## 3 Predictions with Different Turbulence Models
The results shown in Figures 3-5 indicate that the standard adjustment of increasing one of the constants, \(C_{1c}\), from 1.44 (used in the SKE model) to 1.6 (used in the SKE-R model) was not effective in this work or the work by [PERSON] et al. [59]. This was due to the lower density of the co-flow. Differences in density between the jet, pilot, and co-flow are known to affect the mixing behavior of jets. A reasonable qualitative prediction of the flame was achieved for all model combinations. However, in contrast to those results from LES simulations [15, 33, 34, 35, 37, 38], none of them came quantitatively close to the measured data [40, 41, 42, 43], shown with the yellow-filled red circles in Figures 4 and 5. This means the predictions with the turbulence models used in this work could not exactly reproduce the methane-air combustion in the Sandia Flame D. Deviations of predicted and measured values are shown in Table 3 for different turbulence models with GRI-Mech 3.0, where it can be seen that the largest deviation happened to the minor species, i.e., OH. Figure 5: Comparison of the turbulence models of SKE-R, SKE, RKE, SST, and RSM for the Sandia Flame D: mean axial velocity profiles with the detailed chemical kinetic mechanism of GRI-Mech 3.0. \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline
**Scalar Peak and Its Location** & **Exp** & **EDC-S/SKE** & **Deviation** & **EDC-S/SKE-R** & **Deviation** & **EDC-S/SKE** & **Deviation** & **EDC-S/S/RSM** & **Deviation** & **Deviation** \\ \hline Max & & & & & & & & & & & \\ Temperature/axis & 1945 & 2040 & 4.88\% & 2090 & 7.4\% & 2026 & 4.16\% & 2031 & 4.41\% & 2083 & 7.08\% \\ Location/\(d\) = & 45.00 & 34.74 & & 52.79 & 37.83 & & 37.83 & & 44.75 & \\ Max Turbulent & 52.7 & 66.07 & 25.68\% & 37.58 & \(-\)28.52\% & 64.12 & 21.98\% & 60.87 & 15.80\% & 57.50 & 9.38\% \\ Energy/axis & & 26.28 & 20.30 & 31.86 & & 24.42 & & 26.72 & & 29.20 & \\ Min Y/axis & 0.019 & 0.009 & \(-\)51.51\% & 0.003 & \(-\)82.82\% & 0.010 & \(-\)48.76\% & 0.009 & \(-\)51.56\% & 0.006 & \(-\)69.93\% \\ Location/axis, \(d\) = & 45.00 & 29.20 & 44.75 & & 34.74 & & 34.74 & & 41.16 & \\ Max Y/axis/axis & 0.116 & 0.121 & 3.96\% & 0.124 & 6.71\% & 0.119 & 2.84\% & 0.119 & 2.57\% & 0.123 & 6.02\% \\ Location/axis, \(d\) = & 45.00 & 29.20 & 44.75 & & 31.86 & & 34.74 & & 37.83 & \\ Max Y/Caxis & & 0.108 & 0.114 & 5.27\% & 0.119 & 10.59\% & 0.112 & 3.42\% & 0.113 & 4.17\% & 0.118 & 8.95\% \\ Location/axis, \(d\) = & 50.000 & 34.74 & & 57.27 & & 37.83 & & 41.16 & & 48.62 & \\ Max Y/Caxis & & 0.0453 & 0.0514 & 13.57\% & 0.088 & 29.88\% & 0.0529 & 16.73\% & 0.0523 & 15.39\% & 0.0565 & 24.62\% \\ Location/axis, \(d\) = & 40.00 & 26.72 & & 41.16 & & 31.86 & & 31.86 & & 37.83 & \\ Max Y/axis & & 0.00288 & 0.00290 & 0.69\% & 0.0036 & 20.21\% & 0.00287 & \(-\)0.35\% & 0.00285 & \(-\)0.96\% & 0.00343 & 19.07\% \\ Location/axis, \(d\) = & 4.00 & 26.72 & & 41.16 & & 29.20 & & 29.20 & & 34.74 & \\ Max Y/Offaxis & & 0.00148 & 0.00414 & 197.74\% & 0.00389 & 162.92\% & 0.00433 & 192.36\% & 0.00426 & 187.95\% & 0.00401 & 171.18\% \\ Location/axis, \(d\) = & 50.00 & 34.74 & & 52.79 & & 37.83 & & 37.83 & & 44.75 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Deviations of predicted and measured values for different turbulence models with GRI-Mech 3.0 (deviation = (prediction – measurement)/measurement \(\times\) 100%). Taking the temperature peak of experiment as the demarcation point in Figure 4a, before this point, the temperature profile predicted by SKE-R was consistent well with the range of experiment data. However, after this point, the temperature was largely over-predicted. Only the temperature profile predicted by RSM reasonably agreed with the experiment data, and the peak position was properly predicted at \(x/d\)-45. All the temperature peaks were slightly over-predicted, as shown in Table 3, and the largest was about 145 K by SKE-R.