The dynamics of flames in closed rectangular channels under the Darrieus-Landau instability

The onset and nonlinear evolution of the Darrieus-Landau (DL) instability in premixed flames propagating in closed vessels are investigated using a novel confined-hydrodynamic theory. The theory models the flame as a surface propagating in a mixture that is continuously compressed, causing a pressure and temperature rise with a substantial increase in burning rate. The flame speed obtained from the analysis of the flame zone depends on both stretch rate and decrease in flame thickness, and is modulated by a pressure-dependent Markstein length. Multi-dimensional flames are numerically simulated using an embedded-manifold methodology implemented within a massively parallel low-Mach number variable-density Navier-Stokes solver, and the evolving dynamics are examined for various values of Markstein length, heat release, and channel length. We show that the onset of instability is delayed, compared to a freely propagating flame under similar conditions, due to the vortical motion developing in the limited volume of burned gas trapped between the flame and the wall that opposes the vorticity generated in the flame zone which promotes the instability. The unstable flame initially develops a cusp-like intrusion pointing toward the burned gas – a signature of the DL instability. However, the steadily propagating structure observed in freely propagating flames manifests itself only in very long channels. Compression and pressure rise produce a continually evolving morphology, with repetitive cell splitting and merging that alters the flame surface area and its propagation. Although the overall propagation speed decreases due to the flame motion into a progressively denser gas, the highly corrugated conformations produce intermittent flame acceleration synchronous with rapid changes in flame surface area. Consequently, the flame travels faster than the corresponding planar flame. Notable is the dynamic change in Markstein length associated with the pressure rise that leads to unstable flames even for conditions where a freely propagating planar flame would remain stable.