Computation of the mean hydrodynamic structure of gaseous detonations with losses

Abstract

In this paper, computations of gaseous detonations in several configurations are performed and discussed. The objective is to investigate the detonation characteristics, specially the influence of losses, through a quantitative analysis of the mean hydrodynamic structure. The results are divided into two parts: The first one relates to the propagation of ideal detonations without losses and to their characterization for a specific set of thermodynamic parameters, and the second part investigates the influence of losses on the averaged hydrodynamic structure of detonations. This is achieved by means of a specific and canonical configuration, in which a detonation wave propagates in a reactive layer bounded by an inert gas. This topology is investigated through a comparison of the instantaneous flow-field, cellular structure, and averaged quantities such as the mean curvature and the hydrodynamic thickness. This configuration may be regarded as non-ideal, as the detonation experiences losses due to the expansion of the detonation products toward the inert layer. The prediction of the detonation characteristics, as well as the conditions at which quenching occurs, remains a challenge when losses are involved. The main objective is to assess to what extent the hydrodynamic thickness can be used as a relevant length scale in the analysis of the resulting database. The results also highlight the key features of this configuration, which isolates a specific issue related to the rotating detonation engines. The temperature of the inert layer strongly affects the structure of the detonation–shock combined wave. For a high-temperature inert gas confinement, a detached shock appears in the upper layer and a jet of fresh mixture develops downstream of the front. For this condition, we observed a reduction in the critical dimensions by a factor of two. The degree of regularity of the detonation structure, from weakly unstable to mildly unstable cases, through the variation of the reduced activation energy, was investigated. The hydrodynamic thickness appears to be a useful characteristic length scale for the detonation, as it allows a better analysis of the results and a scaling of data. Moreover, the detonation velocity deficit can be globally expressed as a function of the ratio of the hydrodynamic thickness to the mean radius of curvature at the bottom of the wall.

Appendix: Influence of the numerical resolution

The influence of the numerical resolution on the numerical results is investigated. Simulations were conducted for different grid spacings: $N_{1/2}=10$, $N_{1/2}=30$, and $N_{1/2}=60$. The physical configuration is similar to the one investigated in Sect. 3.1.2, i.e., marginal detonations for $E_{{\text {a}}}^*=38.23$. The detonation wave speed is identical in the three cases and is equal to $\bar{D}=2845\,\hbox {m}\,\hbox {s}^{-1}$.

Fig. 18

Temperature field obtained for different numerical resolutions $N_{1/2}$: a $N_{1/2}=10$, b $N_{1/2}=30$, and c $N_{1/2}=60$

Fig. 19

Vorticity field obtained for different numerical resolutions $N_{1/2}$: a $N_{1/2}=10$, b $N_{1/2}=30$, and c $N_{1/2}=60$

Figure 18 presents the temperature field obtained for various values of $N_{1/2}$. This comparison enables the identification of qualitative differences associated with different numerical resolutions. The shape of the detonation front is not affected by the increase in mesh resolution. On the other hand, structures that develop downstream are significantly affected. In particular, the shape of the pockets of fresh gas depends on $N_{1/2}$. In Fig. 18a ($N_{1/2} = 10$), the outline of the pocket is smooth. This is not the case in  Fig. 18b, c, for which the hydrodynamic instabilities deform the interface of the fresh gas pocket and give it a serrated and filamentary shape. This is particularly pronounced for the case with $N_{1/2} = 60$. The flow in its entirety is impacted by these differences, as shown by the temperature fields for the different values of $N_{1/2}$. The fields in Fig. 19 show the vorticity fields associated with the different numerical resolutions. The points of high vorticity foreshadow the trajectory of the triple points along the front, but especially the presence of vortex structures in the downstream flow. Their number is reduced for $N_{1/2} = 10$ and increases significantly for $N_{1/2} = 30$ and $N_{1/2} = 60$.

Figure 20 compares the average profiles obtained for $N_ {1/2} = 10$, 30, and 60 in order to quantify the influence of the numerical resolution on the average flow. While the visualizations of Figs. 18 and 19 demonstrate a significant increase in hydrodynamic instabilities with resolution, the average profiles agree with each other. The pressure and density profiles (Fig. 20a, b) and mass fraction (Fig. 20c) converge to the same solution. It is the same with the fluctuations of pressure and speed. The sonic plane is determined at $7 \pm 1 \, \lambda$ (Fig. 20d). The fluctuations, shown in Fig. 21, have similar levels.

Fig. 20

Evolution of the mean pressure $\bar{p}$ normalized by $p_0$ (a), the mean density $\bar{\rho }$ normalized by $\rho _0$ (b), the mean mass fraction $\widetilde{Y}$ (c), and the mean local Mach number $\widetilde{M}$ (d) as a function of the distance from the leading shock normalized by the half-reaction length for different numerical resolutions $N_{1/2}$

Fig. 21

Evolution of the mean pressure fluctuations $(\bar{p'^{2}})^{1/2}$ normalized by $\bar{p}$ (a), the mean density fluctuations $(\bar{\rho '^{2}})^{1/2}$ normalized by $\bar{\rho }$ (b), the mean velocity fluctuations $(\widetilde{u''^{2}})^{1/2}$ normalized by $\widetilde{u}$ (c), and the mean local Mach number $\widetilde{M_\mathrm{t}}$ (d) as a function of the distance from the leading shock normalized by the half-reaction length for different numerical resolutions $N_{1/2}$

These results demonstrate that the main characteristics of the mean flow are correctly recovered by calculations made with a relatively low resolution ($N_ {1/2} = 10$). The speed of the front, its cellular structure, and the average profiles differ marginally with the spatial resolution of the simulations. On the other hand, the flows are of a degree of complexity and turbulence which evolves with $N_ {1/2}$. This is particularly the case with hydrodynamic fluctuations (Kelvin–Helmholtz, Rayleigh–Taylor, and Richtmyer–Meshkov). These instabilities are intended to play an important role in the mechanism of propagation of highly unstable detonations. The fact that the mean flow characteristics are recovered with a relatively low numerical resolution reinforces the hypothesis that the modeled mixture (${\text {H}}_2 / {\text {O}}_2$ at stoichiometry and pressure of 1 atm) is only moderately unstable at the difference from other mixtures such as ${\text {CH}}_4{-}{\text {O}}_2$ or ${\text {C}}_3 {\text {H}}_8{-}{\text {O}}_2$. On the other hand, the results obtained here as to the influence of the numerical resolution on the propagation of a marginal detonation do not completely prejudge the influence of $N_ {1/2}$ for a different configuration, in particular, when the cell irregularity is increased by losses in the reaction zone.