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 Michel-son-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 E(k,t) in the limit of long times and long waves, viz E(k,t) ˜ (Ω/a) F(lkl t S L )/k 2+d , for fixed kt, where S L is the laminar burning speed, Ω 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; lkl 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 lkl 2+d E(k,t) ˜ (Ω/a) 2 F(lkl 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(∞) ≠ 0. F(∞) ≠ 0 will ultimately lead to an increase in effective burning speed proportional to (Ω/a) 2 F& lpar;∞) 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(∞) ≠ 0 and the logarithmic growth signal a gradual failure of the MS equation and a transition to fractalization, with an excess fractal dimension dp − d ≅ d(Ω/a) 2 F(∞) for (he 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(∞) 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.