2.1 Reactive Euler equations for quasi one-dimensional flow

The exact equations of gas dynamics in multiple dimensions for a $N$ -component system, which neglect transport terms, are the so-called reactive Euler equations. The usual continuity, momentum conservation and energy conservation following a fluid particle are:

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle -\frac{1}{\unicode[STIX]{x1D70C}}\frac{D\unicode[STIX]{x1D70C}}{Dt}=\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}\frac{D\boldsymbol{u}}{Dt}=-\unicode[STIX]{x1D735}p, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{De}{Dt}=-\frac{p}{\unicode[STIX]{x1D70C}^{2}}\frac{D\unicode[STIX]{x1D70C}}{Dt}, & \displaystyle\end{eqnarray}$$

where $D/Dt=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ is the material derivative. Here $u$ is the velocity vector, $p$ is the pressure, $\unicode[STIX]{x1D70C}$ is the density and $t$ is the time variable. These need to be complemented by an appropriate equation of state for the internal energy $e$ of the form

(2.4) $$\begin{eqnarray}\displaystyle e=e(p,\unicode[STIX]{x1D70C},Y_{1},Y_{2},\ldots ,Y_{i},\ldots ,Y_{N}) & & \displaystyle\end{eqnarray}$$

and the appropriate kinetics for the evolution of the mass fractions $Y_{i}$ of each of the system’s $N$ components:

(2.5) $$\begin{eqnarray}\displaystyle \frac{DY_{i}}{Dt}=\frac{\unicode[STIX]{x1D714}_{i}}{\unicode[STIX]{x1D70C}}. & & \displaystyle\end{eqnarray}$$

When a quasi-one-dimensional formulation is sought in a duct (or streamtube) of cross-section $A(x)$ , the variables take the meaning of transverse averages (or streamtube variables) and the equations remain unchanged, with $D/Dt=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+u\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x$ . The divergence of the velocity field in the continuity equation (2.1), which describes the rate of volume change of a fluid element per unit volume, can be expressed as two contributions:

(2.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}x}+\dot{\unicode[STIX]{x1D70E}}_{A}. & & \displaystyle\end{eqnarray}$$

The first is the usual rate of strain of a fluid element in the flow direction $x$ , and the second, $\dot{\unicode[STIX]{x1D70E}}_{A}$ , is the rate of strain in the transverse direction (Whitham Reference Whitham1974; Williams Reference Williams1985):

(2.7) $$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D70E}}_{A}\equiv u\frac{\text{d}(\ln A)}{\text{d}x}. & & \displaystyle\end{eqnarray}$$

This term plays a fundamental role in both inert (Whitham Reference Whitham1974) and reactive gas dynamics (Fickett & Davis Reference Fickett and Davis1979), as it provides the rate of expansion or compression of a fluid element due to geometrical effects. For such quasi-one-dimensional flows, the conservation laws are best cast in characteristic form, which, after some manipulation and basic thermodynamic identities (Wood & Kirkwood Reference Wood and Kirkwood1957; Williams Reference Williams1985; Kao Reference Kao2008), yield

(2.8) $$\begin{eqnarray}\displaystyle \frac{1}{\unicode[STIX]{x1D70C}c^{2}}\frac{D_{\pm }p}{Dt}\pm \frac{1}{c}\frac{D_{\pm }u}{Dt}=\dot{\unicode[STIX]{x1D70E}}-\dot{\unicode[STIX]{x1D70E}}_{A}, & & \displaystyle\end{eqnarray}$$

where the derivatives

(2.9) $$\begin{eqnarray}\displaystyle \frac{D_{\pm }}{Dt}=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+(u\pm c)\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x} & & \displaystyle\end{eqnarray}$$

are taken along the path of forward facing $C_{+}$ and rear facing $C_{-}$ characteristics, given respectively by $\text{d}x/\text{d}t=u\pm c$ . The thermicity $\dot{\unicode[STIX]{x1D70E}}$ denotes the gas dynamic effect of chemical reactions or other relaxation phenomena on the rate of pressure and speed changes along the family of characteristics. We immediately remark that the effect of transverse flow divergence $\dot{\unicode[STIX]{x1D70E}}_{A}$ has precisely the opposite effect to the thermicity $\dot{\unicode[STIX]{x1D70E}}$ on modulating the strength of acoustic waves along the characteristic paths.

