm{CJ}}$, which can be expressed concisely as

$$\begin{aligned} P^2-2bP+c=0, \end{aligned}$$

where

$$\begin{aligned} \begin{array}{l} \displaystyle {b(\lambda ) = \frac{\rho _1w_1}{2w_1+1} \left[ D_{\mathrm{CJ}}^2 - \frac{(1+w_1)p_1+w_2}{\rho _1w_1} \right] ,} \\ \displaystyle {c(\lambda ) = \frac{2\rho _1^2(D_{\mathrm{CJ}})^2}{2w_1+1}\left[ \lambda q_\mathrm{s} - \frac{ \bigl (w_1-w^{(u)}_1\bigr ) p_1 + \bigl (w_2-w^{(u)}_2\bigr ) }{\rho _1} \right] .} \end{array} \end{aligned}$$

Fig. 25

Reaction-zone structure, as determined by the profile of $\lambda$ against $\sigma x$

In these equations, $\mathbf {w} = (1-\lambda )\mathbf{w}^{(u)}+ \lambda \mathbf{w}^{(r)}.$ Based on the results for an ideal gas, one expects $b(\lambda )$ to be positive and $c(\lambda )$ to be monotonically increasing from zero as $\lambda$ increases through the reaction zone from zero to unity. The quadratic for P has the roots

$$\begin{aligned} P = b(\lambda ) \pm \sqrt{b^2(\lambda )-c(\lambda )}. \end{aligned}$$

We choose the plus sign, which corresponds to a shock at the head of the reaction zone and a decreasing pressure through the reaction zone. With P known, $p=p_1+P$ yields the pressure p and then, u is given by

$$\begin{aligned} u = \frac{P}{\rho _1D_{\mathrm{CJ}}}. \end{aligned}$$

One is now in a position to integrate (23) for $\lambda (x).$ The resulting structure appears in Fig. 25, as the graph of $\lambda$ against the scaled spatial coordinate $\sigma x$. We note, from the figure, that the dimensionless reaction-zone length $x_{\mathrm{rz}}$ is given by

$$\begin{aligned} \sigma x_{\mathrm{rz}} = 2 \times 10^{-4}, \end{aligned}$$

(24)

where we have assigned the end of the reaction zone to correspond to $\lambda =0.99$.