Interface dynamics, pole trajectories, and cell size statistics

We demonstrate the possibility to reproduce the experimental evolution of an interface, here a flame front, through the trajectory of a few poles whose position in the complex plane expresses the interface shape. These poles are analytical solutions of the Sivashinsky equation and they evolve according to an ordinary differential equation. The direct comparison with experimental flame fronts propagating in a quasi-two-dimensional configuration is made at the nonlinear but deterministic stages of the front dynamics, reproducing a cell creation and fusion process. At later times, when the front is sensitive to noise as in the Kardar-Parisi-Zhang equation, we demonstrate that the cell size distribution is still ruled by the pole attractive nature.

Figure 1

Left: Sketch of the vertical Hele-Shaw cell. Right: Zoom on a lean propane-air flame front at different heights (inverted colors). The flame is initially anchored on the upper edge. When the upstream flow of reacting gas is stopped the flame moves downward and is destabilized.

Figure 2

(a) Downward propagation of an experimental propane-air flame front (equivalence ratio is 0.9). Red dots are placed on the cusps in order to track their trajectory. (b) Poles evaluated from the initial condition are represented by blue points. (c) Evolution of the pole solutions according to the system of ODEs (4).

Figure 3

Measurements of growth rate $\sigma$ for lean propane air flame at equivalence ratio 0.9. The solid line is the least-square parabolic fit in accordance with the dispersion relation (2) exhibiting the maximum ${\sigma }_M$ and cutting the horizontal axis at $k_c$.

Figure 4

Detail of a two-cusp fusion process in experiments. Field of view is 22 mm large. Inset: evolution of the squared distance between cusps in time (normalized by initial distance).

Figure 5

Distribution of distances $p(\xi =d/〈d〉)$ [and $p\left(d\right)$ in the inset] between successive cusps for propane-air flames with equivalence ratios between 0.9 and 1.4, in linear scale (a) and semilogarithmic scale (b). The result of the numerical integration of the MS equation is also reported. In the plots of the distribution of normalized distances $\xi$, the solid line is the gamma distribution with order $\nu =2$.