Dynamics in a rod model of solid flame waves revisited

Abstract

We consider gasless solid fuel combustion in a cylinder of radius $\tilde R$, associated with the SHS process, which employs combustion waves to synthesize materials. We model the full three dimensional (3D) problem in cylindrical geometry by considering an array of interacting 1D rods, each of which supports propagating waves in the axial direction. We consider a configuration of 8 equally spaced rods on the surface of the cylinder, 8 equally spaced interior rods at $\tilde R$/2 and 1 axial rod (8-8-1 model). The rods interact via heat transfer, with the heat transfer terms corresponding to a discretization of the transverse Laplacian. The rod model is designed to provide insight into the qualitative behavior of the full 3D problem, employing vastly reduced computational resources. We previously considered a coarser grained 3-1 model. Whether or not the rod model describes behavior occurring in the full 3D problem may well depend on the number of rods as well as on the symmetries of their arrangement. Therefore, here we consider a larger number of rods exhibiting different symmetries.

Both the full 3D model and the rod model allow for a uniformly propagating planar combustion wave. The dispersion relation for this solution is determined for both models and shown to be equivalent when certain parameters in the two models are identified. The 8-8-1 model admits analogs of spin and radial modes which are known to exist experimentally and as solutions of the 3D model. Unlike the previously studied 3-1 model, the 8-8-1 model is sufficiently fine-grained that the interior structure of these modes can be differentiated. Furthermore, in contrast to the 3-1 model, which can only describe 1-headed spins, analogues of multi-headed spin modes can now be simulated. We find an extensive variety of both time periodic and quasiperiodic (QP) modes, many not previously described.

We determine solution behavior as a function of R, the nondimensionalized cylindrical radius, for selected values of the Zeldovich number Z = N(1 − σ)/2, where N is a nondimensionalized activation energy and σ is the ratio of the unburned to the burned temperature.

In addition to new modes that we find in the more fine-grained rod model, there are two general conclusions, that were drawn from the 3-1 model which also follow from the 8-8-1 model, suggesting that they are valid for the full 3D model. Specifically, we find that (i) with only one exception, all QP modes can be interpreted as a combination of spin and radial modes associated with the linear stability analysis of the uniformly propagating solution which in many cases are not stable by themselves and (ii) for Z above the stability boundary as R increases, generator frequencies and indeed the frequency spectrum, hence the temporal behavior of all modes, approach those of the singly periodic pulsating planar (PP) solution. Thus, in the time domain each rod exhibits the temporal behavior of the PP mode (appropriately phase shifted in time) over intermediate timescales. Over long timescales there is a phase, but not amplitude, modulation in the solution. This result suggests that for large samples, which are important for technological applications of the SHS process and which are very difficult to simulate for the full 3D problem, the temporal (but not spatial) behavior is that of the PP mode.