Formation of wrinkles in outwardly propagating flames

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Abstract
Numerical integrations of the partial differential equation proposed by Filyand, Sivashinsky, and Frankel [Physica D 72, 110 (1994)] to describe the dynamics of outward accelerating flames are presented. The computational results reported by Filyand, Sivashinsky, and Frankel are confirmed: as time increases, a repetitive formation of cusps, as well as a rapid (power-law) expansion of the mean flame radius, are observed. However, the identification of invariant subspaces for the equation shows that even when the initial condition belongs to such subspaces, numerical round-off errors are responsible for excursions of the solution outside these subspaces. In Fourier space, this corresponds to the generation of spurious Fourier modes that grow as time increases. This computational error is controlled here by a filter that forces the solution, at each time step, to stay inside the invariant subspaces. The results of our filtered simulations are very similar to those resulting from unfiltered integrations, showing that both the formation of cusps and the rapid acceleration of the flame front are independent of the growth of spurious Fourier modes. The connection between such dynamics and exact pole solutions of the equation (in which the number of poles is fixed) is investigated. It is found that the latter are unstable and the more complicated (stable) dynamics consists of successive instabilities through which the flame front closely follows a (2N+1)-pole solution before approaching a (2N+3)-pole solution. These migrations are responsible for both the formation of new cusps and the rapid power-law acceleration of the mean front.