The thermicity in its most general form is given by

(2.10) $$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D70E}}=-\frac{\unicode[STIX]{x1D70C}}{c_{p}}\left(\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}T}\right)_{p,Y_{i}}\mathop{\sum }_{i=1}^{N}\left(\frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}Y_{i}}\right)_{p,\unicode[STIX]{x1D70C},Y_{j,j\neq i}}\frac{DY_{i}}{Dt} & & \displaystyle\end{eqnarray}$$

(Fickett & Davis Reference Fickett and Davis1979; Williams Reference Williams1985; Kao Reference Kao2008). For a mixture of ideal gases, the thermicity reduces to

(2.11) $$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D70E}}=\mathop{\sum }_{i=1}^{N}\left(\frac{\bar{W}}{W_{i}}-\frac{h_{i}}{c_{p}T}\right)\frac{DY_{i}}{Dt}, & & \displaystyle\end{eqnarray}$$

where $W_{i}$ is the molecular weight of the $i$ th component, $\bar{W}$ is the mean molecular weight of the mixture, $h_{i}$ is the specific enthalpy of the $i$ th specie and $c_{p}$ is the mixture frozen specific heat. This expression takes a simple and more familiar form for a reacting perfect gas with an equation of state of the form

(2.12) $$\begin{eqnarray}\displaystyle & \displaystyle p=\unicode[STIX]{x1D70C}RT, & \displaystyle\end{eqnarray}$$

(2.13) $$\begin{eqnarray}\displaystyle & \displaystyle e(p,\unicode[STIX]{x1D70C},Y_{R},Y_{P})=\frac{1}{\unicode[STIX]{x1D6FE}-1}\frac{p}{\unicode[STIX]{x1D70C}}+(1-Y_{P})Q, & \displaystyle\end{eqnarray}$$

where $Y_{R}$ and $Y_{P}$ are the mass fractions of the reactants and products, respectively, $\unicode[STIX]{x1D6FE}$ is the isentropic exponent and $Q$ is the chemical energy released during the reaction (Fickett & Davis Reference Fickett and Davis1979):

(2.14) $$\begin{eqnarray}\displaystyle \dot{\unicode[STIX]{x1D70E}}=\frac{(\unicode[STIX]{x1D6FE}-1)\unicode[STIX]{x1D70C}Q}{\unicode[STIX]{x1D6FE}p}\frac{DY_{P}}{Dt}. & & \displaystyle\end{eqnarray}$$

In the present paper, we will deal with flows for which the source term appearing in the quasi-one-dimensional formulation for a streamtube or in a tube of cross-sectional area $A(x)$ , namely

(2.15) $$\begin{eqnarray}\displaystyle K\equiv \frac{\text{d}(\ln A)}{\text{d}x} & & \displaystyle\end{eqnarray}$$

is a constant. The source term being a constant permits us to establish a steady (on average) detonation structure, as can be anticipated since the source is neither a function of space nor of time. In the current work, we study the dynamics of detonations in a channel with a constant logarithmic derivative $K$ , i.e. a channel with a cross-section varying exponentially with distance. This is known as an exponential horn in acoustics, since, in the absence of chemical reactions, acoustic waves will have constant amplification factors by virtue of (2.8).

2.2 Steady wave solution

The steady travelling wave solution in the presence of lateral mass divergence for quasi-one-dimensional flow is obtained in the usual way (Wood & Kirkwood Reference Wood and Kirkwood1954; He & Clavin Reference He and Clavin1994; Klein et al. Reference Klein, Krok and Shepherd1995; Yao & Stewart Reference Yao and Stewart1995; Short & Bdzil Reference Short and Bdzil2003; Kasimov & Stewart Reference Kasimov and Stewart2005; Bdzil & Stewart Reference Bdzil and Stewart2007). Typically, the flow divergence in the detonation frame of reference is kept constant. In our experiments, however, the logarithmic derivative of the streamtube $A$ in the laboratory frame of reference, given by (2.15), is kept constant, to reflect the exponential horn geometry.

