Photographic investigation into the mechanism of combustion in irregular detonation waves

Abstract

Irregular detonations are supersonic combustion waves in which the inherent multi-dimensional structure is highly variable. In such waves, it is questionable whether auto-ignition induced by shock compression is the only combustion mechanism present. Through the use of high-speed schlieren and self-emitted light photography, the velocity of the different components of detonation waves in a ${\text{ CH}}_4+2\text{ O}_2$ mixture is analyzed. The observed burn-out of unreacted pockets is hypothesized to be due to turbulent combustion.

Appendices

Appendix A: Induction time calculations

The induction time calculations performed in the present work were zero-dimensional calculations. We present here an approximate solution for the case of a single-step kinetic mechanism, against which results generated by integrating detailed chemistry were compared. The problem involving a detailed chemical kinetic scheme parallels the single-step case, only with a set of ODE analogous to (1) instead of a single ODE, and with the heat release term of (11) comprising the sum of the heat release of individual kinetic steps rather than a single global heat release. We follow mostly the derivation of [7], which we modify slightly for the case of the induction time of a particle in an expansion. For a constant volume explosion, we have the kinetic rate equation

$$\begin{aligned} \frac{\mathrm{d}\left[\text{ Fuel}\right]}{\mathrm{d}t}&= -\omega \end{aligned}$$

(1)

where

$$\begin{aligned} \omega&= \left[\text{ Fuel}\right]^a\left[\text{ Oxidizer}\right]^b A \mathrm{e}^\frac{-E_a}{{\fancyscript{R}}T}\end{aligned}$$

(2)

$$\begin{aligned} \left[\text{ Fuel}\right]&= \frac{P x_F}{{\fancyscript{R}}T}\end{aligned}$$

(3)

$$\begin{aligned} \left[\text{ Oxidizer}\right]&= \frac{P x_O}{{\fancyscript{R}}T}\end{aligned}$$

(4)

$$\begin{aligned} x_O&= 1-x_F \end{aligned}$$

(5)

in the simple case of a one-step Arrhenius kinetic model with a single reactant and single product. Substitution yields

$$\begin{aligned} \frac{\mathrm{d}\left[\text{ Fuel}\right]}{\mathrm{d}t}&= \left(\frac{P}{{\fancyscript{R}}T}\right)^{a+b}x_F^a x_O^b A \mathrm{e}^{\left(\frac{-E_a}{{\fancyscript{R}}T}\right)}. \end{aligned}$$

(6)

Using conservation of energy and assuming a calorically perfect gas, we obtain

$$\begin{aligned} \frac{\mathrm{d}\left(C_v T\right)}{\mathrm{d}t}&= \frac{Q}{\rho }\left(\frac{P}{{\fancyscript{R}}T}\right)^{a+b} x_F^a x_O^a A \mathrm{e}^\frac{-E_a}{{\fancyscript{R}}T}. \end{aligned}$$

(7)

Constant volume explosions are characterized by a slow initial change in properties, which enables us to assume

$$\begin{aligned} x_F&\approx x_{F,0},\nonumber \\ x_O&\approx x_{O,0},\nonumber \\ C_v&\approx C_{v,0},\\ P&\approx P_0,\nonumber \\ \rho&\approx \rho _0.\nonumber \end{aligned}$$

(8)

Furthermore, we assume that $T \approx T_0$, but only in the geometric term and not in the exponential term. The reasoning is that even a small change in temperature causes a large change in the exponential term and, so, we retain the exponential dependence on temperature over the geometric dependence from $T^{-\left(a+b\right)}$. The end result is the ODE

$$\begin{aligned} \frac{\mathrm{d}T}{\mathrm{d}t}&= k_1 \mathrm{e}^{\frac{-E_a}{{\fancyscript{R}} T}}\end{aligned}$$

(9)

$$\begin{aligned} k_1&= \frac{1}{C_{v,0}}\frac{Q}{\rho _0}\left(\frac{P_0}{{\fancyscript{R}} T_0}\right)^{a+b} x_{F,0}^a x_{O,0}^b A. \end{aligned}$$

(10)

In the case of a particle in an expansion, we modify (9) to include an approximate effect of the expansion on temperature. Equation 9 becomes

$$\begin{aligned} \frac{\mathrm{d}T}{\mathrm{d}t}=k_1 \mathrm{e}^{\frac{-E_a}{{\fancyscript{R}} T}}+\left(\frac{\mathrm{d}T}{\mathrm{d}t}\right)_\mathrm{expansion}, \end{aligned}$$

(11)

where the second term is the time rate of change of temperature as determined by a self-similar fit to the shock decay. The self-similar solution of a blast wave is given in Appendix B. While (11) was integrated numerically, an approximate solution to (9) can be found. A simple change of variables and successive integration by parts yield the series solution

$$\begin{aligned} \frac{e^\alpha }{z^2}\sum _{n=0}^{\infty }\left(\frac{n!}{\alpha ^n}\right)-\frac{e^{\alpha /\theta }}{\left(\alpha /\theta \right)^2}\sum _{n=0}^{\infty }\left(\frac{n!}{\left(\alpha /\theta \right)^n}\right) = \frac{k_1}{\alpha T_0}t \end{aligned}$$

(12)

where

$$\begin{aligned} \alpha&= \frac{E_a}{{\fancyscript{R}} T_0},\end{aligned}$$

(13)

$$\begin{aligned} \theta&= \frac{T}{T_0},\end{aligned}$$

(14)

$$\begin{aligned} z&= \alpha /\theta . \end{aligned}$$

(15)

Only the first term of the series is retained, which is an error of only 15 % [7], giving the final ignition criterion

$$\begin{aligned} \frac{k_1}{\alpha T_0} t \frac{\alpha ^2}{e^\alpha } = 1. \end{aligned}$$

(16)

Appendix B: Properties along the particle path in a blast wave flow field

While fluid particles crossing a constant velocity shock will remain at a constant thermodynamic state, particles crossing a decaying shock will expand and their thermodynamic state will change accordingly. The variation of properties with time for a given fluid particle can be found approximately using a relatively simple analytic solution. For a broader treatment of the topic, see [27].

