Theoretical framework for knock onset: ZFK–FKPP-transition-based analysis of flame acceleration with a chemically unfrozen preheat zone

Engine knocking remains a critical barrier to improving the thermal efficiency of internal combustion engines. In this context, this study investigated the mode transition between two flame propagation regimes — from the Zel’dovich–Frank-Kamenetskii (ZFK) mode, characteristic of large activation energy, to the Fisher–Kolmogorov–Petrovsky–Piskunov (FKPP) mode, associated with small activation energy — as the overall activation energy decreases. A linear eigenvalue analysis of the master equation revealed that this mode transition occurred when reactions in the preheat zone exceeded a certain threshold. Subsequently, direct numerical simulations (DNS) of knocking phenomena for both n -heptane/O 2 /Ar and primary reference fuel (PRF)/air mixtures, previously validated against experiments, were reanalyzed. From the ZFK to the FKPP, transition conditions for target mixtures were numerically obtained. The transition for the n -heptane/O 2 /Ar mixture occurred when the end-gas condition exceeded approximately 1163 K and 2.1 MPa in two-dimensional DNS, accompanied by pronounced acceleration of the burning velocity prior to knock onset. This temperature and pressure threshold accurately corresponded to the knock onset condition of the same mixture. Furthermore, the thresholds of PRF80, 90, and 100/air mixtures were consistent with the knock onset conditions observed in one-dimensional DNS for the corresponding mixtures. In conclusion, these findings demonstrate that the fundamental transition from ZFK to FKPP modes captures the knock onsets, providing a robust and physically grounded criterion for knock prediction across a wide range of fuels. Novelty and significance statement This study reveals a novel and unexpected connection between a macroscopic theoretical framework and detailed numerical predictions of knock onset. Although the ZFK–FKPP transition represents a coarse, macroscopic framework that does not resolve fine-scale details, the transition conditions computed using detailed chemical reaction mechanisms for the specific target mixtures are consistent with the knock onset obtained from the DNS. This remarkable agreement demonstrates that, by numerically analyzing the ZFK–FKPP transition with realistic chemistry for a given mixture, the DNS knock onset can be predicted, which is an original finding.