<div>Studies of turbulent combustion in the past mainly focus on problems with single-regime combustion. In practical combustion systems, however, combustion rarely occurs in a single regime, and different regimes of combustion can be observed in the same system. This creates a significant gap between our existing knowledge of combustion in single regime and the practical need in multi-regime combustion. In this work, we aim to extend the traditional single-regime combustion models to problems involving different regimes of combustion. Among the existing modeling methods, Transported Probability Density Function (PDF) method is attractive for its intrinsic closure of treating detailed chemical kinetics and has been demonstrated to be promising in predicting low-probability but practically important combustion events like local extinction and re-ignition. In this work, we focus on the model assessment and advancement of the Large Eddy Simulation (LES)/ PDF method in predicting turbulent multi-regime combustion.</div><div><br></div><div><div>Two combustion benchmark problems are considered for the model assessment. One is a recently designed turbulent piloted jet flame that features statistically transient processes, the Sydney turbulent pulsed piloted jet flame. A direct comparison of the predicted and measured time series of the axial velocity demonstrates a satisfactory prediction of the flow and turbulence fields of the pulsed jet flame by the employed LES/PDF modeling method. A comparison of the PLIF-OH images and the predicted OH mass fraction contours at a few selected times shows that the method captures the different combustion stages including healthy burning, significant extinction, and the re-establishment of healthy burning, in the statistically transient process. The temporal history of the conditional PDF of OH mass fraction/temperature at around stoichiometric conditions at different axial locations suggests that the method predicts the extinction and re-establishment timings accurately at upstream locations but less accurately at downstream locations with a delay of burning reestablishment. The other test case is a unified series of existing turbulent piloted flames. To facilitate model assessment across different combustion regimes, we develop a model validation framework by unifying several existing pilot stabilized turbulent jet flames in different combustion regimes. The characteristic similarity and difference of the employed piloted flames are examined, including the Sydney piloted flames L, B, and M, the Sandia piloted flames D, E, and F, a series of piloted premixed Bunsen flames, and the Sydney/Sandia inhomogeneous inlet piloted jet flames. Proper parameterization and a regime diagram are introduced to characterize the pilot stabilized flames covering non-premixed, partially premixed, and premixed flames. A preliminary model assessment is carried out to examine the simultaneous model performance of the LES/PDF method for the piloted jet flames across different combustion regimes.</div><div><br></div><div>With the assessment work in the above two test cases, it is found that the LES/PDF method can predict the statistically transient combustion and multi-regime combustion reasonably well but some modeling limitations are also identified. Thus, further model advancement is needed for the LES/PDF method. In this work, we focus on two model advancement studies related to the molecular diffusion and sub-filter scale mixing processes in turbulent combustion. The first study is to deal with differential molecular diffusion (DMD) among different species. The importance of theDMD effects on combustion has been found in many applications. However, in most previous combustion models equal molecular diffusivity is assumed. To incorporate the DMD effects accurately, we develop a model called Variance Consistent Mean Shift (VCMS) model. The second model advancement focuses on the sub-filter scale mixing in high-Karlovitz (Ka) number turbulent combustion. We analyze the DNS data of a Sandia high-Ka premixed jet flame to gain insights into the modeling of sub-filter scale mixing. A sub-filter scale mixing time scale is analyzed with respect to the filter size to examine the validity of a power-law scaling model for the mixing time scale.</div></div>