The starting point is the non-dimensional governing equations for the point blast problem in the strong shock limit

$$\begin{aligned}&(\phi -z)\psi ^{\prime }+\psi \phi ^{\prime }+\frac{j\psi \phi }{z}=0, \end{aligned}$$

(17)

$$\begin{aligned}&(\phi -z)\phi ^{\prime }+\frac{f^{\prime }}{\psi }+\theta \phi =0, \end{aligned}$$

(18)

$$\begin{aligned}&(\phi -z)\left(\frac{f}{(\gamma -1)\psi } \right)^{\prime }+2\theta \frac{f}{(\gamma -1)\psi }\nonumber \\&\qquad +\frac{f}{\psi }\phi ^{\prime }+\frac{jf\phi }{z\psi }=0,\end{aligned}$$

(19)

$$\begin{aligned}&\theta =\frac{-(j+1)}{2}=\frac{\beta -1}{\beta } \end{aligned}$$

(20)

$$\begin{aligned}&\varphi (1)=\frac{2}{\gamma +1}\end{aligned}$$

(21)

$$\begin{aligned}&\psi (1)=\frac{\gamma +1}{\gamma -1}\end{aligned}$$

(22)

$$\begin{aligned}&f(1)=\frac{2}{\gamma +1} \end{aligned}$$

(23)

where the non-dimensionalized variables are the velocity $\phi$, the density $\psi$, the pressure $f$, and the coordinate variable $z$. The scales are, respectively, the shock velocity $\dot{R}_s$, the initial density $\rho _0$, a pressure scale $\rho _0 \dot{R}_s^2$, and the shock radius $R_s$. The main simplifying assumption is that of a linear velocity profile

$$\begin{aligned} \phi =\frac{2}{\gamma +1}z. \end{aligned}$$

(24)

If the above conservation laws are solved numerically, one finds that the velocity profile deviates only slightly from a linear profile with the deviation increasing with the geometric index $j$.

The assumption of a linear velocity profile gives an expression for the density and pressure profiles as a function of the self-similar coordinate variable $z$

$$\begin{aligned} \psi&= \frac{\gamma +1}{\gamma -1}z^\frac{2\left(j+1\right)}{\gamma -1},\end{aligned}$$

(25)

$$\begin{aligned} f&= \frac{2}{\gamma +1}z^{\frac{2}{\gamma -1}[\gamma (j+1)+\theta (\gamma +1)]}. \end{aligned}$$

(26)

The above expressions are transformed back to the physical coordinates $(P,\rho ,u,r,t)$ and the velocity of a particle can then be integrated to obtain the particle path, using

$$\begin{aligned} u(r,t)=\frac{\mathrm{d}r}{\mathrm{d}t}. \end{aligned}$$

(27)

This yields

$$\begin{aligned} r_p(t)=r_0\left(\frac{t}{t_0}\right)^{\frac{2}{\gamma +1}\beta } \end{aligned}$$

(28)

where $(t_0,r_0)$ refer to the time and position at which a particle of interest was initially compressed by the decaying shock wave. The expression for the particle path is further substituted into the expressions for pressure and density, yielding

$$\begin{aligned} \frac{\rho _p}{\rho _0}&= \left(\frac{t}{t_0}\right)^{\left(B-1\right)C\beta }, \end{aligned}$$

(29)

$$\begin{aligned} \frac{T_p(t)}{T_0}&= \left(\frac{t}{t_0}\right)^{2\beta -2+\left(B-1\right)E\beta -\left(B-1\right)C\beta }. \end{aligned}$$

(30)

An ideal equation of state is assumed to obtain the temperature variation. The constants $B$, $C$, and $E$ are given by

$$\begin{aligned} B&= \frac{2}{\gamma +1}, \end{aligned}$$

(31)

$$\begin{aligned} C&= \frac{2\left(j+1\right)}{\gamma -1}, \end{aligned}$$

(32)

$$\begin{aligned} E&= \frac{\left(B-1\right)C\gamma -2\theta }{B-1}. \end{aligned}$$

(33)

We need to determine two constants, $\alpha$ and $\beta$. These are defined from the shock path $R_s=\alpha t^\beta$, which is matched to experiments. The trajectory of the leading shock waves (i.e., both the Mach stem and incident shock) is measured experimentally from the framing and streak records and used to match $\beta$. From this, a fitted value of the geometric exponent, $j$, is found. Normally, the geometric index takes on set values of $j=0$ (planar), $j=1$ (cylindrical) or $j=2$ (spherical). In the present experiments, the decay was less severe even than for a planar blast and, typically, $j<0$ for the present experiment. In this application, the geometric index, $j$ is a parameter fitted to the shock decay rather than a parameter which represents the symmetry of the problem. A negative value of $j$ simply means the shock wave had a slower decay than a planar blast. The slower decay is due to the heat released behind the shock.