# Application of Tucker Decomposition in Temperature Distribution Reconstruction
## 4 Simulation Experiment
### Numerical Simulation Calculation Results Analysis  
Sandia Labs has released a measurement data set of Flame D, taking the cross-sections formed by +z and +x in the model as the research object. Compare the simulation data with the measured data to analyze the accuracy of the simulation.  
The comparison between the simulation calculation data of the \(z\)-axis observation point and the temperature measurement data of the corresponding position is shown in the following figure.  
It can be seen from Figure 11 that there is a certain deviation between the simulation results and the measured data. Analyzing the influencing factors, the error between the calculation result and the measured data mainly comes from the solution process and the model building process. Since the grid independence analysis has been carried out, the influence of grid division on the calculation results can be ignored, so the error in the solution process mainly comes from the discrete process of the governing equations. In the process of model establishment, the selection of turbulence model, chemical reaction model and parameter setting, as well as the simplification and assumption in the process of model building, are the main factors that cause the deviation between the calculated results and the measured data.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter Name** & **Fuel** & **Heat Flow** & **Air** \\ \hline Velocity (m/s) & 49.6 & 11.4 & 0.9 \\ Pressure (atm) & 0.993 & 0.993 & 0.993 \\ Temperature (K) & 294 & 1880 & 291 \\  & & N\({}_{2}\): 0.7342 & \\  & & O\({}_{2}\): 0.054 & \\ Component Proportion(\%) & \
\begin{tabular}{c} CH\({}_{4}\): 0.1575 \\ air: 0.1575 \\ \end{tabular} &
\begin{tabular}{c} H\({}_{2}\)O: 0.0942 \\ CO: 0.00407 \\ CO\({}_{2}\): 0.1098 \\ \end{tabular} & \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of Flame D [32].  
Figure 10: Geometry of the simulation model.  
Nevertheless, as shown in Figure 11, the temperature calculated values at different positions are basically consistent with the general trend of the actual temperature distribution, that is, the simulation calculation results are close to the actual measured values.  
In addition, the \(z\)-axis coordinates are taken as 30 d, 45 d, 60 d and 75 d for the radial observation point temperature data comparison as follows.  
Similar to the conclusion in Figure 11, the calculated temperature values at different radial positions shown in Figure 12 are basically consistent with the overall trend of the actual temperature distribution. Therefore, the simulation model can better simulate the temperature distribution of Flame D.  
Figure 11: Temperature comparison in \(z\)-axis.  
Figure 12: Temperature comparison in different locations: (**a**) Temperature comparison in z = 30 d; (**b**) Temperature comparison in z = 45 d; (**c**) Temperature comparison in z = 60 d; (**d**) Temperature comparison in z = 75 d.  
In summary, although the numerical calculation method has approximations and assumptions that cannot be avoided in the process of solving and model building, it can still accurately simulate the basic properties of the simulation object and meet the actual needs of engineering applications. The methane combustion simulation model established in this paper can simulate the real temperature distribution of Flame D accurately.