Note that the logarithmic derivative of the streamtube cross-sectional area is also the curvature of the detonation front $\unicode[STIX]{x1D705}$ (Fickett & Davis Reference Fickett and Davis1979),

(2.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}=K=\frac{\text{d}(\ln A)}{\text{d}x}, & & \displaystyle\end{eqnarray}$$

such that we seek a travelling wave solution that maintains a constant front curvature.

A physically relevant derivation of the steady wave solution starts with the governing equations for quasi-one-dimensional gas dynamics written in characteristic form (2.8). Seeking a travelling wave solution with speed $D$ , we change to a wave-fixed frame with the change of variables $x^{\prime }=Dt-x$ , $t^{\prime }=t$ and $v=D-u$ and requiring that the unsteady terms vanish (for the steady travelling wave), the differential operators transform as

(2.17) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{D_{\pm }}{Dt}=(v\mp c)\frac{\text{d}}{\text{d}x^{\prime }}, & \displaystyle\end{eqnarray}$$

(2.18) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{D}{Dt}=v\frac{\text{d}}{\text{d}x^{\prime }}. & \displaystyle\end{eqnarray}$$

With this transformation, the characteristic equation (2.9) becomes

(2.19) $$\begin{eqnarray}\displaystyle \frac{1}{\unicode[STIX]{x1D70C}c^{2}}\frac{\text{d}p}{\text{d}x^{\prime }}\mp \frac{1}{c}\frac{\text{d}v}{\text{d}x^{\prime }}=\frac{\dot{\unicode[STIX]{x1D70E}}-\dot{\unicode[STIX]{x1D70E}}_{A}}{v\mp c}, & & \displaystyle\end{eqnarray}$$

with the upper and lower signs for the $C_{+}$ and $C_{-}$ families of characteristics, respectively. Solving for the derivatives, we obtain the usual relations for the steady ZND structure of detonations,

(2.20) $$\begin{eqnarray}\displaystyle \frac{\text{d}v}{\text{d}x^{\prime }}=-\frac{1}{\unicode[STIX]{x1D70C}v}\frac{\text{d}p}{\text{d}x^{\prime }}=\frac{\dot{\unicode[STIX]{x1D70E}}-\dot{\unicode[STIX]{x1D70E}}_{A}}{1-M^{2}}, & & \displaystyle\end{eqnarray}$$

with a similar expression for the density derivative,

(2.21) $$\begin{eqnarray}\displaystyle \frac{\text{d}\unicode[STIX]{x1D70C}}{\text{d}x^{\prime }}=-\frac{\unicode[STIX]{x1D70C}}{v}\left(\frac{\text{d}v}{\text{d}x^{\prime }}+\dot{\unicode[STIX]{x1D70E}}_{A}\right) & & \displaystyle\end{eqnarray}$$

and the evolution of the chemical composition of the mixture,

(2.22) $$\begin{eqnarray}\displaystyle \frac{\text{d}Y_{i}}{\text{d}x^{\prime }}=-\frac{\unicode[STIX]{x1D714}_{i}}{\unicode[STIX]{x1D70C}v}. & & \displaystyle\end{eqnarray}$$

