A dynamical system approach for solving the steady structure of high speed deflagrations

Abstract

High-speed deflagrations have burning velocities much higher than laminar ones, and compressible effects become important. In the present paper, we study the structure of such high speed deflagrations in the thickened flame limit for a one-step Arrhenius rate law, whereby the transport coefficients are increased to give rise to buring velocities of finite Mach number. We study their steady structure and compare with the laminar low-Mach classical structure. The singular nature of both the fresh and burned gases conditions in the compressible regime, i.e., for large values of flame propagation Mach number, which are both saddle points, precludes the application of simple shooting methods to obtain their structure. The steps leading to the flame structure borrow techniques used to treat mathematical features commonly found in the study of dynamical systems (phase portrait of the flame structure equations system, eigenvalue decomposition of its linearized version), that can also be used to further comment on the nature of high-speed flames. The method proposed permits to determine the structure of deflagrations propagating up to speeds of aproximately 0.95 of the CJ-deflagration burning velocity, for a wide range of gas parameters commonly found in the field of numerical simulation of accelerating flames and their transitions to detonations. We comment on how the increasing role of compressibility modifies the structure of the laminar flame.