Variance consistent mean shift particle model for treating differential molecular diffusion in transported PDF methods for turbulent reactive flows

A mean shift (MS) particle model has been developed before to account for molecular diffusion in the transported probability density function (PDF) methods for turbulent reactive flows (McDermott and Pope, Journal of Computational Physics, 2007, 226, 947-993). The MS model is superior over the traditional random walk model for its capabilities to incorporate differential molecular diffusion and to yield a consistent DNS limit of scalar variance production/dissipation. The MS model, however, yields inconsistent transport of the scalar variance because of the missing molecular diffusion in the resultant variance transport equation. To yield fully consistent transport of scalar mean and variance, a variance consistent mean shift (VCMS) model is developed in this work. The VCMS model formulations are developed for both a single-scalar system and a multi-scalar system. Boundedness violation of particle scalars exists in both the MS and VCMS models and causes a realizability issue of the models. A correction algorithm (called MP correction by McDermott and Pope) has been developed before to correct the boundedness violation in the MS model. The MP correction is found to produce overcorrection to the particles, resulting in an underprediction of the scalar variance. In addition, it is only applicable to the IEM mixing model. In this work, three new boundedness correction algorithms are developed based on the strategies of a least-decay factor (LDF), clipping at scalar bounds (CSB), and clipping with paired mixing (CPM). The accuracy and consistency of the VCMS models are examined and validated thoroughly in a turbulent mixing layer problem. The effectiveness and accuracy of the different boundedness correction algorithms are also examined and assessed in the mixing layer problem. The applicability of the CPM correction to other mixing models (MCurl and EMST) is also demonstrated.