With the appropriate kinetics to close the thermicity via (2.10), (2.11) or (2.14), the structure of the travelling wave solution is readily obtained by integrating (2.20)–(2.22) from the shock at $x^{\prime }=0$ . For a given shock speed $D$ , the post-shock state where the integration starts is given by the shock jump relations. The flow begins subsonic, but tends to increase its speed $v$ due to energy release (positive thermicity). The lateral flow divergence has the opposite effect. As the flow may eventually become sonic, hence gas-dynamically isolating the reaction zone from the rear flow, singular behaviour (marking non-steady dynamics) in (2.19), (2.21) and (2.20) is only prevented by the local balance between the thermicity $\dot{\unicode[STIX]{x1D70E}}$ and the rate of lateral strain $\dot{\unicode[STIX]{x1D70E}}_{A}$ . This is the so-called generalized Chapman–Jouguet (CJ) criterion. This condition provides the rear boundary condition. The particular values of detonation speed that yield post-shock flows which will satisfy the generalized CJ condition somewhere in the reaction zone behind the lead shock will mark the possible steady state travelling wave solutions. Note that an interesting interpretation of this criterion can be physically interpreted by invoking the characteristic structure given by (2.19): the reaction zone structure can be seen to consist of a region where forward facing characteristics communicate pressure wave amplification forward to sustain the lead shock, while the limiting characteristic travels at precisely the lead shock speed, along which the vanishing of the overall thermicity ensures that the waves travelling forward are not amplified.

The structure of the travelling wave solution and the dependence between lateral flow divergence rate $K$ and detonation speed has been obtained in the present study for a one-step perfect reacting gas, and also for a mixture of ideal gases with realistic thermal and chemical kinetic properties. For the simple one-step model, the ordinary differential equations were integrated using the numerical ODE solvers provided by Mathematica. For the real gas calculations, a custom Python code was developed. It uses Cantera’s framework to evaluate the relevant thermo-chemical data (Goodwin, Moffat & Speth Reference Goodwin, Moffat and Speth2017) and the Shock and Detonation Toolbox for evaluating the shock jump conditions (Kao & Shepherd Reference Kao and Shepherd2008). The resulting system of ODEs was integrated using the variable-coefficient ODE solver (VODE) provided by SciPy. The shooting procedure to obtain the steady detonation structure with lateral mass divergence follows the method outlined by Klein et al. (Reference Klein, Krok and Shepherd1995). Our code was verified by comparing the results with those provided by Klein et al. (Reference Klein, Krok and Shepherd1995) using their model for the streamtube area variation.

Figure 1 shows, for example, the numerically obtained $D(K)$ curves for the two mixtures investigated in the present study, namely $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ and $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ at 8.6 kPa initial pressure using the San Diego chemical mechanism (Williams Reference Williams2014). Results are also provided using two other chemical kinetic mechanisms widely used in the literature and also validated against published ignition delay data for these mixtures: the Wang mechanism (Wang et al. Reference Wang, You, Joshi, Davis, Laskin, Egolfopoulos and Law2007) and the CaltechMech mechanism (Blanquart Reference Blanquart2018). The $D(K)$ curves show minimal influence on which kinetic mechanism was used and display the characteristic decay of detonation speed with increasing lateral flow divergence, as well as the turning point denoting the maximum permissible lateral flow divergence. In this paper, further calculations will use the San Diego chemical mechanism.

Figure 1.

Relation between the detonation speed and lateral flow divergence rate $K$ for $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ and $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ at 8.6 kPa initial pressure obtained with different chemical kinetic mechanisms. Dotted lines represent the ideal Chapman–Jouguet (CJ) speed.

4.1 Experimental set-up

Figure 8 shows the schematic of the 3.4 m long rectangular channel used for the experiments in this paper. The channel is made of aluminium. The internal height and width of the channel are 203 mm and 19 mm respectively. This internal width was found to be the optimum value to suppress the transverse perturbations and thus approximates the detonation cellular dynamics as two-dimensional (Bhattacharjee Reference Bhattacharjee2013).

Figure 8.

A schematic of the experimental apparatus to study diverging detonations.

The channel had three identical sections (figure 8). The diverging ramp was placed in the third section, which was equipped with glass panels allowing visualization of the detonation front evolution. Two exponentially diverging ramps were used for the tests, which permitted us to correct for boundary layer losses in a thin channel geometry, as described later. A long ramp (with 1 m length and $K=(1/A)(\text{d}A/\text{d}x)=2.302~\text{m}^{-1}$ ) and a short ramp (with 0.5 m length and $K=(1/A)(\text{d}A/\text{d}x)=4.605~\text{m}^{-1}$ ) were used. At the entrance of the diverging section, the initial opening was 23 mm in height. A ‘cookie-cutter’ entrance was used in order to isolate the detonation front entering the diverging section. The diverging ramp was machined from a sheet of polyvinyl-chloride (PVC).

