A theoretical study on the relationship between pressure rise and the Damköhler number of end-gas auto-ignition in spark-ignited engines

The pressure rise caused by end-gas auto-ignition in spark-ignited engines is discussed using numerical simulation and theoretical approaches. The main objective of this study is to explain the mechanism by which the end-gas expansion during auto-ignition suppresses the pressure rise in spark-ignited engines, and theoretically to demonstrate using asymptotic analysis that the pressure rise depends on the Damköhler number. A one-dimensional direct numerical simulation (DNS) of end-gas auto-ignition is performed, and the modelling assumptions for it are discussed based on the DNS results. The Damköhler number, defined as the acoustic time scale and the characteristic time scale of the chemical reaction, is introduced in the modelling. The end-gas auto-ignition model is solved numerically, and it is shown that the pressure rise increases with the Damköhler number. Additionally, it is shown that the tendency of the pressure rise is due to the balance between the propagation rate of the expansion wave generated in the end gas and the reaction rate at auto-ignition, which varies with the Damköhler number. To derive the analytical solution of the relationship between the pressure rise and Damköhler number, the end-gas auto-ignition model is simplified based on the numerical results. The simplified model for end-gas auto-ignition is then solved using Newton’s method, and the analytical solution of the pressure rise is derived.