# Fast Temperature Field Extrapolation Under Non-Periodic Boundary Conditions
## 3 Model Description
### Non-Periodic Thermal Boundary  
The non-periodic thermal boundary concept is widespread in various scientific and engineering disciplines. In order to reveal the influence of non-periodic thermal boundary conditions on a thermal field, the flame boundary model is selected in this work. The pool of fire is the most effective scenario for modeling a flame boundary. The large eddy simulation has been applied for the flame simulation and gradient diffusion is the turbulence model used to close the scalar flux and the SGS momentum terms in the large eddy simulation. Radiation is introduced into the energy equation via the source terms, based on the Stefan-Boltzmann law. In our method, the time step is dynamically determined by an automated algorithm to simultaneously satisfy the Courant-Friedrichs-Lewy (CFL) condition governing convective stability and the Fourier number criterion for diffusive processes. Therefore, the flame boundary and the complex structure with explosives are coupled. Figure 3a shows the coupled model and the structure with explosives is completely surrounded by the pool fire. Eight temperature measurement points are arranged in the model to track the gas phase temperature of the flame, as shown in Figure 3b. The temperature of the coupled model in the pool of fire for the first 500 s is calculated, and the flame gas phase temperature of each measurement point is obtained. Figure 4 shows a graph of the temperature over time. \begin{table}
\begin{tabular}{c c c} \hline
**Material** & **Thickness/mm** &
\begin{tabular}{c} **Thermal** \\ **Conductivity/(W/m\(\cdot\)K)** \\ \end{tabular} \\ \hline Carbon-phenolic & 30 & 0.48 \\ TC11 & 10 & 7.5 \\ Air & 35 & 0.038 \\
304 stainless steel & 5 & 19.5 \\ PBX-9503 & \(\phi 100\times 200\) & 0.302 \\ \hline \end{tabular}
\end{table} 
Table 1: Thermal conductivities and thicknesses of the materials in each layer. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline \(\rho\) & 1845 kg/m\({}^{3}\) \\ \(Q\) & \(4.78\times 10^{5}\) J/kg \\ \(Z\) & 4780/s \\ \(E\) & 143, 900 J/mol \\ \(R\) & 8.314 J/(mol \(\cdot\) K) \\ \hline \end{tabular}
\end{table} 
Table 2: The specific related parameters of the PBX-9503 explosive [44]. The results show that the flame temperature on the same surface of the structure is consistent; there is a great difference in the flame temperature on different surfaces, such as when, after 300 s, the temperature of measurement point 4 is 350 K higher than that of measurement point 6. The reason for this difference in temperature on the various surfaces is the different combinations of fuel steam and oxygen. Measurement points 4 and 5 are located on the side of the structure (side surface) where the fuel steam and oxygen supply are sufficient. Full combustion leads to high temperatures. Measurement points 6, 7, and 8 are located on the upper surface of the structure (rear fire surface), and are not directly reached by the flames; therefore, the temperature is low. Measurement points 1, 2, and 3 are located on the lower surface of the structure (facing-fire surface), which are directly in contact with the flame. The combustion on the facing-fire surface is insufficient due to an inadequate oxygen supply, meaning that the temperature of the facing-fire surface is between the rear fire surface and the side surface. According to the above conclusions, the pulsating flame temperatures measured on the different surfaces can be assigned to the structure. Then, the simulation of heat transfer between the flame and the wall and the simulation of heat transfer inside the complex structure with explosives can be conducted. The heat transfer model of the structure boundary is shown in Figure 5, which is mainly divided into two parts: the thermal radiation \(\hat{q}^{{}^{\prime\prime}}_{\sigma}\) of the high-temperature flame and the heat convection \(\hat{q}^{{}^{\prime\prime}}_{\sigma}\) of combustion. Figure 4: The temperature of each measurement point with time.