The detonation was initiated by two different methods in the first section of the channel. In the first method, a powerful spark from capacitor discharge was used. Its discharge characteristics are described in detail by Bhattacharjee (Reference Bhattacharjee2013). The capacitor bank nominally stored approximately 1000 J at a charging voltage of approximately $V_{0}=32~\text{kV}$ . However, a useful energy of only approximately 5–10 J was discharged in approximately $5~\unicode[STIX]{x03BC}\text{s}$ . This generated a strong blast wave giving rise to a high-speed deflagration wave. A wire-mesh grid was installed in the first section of the tube in order to ensure the rapid transition of the deflagration wave to a detonation. This method, however, was not successful in the experiments conducted with the acetylene mixture at initial pressures below 8 kPa. For these conditions, a more sensitive reactive driver was used in the first section of the channel, which was separated from the test mixture by a thin Mylar diaphragm (see figure 8). The driver gas used was $\text{C}_{2}\text{H}_{4}+3\text{O}_{2}$ . The driver gas was ignited using the same capacitor discharge as described above, in which a detonation was established. This incident detonation wave was found to transmit successfully into the test mixture.

The mixtures tested were $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ and $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ . The mixtures were prepared by the method of partial pressures in a separate mixing tank. The mixtures were typically left to mix over a period of at least 24 h prior to an experiment. Before an experiment, the channel was first evacuated to a pressure below 100 Pa prior to the injection of the test mixture. The experiments were performed at ambient room temperature of $20\pm 2\,^{\circ }\text{C}$ . The sensitivity of the test mixtures was controlled by varying the initial pressure of the gases.

The visualization of the detonation wave front evolution in the diverging channels was performed using a large-scale shadowgraph system described by Dennis et al. (Reference Dennis, Maley, Liang and Radulescu2014). A high-speed Phantom V1210 camera was used for visualization. Images were recorded at a resolution of 1152 by 256 and a framing rate of approximately 42 000 frames per second, yielding a time resolution of approximately $24~\unicode[STIX]{x03BC}\text{s}$ . An exposure of $1~\unicode[STIX]{x03BC}\text{s}$ was used. For reference, typically 20 consecutive frames were acquired to monitor the detonation passage through the test section. These frames were then post-processed in order to evaluate the speed of the lead shock between successive frames. These were recorded along the top wall only. The typical errors in evaluating these speeds, derived from the image resolution and the ability to identify the lead shock position in the photographs was approximately $\pm 2\,\%$ (Borzou Reference Borzou2016).

4.2 Results in $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$

Figure 9 shows the evolution of the detonation wave propagation for five different experiments in $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ at different initial pressures. Each record was obtained by overlaying the shadowgraph images obtained from the high-speed visualization videos of the same experiment. The detonation wave propagated from left to the right in the diverging domain. The leading shock front is identified by the thick black line, followed by a bright area, corresponding to the strong chemi-luminescence of the gas at high temperatures, which is overlaid on the shadowgraph record.

At sufficiently high pressures (figure 9 a,b), the detonation front is textured with a large number of small sized cells. Traces of the transverse waves starting from the triple points on the wave front and extending back across the reaction zone are also observed. As the detonation progresses in the enlarging section, it acquires the expected curvature due to the geometrical divergence of the gases behind the front. With decreasing pressure, as shown in figure 9(c,d), the cellular structure of the front is enlarged, partly due to the lower reactivity of the gas at lower pressures (see below) and partly due to the stronger propensity for the detonation wave to be attenuated by the diverging geometry. A somewhat thicker reaction zone is also observed. The cellular structure is characterized by transverse detonations consuming the gas behind the incident shock portions of the front. This can be clearly seen in figure 9(d), where most of the gas accumulating as non-reacted behind the weaker portion of the shock near the bottom wall reacts very rapidly behind a transverse wave, which takes the form of a transverse detonation wave with fine cellular structure typical of near- limit detonations reported by Gamezo et al. (Reference Gamezo, Vasil’ev, Khokhlov and Oran2000). Very few non-reacted pockets are observed, with negligible volumes, as shown in figure 9(d). When the initial pressure was lowered below a critical value, the detonation was extinguished in the diverging channel, with a quasi-1-D reaction zone decoupling from the lead shock, as shown in figure 9(e).

