Landau-Darrieus instability and the fractal dimension of flame fronts

Nonlinear dynamics of a slow laminar flame front subject to the Landau-Darrieus instability is investigated by means of numerical simulations of the Frankel equation, when the expansion degree $\gamma =\frac{({\rho }_u-{\rho }_b)}{{\rho }_u}$ is small (here ${\rho }_u$ and ${\rho }_b$ are the densities of the unburned and burned "gases," respectively). Only burning in two-dimensional space is considered in our simulations. The observed acceleration of a front wrinkled by the instability can be ascribed to the development of a fractal structure along the front surface with typical spatial scales being between the maximum and the minimum truly unstable wavelengths. It is found that the fractal excess $\Delta D=D-1\mathrm{}$ decreases rapidly with decreasing of $\gamma$, to a first approximation as $\Delta D=D_0{\gamma }^2$, where $D$ is the fractal dimension of the front. Our rough estimation of $D_0$ gives $D_0\approx 0.3$. The low accuracy of the $D_0$ estimation is caused by certain peculiarities of the Frankel equation that lead to extreme difficulties of its simulation even with the aid of supercomputers when $\gamma \lesssim 0.3-0.4$. It is shown, however, that $D_0$ can be calculated also from the statistical properties of the Sivashinsky equation, which is easier to simulate, though the fractal excess for the Sivashinsky equation itself is equal to 0 (in a certain sense). The other important result of our simulations is that the front self-intersections play an extremely weak role when $\gamma$ is small.