On a scaling law for coarsening cells of premixed flames: an approach to fractalization

We consider thin unstable premixed flames which are planar on average and evolve spontaneously from weak, random initial conditions. The underlying dynamics is chosen to follow a Michelson-Sivashinsky equation, and attention is focused on the statistical properties of its solutions. Generalizing a suggestion of Blinnikov & Sasorov (Phys. Rev. E, 53, p. 4827, 1996) we propose an asymptotic law for the ensemble-averaged power density spectrum of wrinkling EO(k,t) in the limit of long times and long waves, viz E(k,t) similar to (Omega /a)F-2(/k/ t S-L)/k(2+d), for fixed kt, where S-L is the laminar burning speed, Omega and a are known functions of the burnt-to-unburnt density ratio, F(.) is a numerically-determined function, d + 1 = 2 or 3 is the dimension of the ambiant space through which propagation takes place; /k/ is the current wavenumber of wrinkling. Our proposal and the above authors' are tested against extensive, high-accuracy integrations of the MS equation. These ssugest, after ensemble-averaging, a corrected law of the form /k/(2+d) E(k,t) similar to (Omega /a)F-2(/k/ t S-L)e(-k/k*+t*/t) (valid for any k); here k* and t* are constants and F(.) is the same as above. Our results also indicate that F(infinity) not equal 0 .F(infinity) not equal 0 will ultimately lead to an increase in effective burning speed proportional to (Omega /a)(2) F(infinity) Log (t/t*), yet only for very long times; the latter are hardly accessible directly, due to round-off jitter. According to the aforementionned reference, F(infinity) not equal 0 and the logarithmic growth signal a gradual failure of the MS equation and a transition to fractalization, with an excess fractal dimension d(F) - d 2 d(Omega /a)(2) F(infinity) for the solutions to the coordinate-free generalization of the MS-equation; however the solutions to the MS equation are not themselves fractal over the time-wise domain when the latter is valid. The value of F(infinity) we estimate is too low, however, to match what other numerical experiments gave: some important ingredient seems to be missing in the theoretical interpretations of the latter, e.g. explicit consideration of external forcing by numerical noise.