UDC 534.222.2

MECHANICS

S. K. ASLANOV, V. N. BUDZIROVSKII,
Corresponding Member of the USSR Academy of Sciences K. I. SHCHELKIN

INSTABILITY CRITERIA FOR DETONATION WAVES

In [1] a method was proposed for calculating the stability of a detonation wave on the basis of its multistage model. Investigation of the characteristic equation \(D(z)=0\), obtained in [1], makes it possible to find criteria for the instability of detonation waves. In the case \(L_1 \approx L_2 \approx \cdots \approx L_n \ne 0\), for perturbations of small relative wavelengths \((kL_j \gg 1)\), with accuracy up to \(e^{-\delta}\), where \(\delta=\min\{kL_j\gamma_{j1}\}\), the determinant

\[ D(z)=\prod_{l=0}^{n} D_l(z). \]

The characteristic equation, consequently, splits into a series of equations

\[ D_0(z)=0,\qquad D_j(z)=0,\qquad j=1,2,\ldots,n, \]

as was indicated earlier in [2], where

\[ D_0(z)=(\alpha_1-1) \begin{vmatrix} 1-M_1^2(z_1/\gamma_{12}+1) & 1 & -1 & z_1\alpha_1\\ 2-(1+M_1^2)(z_1/\gamma_{12}+1) & 2 & -1 & 0\\ 1/\gamma_{12} & -z_1 & 0 & 1\\ -z_1/\gamma_{12} & 1 & 1/(\alpha_1-1)M_1^2 & -z_1 \end{vmatrix}, \tag{1} \]

\[ D_j(z)= \begin{vmatrix} a_{11}^j & a_{12}^j & 1 & -1 & z_{j+1}\alpha_{j+1}\\ a_{21}^j & a_{22}^j & 2 & -1 & 0\\ \dfrac{1}{\gamma_{j1}} & \dfrac{1}{\gamma_{j+1,2}} & -z_{j+1} & 0 & 1\\ -\dfrac{z_j}{\alpha_j\gamma_{j1}} & -\dfrac{z_{j+1}}{\gamma_{j+1,2}} & 1 & \dfrac{1}{(\alpha_{j+1}-1)M_{j+1}^2} & -z_{j+1}\\ a_{51}^j & 0 & 0 & 0 & \dfrac{1}{\alpha_{j+1}-1} \end{vmatrix} \tag{2} \]

\[ a_{11}^j=\left[1-M_j^2\left(\frac{z_j}{\gamma_{j1}}+1\right)\right]\alpha_j;\qquad a_{12}^j=1-M_{j+1}^2\left(\frac{z_{j+1}}{\gamma_{j+1,2}}+1\right); \]

\[ a_{21}^j=2-(1+M_j^2)\left(\frac{z_j^2}{\gamma_{j1}}+1\right);\qquad a_{22}^j=2-(1+M_{j+1}^2)\left(\frac{z_{j+1}}{\gamma_{j+1,2}}+1\right); \]

\[ a_{51}^j=\frac{m_jM_j^2}{\gamma_{j1}}+\frac{1}{(\gamma_{j1}+z_j)};\qquad j=1,2,\ldots,n \]

(\(n\) is the number of intervals of the stepwise scheme).

The characteristic equation \(D_0(z)=0\) for the shock front, as shown in [3], has no roots with positive real part. Analysis of the remaining equations makes it possible to obtain a sufficient criterion for instability of a detonation wave to perturbations of small relative wavelength in the form of restrictions on the chemical kinetics; namely, it is sufficient that at least one of the inequalities be satisfied.

\[ \eta_{1j}<m_j<\eta_{2j}, \tag{3} \]

where

\[ \eta_{1j}= \frac{1+\dfrac{j}{(\alpha_{j+1}-1)M_j^2}} {\left\{1-\dfrac{1+M_jM_{j+1}} {(1+M_j)\left[1+M_{j+1}+\dfrac{\alpha_{j+1}-1}{\alpha_{j+1}}(\varkappa_{j+1}-1)M_{j+1}^2\right]}\right\}}; \tag{4} \]

\[ \eta_{2j}= \frac{2-\alpha_{j+1}}{(\alpha_{j+1}-1)M_j^2} + \frac{\sqrt{1-M_j^2}} {(\alpha_{j+1}-1)M_j^2\sqrt{1-M_{j+1}^2}} \times \]

\[ \times\left\{1+(\alpha_{j+1}-1)\left[2+(\varkappa_{j+1}-1)M_{j+1}^2\right]\right\}, \]

i.e., the detonation wave will be unstable if at least one of the values \(m_j\) lies in the band bounded by the broken curves \(\eta_{1j}, \eta_{2j}\) (Fig. 1).

