Dynamical system analysis of ignition phenomena using the Tangential Stretching Rate concept

We analyze ignition phenomena by resorting to the stretching rate concept formerly introduced in the study of dynamical systems. We construct a Tangential Stretching Rate (TSR) parameter by combining the concepts of stretching rate with the decomposition of the local tangent space in eigen-modes. The main feature of the TSR is its ability to identify unambiguously the most energetic scale at a given space location and time instant. The TSR depends only on the local composition of the mixture, its temperature and pressure. As such, it can be readily computed during the post processing of computed reactive flow fields, both for spatially homogeneous and in-homogenous systems. Because of the additive nature of the TSR, we defined a normalized participation index measuring the relative contribution of each mode to the TSR. This participation index to the TSR can be combined with the mode amplitude participation Index of a reaction to a mode - as defined in the Computational Singular Perturbation (CSP) method - to obtain a direct link between a reaction and TSR. The reactions having both a large participation index to the TSR and a large CSP mode amplitude participation index are those contributing the most to both the explosive and relaxation regimes of a reactive system. This information can be used for both diagnostics and for the simplification of kinetic mechanisms. We verified the properties of the TSR with reference to three nonlinear planar models (one for isothermal branched-chain reactions, one for a non-isothermal, one-step system, and for non-isothermal branched-chain reactions), to one planar linear model (to discuss issues associated with non-normality), and to test problems involving hydro-carbon oxidation kinetics. We demonstrated that the reciprocal of the TSR parameter is the proper characteristic chemical time scale in problems involving multi-step chemical kinetic mechanisms, because (i) it is the most relevant time scale during both the explosive and relaxation regimes and (ii) it is intrinsic to the kinetics, that is, it can be identified without the need of any ad hoc assumption.