# Unsteady Combustion of the Heptane-in-Water Emulsion Foamed with Hydrogen-Oxygen Mixture
## 3 Results and Discussion  
### Flame Dynamics  
First, let us consider the main features of non-steady flame behavior inside the tube filled with the combustible foamed emulsion. Figure 2 demonstrates the multidimensional (two-dimensional) flame front propagating downwards. As one can clearly see, the flame front is extremely corrugated. There are lots of small-scale structures in the background of the general shape of the flame elongated along the channel walls. That flame corrugation is due to the fact that the flame front is permanently subjected to small-scale perturbations originating from the coarse droplets interactions with the flame front. Such a permanent dynamic impact on the flame front causes a continuous increase in the area of the flame surface. Each perturbation of the flame surface gives rise to the gas-dynamic instability of the flame front, so the flame consumes the fresh medium faster. A larger flame area and its faster propagation in space define more frequent interactions of the flame with the droplets. That, in turn, leads to more intense development of the flame surface, etc. Therefore, there is a mechanism of positive feedback according to which the flame acceleration can be precisely described by exponential law \(U_{f}\propto exp(\alpha t)\). The increment \(\alpha\) is an intrinsic parameter of each foam. Thus, as one can see from Figure 3, \(\alpha\) occurs to be almost twice larger in the case of more reactive foam containing stoichiometric hydrogen-oxygen mixture as a foaming gas (Figure 3b) compared with the case of lean hydrogen-oxygen mixture (Figure 3a). In some cases, especially when a stoichiometric hydrogen-oxygen mixture is used as a foaming gas (Figure 3b), the flame acceleration can lead to the transition to detonation. Herewith, the detonation wave occurs to be unstable when propagating through the foam, it decays, and further combustion propagates at a much lower speed (Figure 3b). As a result, the average combustion speed, in this case, occurs to be much lower than the detonation one. This mode was studied in detail previously in [20; 27], where pure hydrogen-oxygen microfoam was considered. When carrying out calculations of flame propagation through the foam according to the proposed model, it is useful to analyze how the choice of initial conditions (Figure 1) affects the solution. A series of three calculations with a different (random) initial distribution of bubbles is carried out to perform such an analysis. The discrepancy of the obtained results is denoted in Figure 3 by a red region between two red lines representing minimal and maximal values obtained numerically. As one can clearly see, the discrepancy is rather small and does not exceed the error of flame speed estimation based on the experimental data. That is due to the fact that the flame acceleration is driven mainly by the local effect of flame front perturbation by the coarse droplets. As a result, the flame evolution depends weakly on the particular spatial distribution of bubbles. The same feature was also illustrated in [27] by changing the width of the channel. It was shown that since the local effect plays a predominant role, there is no obvious dependence of the flame dynamics on the channel width. When analyzing the experimental data obtained in [19], the average flame speed was introduced as an integral parameter characterizing the non-steady process of flame propagation through the channel filled with the foam. Here, a set of calculations with various foam compositions are carried out and analyzed together with the experimental points from [19]. Both sets of experimental and computed values are gathered in Figure 4. As one can see, the model used in the calculations reproduces experimental results quite well, at least on the qualitative level. Quantitatively, the combustion limits are not the same as in the experiment.