Fig. 1

Fig. 2

For Chapman—Jouguet detonation, as \(x=L\) is approached, the upper boundary of the instability region \(\eta_{2j}\to+\infty\). Carrying out the limiting transition as \(k\to0\) \((z\to\infty\), while \(\xi=kz\) remains finite), we obtain the characteristic equation for one-dimensional perturbations \(A(\xi)=0\). In unexpanded form \(A(\xi)\) is represented by a determinant of order \((4n+3)\). Calculation of \(A(0)\) gives the following:

\[ A(0)=(-1)^{\beta+1} \left(1-\frac{\rho_0}{\rho_{n+1}}\right)^2 \frac{1+M_{n+1}}{M_{n+1}} \prod_{l=0}^{n} A_l, \]

where

\[ A_0= \frac{\rho_0}{\rho_{n+1}}(1+M_{n+1}) \bigg/ \left(1-\frac{\rho_0}{\rho_{n+1}}\right)M_{n+1}^2-1; \]

\[ A_l= \frac{1-M_l^2}{M_l^2}\, \frac{2}{(\varkappa_l-1)M_l}; \qquad l=1,\ldots,n; \qquad \beta=\frac{n(n+1)}{2}. \]

The sign of \(A(\infty)\) is determined by the sign of the coefficient \(B\) at the highest exponent

\[ \exp\left\{\sum_{j=1}^{n}\frac{\xi_jL_j}{1-M_j}\right\}, \]

\[ B=(-1)^{\beta+1}\prod_{k=0}^{n} B_k, \tag{5} \]

where

\[ B_0=(1-\alpha_1)^2\left(1+\frac{1}{M_1}\right) \left[ \frac{\alpha_1(1+M_1)}{(\chi_1-1)(1-\alpha_1)M_1^2}-1 \right]; \]

\[ B_k=\alpha_{k+1} \begin{vmatrix} M_k(\alpha_{k+1}-1) & \alpha_{k+1}-1 & \alpha_{k+1}/(\chi_{k+1}-1)M_{k+1} & 0\\ -(1-M_k) & 1+M_{k+1} & -1 & 0\\ -1 & \alpha_{k+1} & \alpha_{k+1}/(\chi_{k+1}-1)M_{k+1} & -(\alpha_{k+1}-1)\\ M_k[m_kM_k+1] & 0 & 0 & 1 \end{vmatrix}. \]

Using the gas-dynamic relations of detonation theory, it is easy to show that \(A_0\), \(A_k\), \(B_0\), \(B_k\) are positive. Comparing the expressions obtained for \(A(0)\) and \(A(+\infty)\), we find that their signs are opposite (and this is a sufficient condition for the existence of at least one positive eigenvalue) only when the sign of the product

\[ \prod_{k=1}^{n} B_k \]

is negative. From this we obtain a sufficient condition for one-dimensional instability of a detonation wave in the form of the system of inequalities

\[ m_i<\eta_{i1},\qquad m_j>\eta_{j1},\qquad i\ne j, \tag{6} \]

where \(\eta_{j1}\) is determined by formula (4), and the number of the latter inequalities is odd, i.e., the broken line \(m_j\) crosses the broken line \(\eta_{j1}\) from below upward at least once. The range of values of \(m_j\) leading to instability in the case of one-dimensional perturbations is not bounded above, unlike the case of short-wave perturbations.

Fig. 3

Criteria (3) and (6), in contrast to the corresponding criteria for the single-stage model \((^3)\), contain parameters of weaker discontinuities, as a result of which the multistage model proves to be more stable. In the case of different orders of magnitude of \(L_j\) (for example, Fig. 2 represents a distribution characteristic of detonation in a gas), when \(L_i\ll L_j\), \(i\ne j\), \(i\ne 1\), the criteria are the same as if, instead of the \(i\)-th and \(j\)-th fronts of chemical reactions, one resultant front were considered (for the distribution in Fig. 2 the instability criteria coincide with those \((^2)\) for the single-stage model). When \(L_1\ll L_j\), \(j=2,3,\ldots,n\) (Fig. 3), the parameters of the narrow zone 1 will not enter the criterion (in \((^3,^6)\), \(j=2,3,\ldots\)). Thus, a detonation wave with a steep drop in pressure (density) behind the shock front (for example, such a detonation wave in solid explosives may be considered) proves to be much more stable than follows from the single-stage approach \((^2)\).

Odessa State University
named after I. I. Mechnikov

Received
19 IV 1968

CITED LITERATURE

\(^1\) S. K. Aslanov, V. N. Budzirovskii, K. I. Shchelkin, DAN, 182, No. 1 (1968). \(^2\) S. K. Aslanov, Prikl. mekh., 2, issue 7 (1966). \(^3\) S. K. Aslanov, DAN, 169, No. 2 (1966).