The Dynamics of Unsteady Detonation with Diffusion

Abstract

The dynamics of one-dimensional detonations predicted by a one-step irreversible Arrhenius kinetic model with the inclusion of mass, momentum, and energy diffusion were investigated. A series of calculations in which activation energy is varied, holding the length scales of diffusion and reaction constant, was performed. As in the inviscid case, as the activation energy increases, the system goes through a period-doubling process and eventually undergoes a transition to chaos. Within the chaotic regime, there exist regions of low frequency limit cycles. An approximation to Feigenbaum's constant, the rate at which bifurcation points converge, is obtained. The addition of diffusion significantly delays the onset of instability and strongly influences the dynamics in the unstable regime. Because the selected reaction and viscous length scales are representative of real physical systems, the common use of reactive Euler equations to predict detonation dynamics in the unstable and marginally stable regimes is called into question; reactive Navier-Stokes may be a more appropriate model in such regimes.

Mean structure of one-dimensional unstable detonations with friction

“…A fine and uniformly spaced mesh of approximately 132 pts/∆ 1/2 is used to capture the shock front and the reaction zone. This choice of grid resolution is largely based on the findings of Romick et al (2011aRomick et al ( , 2012. The mesh is then stretched in the upstream region (unburnt gas) and in the downstream expansion waves to allow for the flow relaxing towards the far-field boundary conditions.…”

“…The bifurcation diagram was also studied by Kasimov et al (2013) and Henrick, Aslam & Powers (2006). From a numerical viewpoint, Romick, Aslam & Powers (2011a, 2012 have indicated that a shock-capturing strategy can retrieve most of the transient features of a weakly and mildly unstable detonation, at the expense of a much greater fine-mesh resolution, as compared to the shock-fitting of Henrick et al (2006).…”

“…The issue of including viscous effects was discussed in Fickett & Davis (1979), and in Romick et al (2011a), Romick, Aslam & Powers (2011b) and Romick et al (2012). The latter studies investigated the effects of mass, momentum and energy diffusion on the dynamics of unsteady detonation, with a one-step irreversible Arrhenius kinetic model.…”

Detailed simulation of the pulsating detonation wave in the shock-attached frame

“…A highly accurate fifth-order accurate algorithm is proposed, and it is shown that this order is achieved in the problem of stable DW propagation. However, it is noted that, in the case of strongly unstable DWs with strong internal shocks behind the LW front, the advantages of the proposed approach disappear; [8] and the later papers (see [10,11]) of the same authors do not contain examples of simulating the propagation of strongly unstable DWs. In contrast, the approach proposed in [9], is effective in the example of the secondary shock that overtakes and interacts with the LW.…”

Recent Results on Stability of Planar Detonations

“…In Figure 7(LEFT) we display the eigenvalue distribution at E ≈ 7.1, for which there are 48 unstable roots together with the translational eigenvalue at λ = 0; further increases in E lead to further unstable eigenvalues. In Figure7(RIGHT) we display for contrast the behavior of rNS eigenvalues for the value of viscosity ν = 0.1 considered in [RAP1,RAP2], tracking the unstable eigenvalues as E is varied through the stability transition region. For this viscous case, we find that there are just two pairs of unstable eigenvalues in total, which after crossing the imaginary axis to the right turn back and rather quickly restabilize by crossing back into the stable half-plane; meanwhile, the nearby inviscid eigenvalues plotted in the same figure may be seen to continue to the right.…”