Figure 9.

Detonation front evolution in $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ along the long ramp at initial pressures of (a) 12.1 kPa, (b) 9 kPa, (c) 6.2 kPa, (d) 4.9 kPa and (e) 4.2 kPa.

The speed of the leading front recorded along the top wall from the sequential frames is shown in figure 10 for the experiments shown in figure 9. At the higher pressures, the framing rate does not resolve the cellular structure of the front, and an essentially time-averaged speed is recorded. The experiments indicate that this speed remains constant for the entire detonation travel in the diverging channel. This confirms the concept that an exponential horn can maintain a steady travelling detonation.

At the lower pressures of 6.2 and 4.9 kPa, the large cellular structures shown in figure 9(c,d) are now better resolved. The cellular dynamics gives rise to speed fluctuations of approximately $500~\text{m}~\text{s}^{-1}$ , or 30 % of the average propagation speed. In spite of these very large fluctuations typical of cellular dynamics, the average speed is approximately constant. Finally, at an initial pressure of 4.2 kPa, the lead shock is seen to continuously decay. A detonation wave can no longer remain self-sustained in the diverging section; see also figure 9(e) showing the absence of cellular structures.

Figure 10.

Speed of detonation front recorded along the top wall in $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ along the long ramp at initial pressures of (a) 12.1 kPa, (b) 9 kPa, (c) 6.2 kPa, (d) 4.9 kPa and (e) 4.2 kPa, corresponding to the experiments shown in figure 9.

The experiments performed with the shorter ramp with double the amount of divergence showed very similar flow fields. Figure 11 shows the evolution of the front of the detonation, while figure 12 shows the recorded speeds along the top wall for four typical experiments showing the range of phenomena observed. While at the higher pressures, the detonation front has a uniform curvature and small cellular structures, the cellular structure enlarges at lower pressures. The records indicate that the cellular structure corresponds to reactive transverse waves, as documented in detail by Gamezo et al. (Reference Gamezo, Vasil’ev, Khokhlov and Oran2000). The speed record of figure 12 shows that for the shorter ramp, a steady detonation can also be established. Once the cellular structure can be resolved by the framing rate used (figures 11 c and 12 c), the front has variations of approximately $500~\text{m}~\text{s}^{-1}$ , or 30 % of the average propagation speed. When the pressure is sufficiently low, the detonation wave cannot propagate in the diverging section. The front consists of a decoupled shock–flame structure (figure 11 d). The lead shock decays to low speeds of approximately $800~\text{m}~\text{s}^{-1}$ by the end of the channel (figure 12 d).

Figure 11.

Detonation front evolution in $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ along the short ramp at initial pressures of (a) 12.1 kPa, (b) 8.3 kPa, (c) 6.2 kPa and (d) 4.9 kPa.

Figure 12.

Speed of detonation front recorded along the top wall in $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ along the short ramp at initial pressures of (a) 12.1 kPa, (b) 8.3 kPa, (c) 6.2 kPa and (d) 4.9 kPa, corresponding to the experiments shown in figure 11.

A summary of average speeds measured in all the experiments in $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ on both ramps is shown in figure 13 as a function of the initial pressure in the mixture. The error bars represent the standard deviation of the measurements for each experiment. As expected, the error bar grows with decreasing pressure, since the cellular dynamics are better resolved. Repeat experiments are also shown. The spread between repeat experiments never exceeded 4 %, which is significantly lower than the standard deviation due to the cellular dynamics fluctuations.

