The fast-time instability map of Liñán’s diffusion-flame regime

Abstract

A detailed spectral map for the fast-time instability in Linan’s diffusion-flame regime is presented in order to clarify the origin of two bifurcations of co-dimension 2, causing the transitions from cellular to uniform-oscillatory instability and from uniform-oscillatory to traveling instability. The role of the real and continuous essential spectrum is found to be pivotal in understanding both transitions. Particular attention is paid to the spectral characteristics in the stable parametric regions, where the interaction with the essential spectrum leads to these transitions. When the Lewis number is increased above unity from below, the discrete real spectrum disappears by submerging below the essential spectrum, and the discrete complex spectrum emerges instead, eventually leading to uniform-oscillatory instability. The transition from uniform-oscillatory to traveling instability, associated with the Bogdanov–Takens bifurcation, involves a phenomenon called gap spectrum. For Lewis numbers slightly greater than unity and Damkohler numbers sufficiently large, the discrete complex spectrum intersects the plane corresponding to the essential spectrum, resulting in a gap in the discrete spectrum for small wave numbers. The discrete complex gap spectrum exhibits a local maximum as the parameter values are modified to approach the Hopf bifurcation boundary. The gap in the discrete complex spectrum disappears and traveling instability emerges when crossing the Hopf bifurcation boundary.