Four Types of Chaotic Dynamics
in Cellular Flames

Theoretical Studies of Ordered States

The first numerical study of the dynamics of cellular flames was made by Michelson and Sivashinsky (1977) using the hydrodynamic model on a one dimensional flame front. Later they extended this study to a wider interval where a fine structure of cells was in constant motion. In a numerical study of the two-dimensional Kuramoto-Sivashinsky equation which arises from the thermodiffusive instability Sivashinsky (1983) considered a square 5 x 5 array of cells. This array evolved in time to a complicated structure with cells of varying sizes that changed their shapes in an irregular manner. Parameters that might correspond to an ordered state were not considered because no such state had been observed experimentally. Later, using a square geometry and periodic boundary conditions, Shtilman and Sivashinsky (1990) showed that an ordered hexagonal array of cells was the stable structure for a spherical surface of large radius. They did not explicitly address the issue of dynamics of this structure.

Bayliss and Matkowsky (1992) and Bayliss, Matkowsky and Riecke (1993) have published studies of numerical simulations using full equations that describe the thermodiffusive instability. In both studies they considered a one-dimensional circular (ribbon) flame concentric with a source of premixed gas directed outward along the radius. Steady ordered states with cell numbers from three to nine were found in the parameter range considered. Our preliminary experiments using a porous plug in the shape of a slot have also found that a one-dimensional ordered array of cells is steady and does not have chaotic motion, suggesting that the motion observed on the circular burner is a two-dimensional effect.

Nicolenko (1993) conducted numerical studies of the two-dimensional Kuramoto-Sivashinsky equation using a square geometry with periodic boundary conditions. His results capture four elements of the phenomenology of ordered states observed in our experiments: the metastability of a state with two inner cells, a maximum of six cells forming around a point, a transition from order to disorder, and chaotic dynamics of ordered states with motion concentrated around the cusps and folds.

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