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=(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 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 approximate to 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 less than or similar to 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. [S1063-651X(96)02605-0]