## 5 Results and Discussion
### Major Species Profiles  
Compared to the experiments, the mean mass fractions of major species, including CH\({}_{4}\), CO\({}_{2}\), H\({}_{2}\)O, N\({}_{2}\), and O\({}_{2}\), are reasonably predicted by all cases, as shown in Figures 6-10. Some under-predicted mean CH\({}_{4}\) mass fractions along the axis are found among all the cases, as shown in Figure 6a, and occur in the zones where the mixtures burn intensely, which means that the mean reaction rates are largely over-predicted. For the radial mean CH\({}_{4}\) mass fraction, the results of case 16 are found to agree well with the experiments, as shown in Figure 6 at \(x/d\) = 7.5 and \(x/d\) = 15. Similarly, the results of case 8 are also acceptable. Compared to the experiments, relatively large differences in the predicted radial mean CH\({}_{4}\) mass fraction occur close to the axis where the mixtures burn intensely, as shown in Figure 6d. The axial profiles of the mean mass fractions of CH\({}_{4}\) predicted by case 8, as shown in Figure 6, agree well with the experiments and are comparable to the results (Figure 12 in [40]) by LES/EDC for flame D. The radial profiles of the mean mass fractions of CH\({}_{4}\) at \(x/d\) = 15 predicted by cases 8 and 9 agree better with the experiments than the results (Figure 5 in [41]) of LES/transported PDF (probability density function) for flame D.  
For CO\({}_{2}\), some over-predicted values along the axis are found in all cases, as shown in Figure 7a, and occur in the zones where the mixtures burn intensely. Cases 8, 17, and 18 aptly predict the experiments. For the radial mean CO\({}_{2}\) mass fraction, the results are found under-predicted upstream of the peak and over-predicted downstream of the peak, as shown in Figure 7 at \(x/d\) = 7.5, 15, and 30. Downstream of the peak temperature, around \(x/d\) = 45, the results are slightly over-predicted, as shown in Figure 7 at \(x/d\) = 45 and 60. In the zones where the reactions of mixtures are almost done, such as the location at \(x/d\) = 60, the predicted results vary slightly among the 20 cases. Compared to the predicted results of other cases, cses 8, 17, and 18 are found to agree well with the experiments for both the axial and radial mean CO\({}_{2}\) mass fraction. The predicted peaks of the mean mass fractions of CO\({}_{2}\) are decreased with the modified EDC constants and the largest is attributed to case 8, as expected. The axial profiles of the mean mass fractions of CO\({}_{2}\), as shown in Figure 7, agree better with the experiments than the LES/EDC results (Figure 12 in [40]) for flame D. The results (at \(x/d\) = 15 and 30, Fig_S5 in [42]) from local modification of EDC constants for flame D show that the peaks are successfully reduced. However, the over-predicted mass fraction of CO\({}_{2}\) at a radius larger than that of the actual peak of mass fraction of CO\({}_{2}\) is improved slightly. Similarly, over-predicted mean H\({}_{2}\)O mass fractions along the axis upstream of the peak and under-predicted profiles downstream of the peak are found in all cases, as shown in Figure 8a, and occur in the zones where the mixtures burn intensely. This means that the mean reaction rates are largely over-predicted. The predicted results of cases 8, 17, and 18 agree reasonably well with the experiments, which indicates that the mean reaction rates are reasonably predicted. For the radial mean H\({}_{2}\)O mass fraction, the results are found to agree quite well upstream of the peak, and are over-predicted downstream of the peak, as shown in Figure 8 for \(x/d\) = 7.5 and 15. At the location of peak temperature, around \(x/d\) = 45, the results are slightly over-predicted, as shown in Figure 8 at \(x/d\) = 45. In the zones where the reactions of the mixtures are almost done, such as at \(x/d\) = 60, the predicted results differ within a small range among the 20 cases. Compared to the predicted resultsof other cases, cases 8, 17, and 18 are found to agree well with the experiments for both the axial and the radial mean H\({}_{2}\)O mass fractions.