Also shown in figure 13 is the ideal Chapman–Jouguet (CJ) detonation speed, which was obtained from chemical equilibrium calculations using the NASA CEA code (McBride & Gordon Reference McBride and Gordon1996). The figure shows that both the CJ speed and the experimentally determined speed drop for both large and small ramp experiments when the initial pressure is lowered. The CJ speed drops because of the enhanced dissociation in the product species at low pressures. Nevertheless, the mean shock speed recorded in the experiments also deviates from the CJ speed with a decrease in the initial pressure. This can be interpreted in terms of the gas sensitivity varying with the initial pressure. Mixtures with lower initial pressures are the ones with slower reaction rates and chemical energy release rate: see figure 21. Therefore, the expansion cooling experienced by the shocked gas particles due to the lateral area divergence is able to decrease the shock strength and speed. Since the CJ speed represents the velocity of a steady detonation wave, the lower initial pressures result in more deviation from the CJ speed.

Figure 13.

Mean detonation wave speeds along the top wall for all initial pressures permitting the sustenance of detonations.

4.3 Results in $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$

The experiments performed in the $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ mixture were qualitatively similar to the ones presented above for the acetylene mixture, although some fundamental differences were observed. Figures 14 and 15 show the results for the long ramp, while figures 16 and 17 show the results for the short ramp. Firstly, for these mixtures, a steady detonation was again observed in the diverging sections, confirming the appropriateness of a mean field description to model the dynamics.

Figure 14.

Detonation front evolution in $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ along the long ramp at initial pressures of (a) 12.1 kPa, (b) 8.7 kPa, (c) 5.2 kPa, (d) 3.4 kPa and (e) 2.8 kPa.

Figure 15.

Speed of detonation front recorded along the top wall in $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ along the long ramp at initial pressures of (a) 12.1 kPa, (b) 8.7 kPa, (c) 5.2 kPa, (d) 3.4 kPa and (e) 2.8 kPa, corresponding to the experiments shown in figure 14.

Figure 16.

Detonation front evolution in $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ along the long ramp at initial pressures of (a) 12.1 kPa, (b) 8.6 kPa, (c) 5.5 kPa and (d) 4.5 kPa.

The most important difference with the earlier mixture was the cellular mode of propagation, as shown in figure 14(d) at a sufficiently low pressure that the cells are large enough to be well resolved. The transverse waves for this mixture appeared as non-reactive, with tongues of non-reacted pockets extending downstream, which pinch off as pockets when triple points collide. This is typical of more unstable detonations (Subbotin Reference Subbotin1976; Austin Reference Austin2003; Radulescu et al. Reference Radulescu, Sharpe, Law and Lee2007b ; Maxwell et al. Reference Maxwell, Bhattacharjee, Lau-Chapdelaine, Falle, Sharpe and Radulescu2017). Nevertheless, reactive transverse waves were also sometimes observed, as shown in figure 16(c). These were however the exception. In order to fully address the mode of propagation of these more unstable detonations, a few schlieren experiments were also performed for the small ramp, since the field of view of these was limited to the dimension of the mirrors (30 cm). The set-up for the schlieren experiments is described elsewhere (Bhattacharjee Reference Bhattacharjee2013). Figure 18 shows part of the front evolution for an initial pressure of 4.8 kPa. As can be clearly seen at this time and space resolution, the detonation wave takes on a characteristic cellular structure with non-reactive transverse waves (Subbotin Reference Subbotin1976; Austin Reference Austin2003; Radulescu et al. Reference Radulescu, Sharpe, Law and Lee2007b ; Maxwell et al. Reference Maxwell, Bhattacharjee, Lau-Chapdelaine, Falle, Sharpe and Radulescu2017). Gases accumulating behind the incident shock, here the central part of the front in figure 18(a), accumulate as non-reacted layers of gas, which are then pinched off from transverse wave collisions (figure 18 c). These pockets are then thinned out from their edges inwards through diffusive phenomena (flames).

Figure 17.

