Doubly conditional moment closure modelling for HCCI with temperature inhomogeneities

This paper presents a doubly conditional moment closure (DCMC) as an a posteriori predictive modelling tool for ignition of mixtures with large thermal stratification in homogeneous charge compression ignition (HCCI) conditions. Double conditioning is applied on enthalpy and its dissipation rate. The performance of the DCMC model is evaluated using a number of previously reported direct numerical simulations (DNSs) with various fuels. The DNSs modelled ignition of various lean homogeneous mixtures with a high level of temperature inhomogeneities. The selected cases exhibit a prevalence of deflagration mode of combustion as opposed to a spontaneous ignition-front mode, which has proven challenging for previous singly CMC. In all simulations, DCMC solver is run in a stand-alone mode with certain terms, such as the probability density functions of enthalpy and dissipation rate, being provided using the DNS input. The DCMC results are in a very good agreement with the DNS data, and are significantly improved compared with a singly conditional moment closure. A set of a posteriori DNS-DCMC tests is also performed to demonstrate importance of various terms in the doubly CMC equations. These tests first reveal that the effects of the cross dissipation and sources of enthalpy and dissipation rate (which lead to convective terms in conditional space) are insignificant and these terms can be safely neglected from the DCMC equations. The significance of this result is that the main unclosed models that would be needed for satisfactory results in a practical simulation of an engine would be the joint probably density function of enthalpy and its dissipation rate and the dissipation rate of dissipation rate.