Landau-Darrieus instability and the fractal dimension of flame fronts

Abstract

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 $\ensuremath{\gamma}=\frac{({\ensuremath{\rho}}_{u}\ensuremath{-}{\ensuremath{\rho}}_{b})}{{\ensuremath{\rho}}_{u}}$ is small (here ${\ensuremath{\rho}}_{u}$ and ${\ensuremath{\rho}}_{b}$ 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 $\ensuremath{\Delta}D=D\ensuremath{-}1$ decreases rapidly with decreasing of $\ensuremath{\gamma}$, to a first approximation as $\ensuremath{\Delta}D={D}_{0}{\ensuremath{\gamma}}^{2}$, where $D$ is the fractal dimension of the front. Our rough estimation of ${D}_{0}$ gives ${D}_{0}\ensuremath{\approx}0.3$. 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 $\ensuremath{\gamma}\ensuremath{\lesssim}0.3\ensuremath{-}0.4$. 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 $\ensuremath{\gamma}$ is small.