## 4 Results  
### Validation on an Air-Piloted Jet Flame (Sandia Flame D)  
In this section, the Sandia flame D is used to support the evaluation of the performance of the LES hybrid FESF/FPV. This is achieved in terms of comparison between numerically obtained results with experimental data for various radial profiles of some scalar quantities, scatter plots and probability density function at three axial positions, z/d = 7.5; 15 and 30. Thereby, different simulations using different numbers of Stochastic Fields (SFi = 4, 6, 8 and 16) are presented, in order to assess the convergence of the number of SFi. Time mean and root mean square (rms) of both axial and radial velocity are displayed in Figure 4. For the axial velocity, the obtained radial profile results feature a good overall agreement with experimental data at different positions for both mean (solid line) and rms quantities (dashed line). However, some under-estimations are noticed in reproducing the behavior of the radial velocity at z/d = 15 while an over-prediction is observed at z/d = 30. By applying various numbers of stochastic fields, it turns out that the mean quantities of the radial velocity obtained with 8 SFi are predicted better at all axial positions. Figure 5 depicts the radial profiles of temperature and mixture fraction at different axial positions. Regarding the mixture fraction evolution, numerical results indicate an excellent agreement for both mean and rms quantities at the axial position z/d = 7.5, and a slight under-estimation at the middle axial location further from the centerline without exhibiting any differences once using varied stochastic field numbers. However, the mean temperature at the centerline was slightly over-predicted at both positions z/d = 7.5 and 15 in Figure 5a, independently from the number of stochastic fields. This over-prediction was mostly pronounced at the centerline only at the further downstream positions, which is explained by the under-estimation of mixture fraction at same axial position. The influence of using different stochastic field numbers is revealed by comparing the rms quantities of the temperature. It appears clearly that the use of 8 SFi allows for an overall better prediction. Such a SFi number agrees with that found by [PERSON] et al. [31]. Further, the reported results proved the efficiency of the suggested novel method once compared to available measurements and to other previous numerical studies in which different flamelet models coupled to the FPV approach have been adopted. In fact, regarding the different computational setups employed in the case of Sandia Flame-D, [PERSON] et al. in [69] used the statistically most likely distribution model (SMLD) coupled to the FPV technique and found the mixture fraction profiles at different axial positions rather slightly under-predicted near the centerline. In [15], the mean temperature profiles were slightly over-predicted in some axial positions far from the centerline. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Flame Jet** & \(f\) & **PV** & **T (k)** & **U\({}_{\text{b}}\) (m/s)** & **v (m2/s)** \\ \hline Fuel jet & 1 & 0 & 300 & 117.8 & 3.271 \(\times\) 10\({}^{-5}\) \\ Oxidizer jet & 0 & 0 & 300 & 0.933 & 3.271 \(\times\) 10\({}^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Conditions for the oxy-fuel jet Flame B3. Figure 3: Flame B3 burner, (**a**) Description of the experimental configuration (**b**) The numerical configuration generated, (**c**) 3D structured grid used with a zoom on the fuel inlet nozzle region where smoother grid is applied. Figure 4: Comparison between experimental data for Sandia Flame-D and large eddy simulation (LES) hybrid Eulerian stochastic field (ESF)/flamelet progress variable (FPV) numerical results for different numbers of stochastic fields (SFi) at different axial positions, z/d = 7.5; 15 and 30. Radial profiles of mean and rms of (**a**) axial velocity, (**b**) radial velocity.