Unfolding the weakly-nonlinear dynamics of polyhedral flames

Under certain conditions, conical lean premixed flames spontaneously break their continuous azimuthal symmetry to form corrugated polyhedral flames with discrete symmetry. In particular, such polyhedral shapes are commonly observed in hydrogen flame experiments. In our earlier works, we have shown that this phenomenon is driven by the intrinsic flame dynamics via a global linear instability with zero frequency that occurs at sufficiently low Lewis number and/or sufficiently high Damköhler number. In this work, we perform a center manifold reduction for this circle-pitchfork (CP) type bifurcation in order to elucidate the weakly-nonlinear dynamics of the transition. To start, we study the case of a single bifurcating polyhedral mode, where the universal unfolding reveals the global spatiotemporal nonlinear dynamics in the vicinity of the critical point. We then move on to a bimodal case where two modes of different azimuthal symmetries simultaneously bifurcate from a codimension-2 double-CP point. In this case, the analysis reveals complex global dynamics that include mixed modes, which may be linked to experimental observations of spontaneously rotating polyhedral flame structures.