Speed of detonation front recorded along the top wall in $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ along the shorter ramp at initial pressures of (a) 12.1 kPa, (b) 8.6 kPa, (c) 5.5 kPa and (d) 4.5 kPa, corresponding to the experiments shown in figure 16.

Figure 18.

Sequence of schlieren records illustrating the front evolution in $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ along the short ramp at 4.5 kPa.

A summary of the average speeds measured in all the experiments in $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ on both ramps is shown in figure 13 as a function of the initial pressure in the mixture, along with the ideal Chapman–Jouguet speeds calculated with the chemical equilibrium code CEA. Similar to the acetylene experiments, the velocity deficit grows with a decreasing pressure until extinction is observed.

4.4 The experimental $D(K)$ curves

The total mass divergence rate experienced by the detonation wave in the experiments of this study is due to the area divergence due to the diverging channel and the divergence of the flow to the boundary layers on the channel walls inherent in a thin channel geometry, by the mechanism discussed in the Introduction. The effective lateral flow divergence rate thus includes both the contribution from the diverging geometry and the contribution of the flow divergence rate due to the boundary layers developing on the walls of the channel, $\unicode[STIX]{x1D719}_{BL}$ :

(4.1) $$\begin{eqnarray}\displaystyle K_{eff}=\frac{1}{A}\frac{\text{d}A}{\text{d}x}+\unicode[STIX]{x1D719}_{BL}. & & \displaystyle\end{eqnarray}$$

Instead of modelling the source term $\unicode[STIX]{x1D719}_{BL}$ , which cannot be done without an empirical constant (Chao et al. Reference Chao, Ng and Lee2009; Camargo et al. Reference Camargo, Ng, Chao and Lee2010; Gao et al. Reference Gao, Zhang, Ng and Lee2016), this loss rate can be directly evaluated from the experiments conducted on the two ramps of different expansion ratios, but constant channel dimension (19 mm), leading to a constant $\unicode[STIX]{x1D719}_{BL}$ . The contribution of the boundary layers losses $\unicode[STIX]{x1D719}_{BL}$ has been obtained experimentally by comparing the experiments performed on the two ramps, for the same mixture and the same velocity deficit and calibrating the effective rate of mass divergence $K_{eff}$ to obtain a unique relation between velocity deficit and loss rate, as expected from theoretical considerations. Figures 19 and 20 show the data obtained after such a reduction of the data shown in figure 13. The induction length used to normalize the lateral divergence parameter for each mixture was calculated as the product of the flow speed at the von Neumann post-shock state in the detonation frame of reference and the ignition delay time calculated for constant volume combustion at the von Neumann state using the San Diego mechanism, which is shown in figure 21. It was found that a unique value of $\unicode[STIX]{x1D719}_{BL}=5.5~\text{m}^{-1}$ permitted the collapse of all data for the two ramps in both mixtures tested. Note that this value is comparable with the divergence provided by the exponential horns themselves. Note also that the reaction time shown in figure 21, calculated as the maximum inverse thermicity in the calculations, varies also proportionally to the induction time in each mixture, such that the normalization by this time scale would have generated the same result.

Figure 19.

The $D{-}K$ characteristic curves for $\text{C}_{3}\text{H}_{8}+5\text{O}_{2}$ obtained experimentally and predicted from the quasi-one-dimensional ZND model using the San Diego chemical mechanism (Williams Reference Williams2014). The solid line is an empirical fit discussed in § 5.2.

Figure 20.

The $D{-}K$ characteristic curves for $2\text{C}_{2}\text{H}_{2}+5\text{O}_{2}+21\text{Ar}$ obtained experimentally and predicted from the quasi-one-dimensional ZND model using the San Diego chemical mechanism (Williams Reference Williams2014). The solid line is an empirical fit discussed in § 5.2.

Figure 21.

Variation of the ignition delay time $t_{i,N}$ and reaction time $t_{r,N}$ for the two mixtures with initial pressure calculated at the post-shock conditions of the steady ZND structure in the absence of losses.