## 4 Temperature and Humidity Fitting Model for Stone Burial Relics
### CFD Airflow Modeling
#### 4.3.2 Turbulence Modeling  
Turbulence modeling is extremely complex, and it is difficult to fully explain turbulent flow patterns, either by theoretical calculations or experimental studies. Existing turbulence numerical simulation methods can be classified into direct and indirect numerical simulation methods. Non-direct numerical simulation methods mainly include turbulent transport coefficient simulation and large-eddy simulation methods.  
The turbulence research object in this paper is the air flow inside the chamber, which belongs to the low-speed turbulent motion category of fluid flow. Turbulence transport coefficient simulation with high accuracy, low computational cost and high reliability was chosen as the solution method in this paper.  
Time-averaged equation  
The turbulent viscosity coefficient \(\mu_{T}\) proposed by [PERSON] is introduced, and the following assumptions are made:  
\[-\rho\overline{u_{i}u_{j}}=\mu_{T}\left(\frac{\partial\overline{u_{i}}}{ \partial x_{j}}+\frac{\partial\overline{u_{j}}}{\partial x_{i}}\right) \tag{11}\]  
The time-averaged momentum transport equation can be obtained as:  
\[\frac{\partial}{\partial t}(\rho\overline{u_{i}})+\frac{\partial\big{(}\rho \overline{u_{i}u_{j}}\big{)}}{\partial x_{j}}=-\frac{\partial\overline{\rho} }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\Bigg{[}(\mu+\mu_{T})\frac{ \partial\overline{u_{i}}}{\partial x_{j}}\Bigg{]}+S_{i} \tag{12}\]  
For all variables, the time-averaged transport equation exhibits the following generic form:  
\[\frac{\partial\big{(}\rho\overline{\phi}\big{)}}{\partial t}+\frac{\partial \big{(}\rho\overline{u_{i}}\overline{\phi}\big{)}}{\partial x_{j}}=\frac{ \partial}{\partial x_{j}}\Bigg{[}(\Gamma+\mu_{T})\frac{\partial\big{(}\rho \overline{\phi}\big{)}}{\partial x_{j}}\Bigg{]}+S \tag{13}\]  
2. Standard k-\(\epsilon\) model  
The analysis of the time-averaged equations indicates that the key to calculating turbulent flow is determining \(\mu_{T}\). Many scholars have proposed many solution methods for this purpose. Through the validation study of [PERSON] et al. [23] and [PERSON] et al. [24] it can be concluded that the \(k-\epsilon\) model achieves a lower computational cost and higher computational accuracy, so the \(k-\epsilon\) model proposed by [PERSON] [23; 24]. The turbulent dissipation rate is denoted in the model and can be defined as follows:  
\[\epsilon=\frac{\mu}{\rho}\overline{\left(\frac{\partial u_{i}}{\partial x_{k} }\right)\left(\frac{\partial u_{i}}{\partial x_{k}}\right)} \tag{14}\]  
The turbulent viscosity coefficient \(\mu_{T}\) can be expressed as a function of the turbulent energy \(k\) and the turbulent dissipation rate \(\epsilon\).  
\[\mu_{T}=\rho C_{\mu}\frac{k^{2}}{\epsilon} \tag{15}\]  
Notably, \(C_{\mu}=0.99\).  
The turbulent kinetic energy \(k\) and turbulent kinetic energy dissipation rate \(\epsilon\) can be obtained as follows:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_ {i})=\frac{\partial}{\partial x_{i}}\Bigg{[}\left(\mu+\frac{\mu_{t}}{\sigma_{ k}}\right)\frac{\partial k}{\partial x_{j}}\Bigg{]}+G_{k}-\rho\epsilon \tag{16}\]  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\frac{\partial}{\partial x_{i}}( \rho\epsilon u_{i})=\frac{\partial}{\partial x_{j}}\Bigg{[}\left(\mu+\frac{ \mu_{t}}{\sigma_{\epsilon}}\right)\frac{\partial\epsilon}{\partial x_{f}} \Bigg{]}+C_{1\epsilon}\frac{\epsilon}{k}G_{k}-C_{2\epsilon}\rho\frac{\epsilon ^{2}}{k} \tag{17}\]  
where \(G_{k}\) is the generating term for the turbulent kinetic energy \(k\) due to the mean velocity gradient.  
\[G_{k}=\mu_{t}S^{2} \tag{18}\]  
\[S=\sqrt{2S_{ij}S_{ij}} \tag{19}\]  
\[S_{ij}=\frac{1}{2}\bigg{(}\frac{\partial u_{i}}{\partial x}+\frac{\partial u_ {j}}{\partial x}\bigg{)} \tag{20}\]Based on experimental validation by [PERSON] et al. and later researchers, the empirical constants of the model take the following values [25]:  
\[C_{1\epsilon}=1.44,\ C_{2\epsilon}=1.92,\ \sigma_{k}=1.0,\ and\ \sigma_{ \epsilon}=1.3\]