Computational analysis of the steady states of the Sivashinsky model of hydrodynamic flame instability

Abstract

In this work a numerical analysis of an integro-differential equation modelling the Darrieus–Landau instability of plane flame fronts was undertaken. A relatively new computational method based on saturated asymptotic approximations was used. Within the considered computational times a steady limiting shape of the flame front was not reached in large enough computational domains of size L > L_c . Instead, a smooth surface of an almost steadily shaped flame is repeatedly disturbed by small perturbations, resembling small cusps, appearing and disappearing randomly in time. The nature of these small cusps as well as of the steady limiting shape of the flame front was studied with a relatively new computational method.

The correlation between the critical length L_c and parameters of the computational algorithm and the computer precision was investigated. The calculations confirmed that, unlike the round-off errors, there is no significant link between the approximation accuracy of the algorithm and L_c . The obtained dependence of the critical length L_c on the magnitude of the round-off errors, considered as an external noise, was compared with the predictions given by other researchers. The agreement supports the idea of high sensitivity of solutions of the Sivashinsky equation to the external noise. A similarity between the appearance of small cusps on the surface of large enough flames governed by the Sivashinsky equation and streamwise streaks accompanying the loss of stability of the classic Hagen–Poiseuille flow was noted.