Ghandhi, J. and Kim, K., "A Statistical Description of Knock Intensity and Its Prediction," SAE Technical Paper 2017-01-0659, 2017, doi:10.4271/2017-01-0659.
Cycle-to-cycle variation in combustion phasing and combustion rate cause knock to occur differently in every cycle. This is found to be true even if the end gas thermo-chemical time history is the same. Three cycles are shown that have matched combustion phasing, combustion rate, and time of knock onset, but have knock intensity that differs by a factor of six. Thus, the prediction of knock intensity must include a stochastic component. It is shown that there is a relationship between the maximum possible knock intensity and the unburned fuel energy at the time of knock onset. Further, for a small window of unburned energy at knock onset, the probability density function of knock intensity is self similar when scaled by the 95th percentile of the cumulative distribution, and log-normal in shape. A methodology to predict the maximum possible knock intensity based on the volumetric expansion rate of the end gas was developed, and the expansion rate was modeled based on blast wave theory, which is proportional to the energy of the initial explosion. The results showed a comparable level of scatter as the measurements.