# Optimization of Airflow Organization in Bidirectional Air Supply Data Centers in China
## 2 Materials and Methods
### Numerical Simulation  
#### 2.2.1 Governing Equations  
All fluid flow processes are governed by the conservation laws of mass, momentum, and energy. The corresponding mathematical models for these physical laws are given by the governing Equations (1)-(5). For incompressible fluids, the mass conservation equation is expressed as follows [10]:  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial \omega}{\partial z}=0 \tag{1}\]  
The momentum conservation equations are formulated as follows [10]:  
\[\frac{\partial u}{\partial t}+\mu\frac{\partial u}{\partial x}+v\frac{\partial u }{\partial y}+\omega\frac{\partial u}{\partial z}=-\frac{1}{\rho}\frac{ \partial p}{\partial x}+\mu(\frac{\partial^{2}u}{\partial x^{2}}+\frac{ \partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}})+S_{u} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Range** & **Accuracy** \\ \hline FLUKE 925 Wind speed recorder (FLUKE, & & \\ Everett, DC, USA) & 0.40 m/s\(-\)25.00 m/s & \(\pm\)0.01 m/s \\ FLUKE 971 Temperature recorder (FLUKE, & & \\ Everett, DC, USA) & \(-\)20 \({}^{\circ}\)C\(-\)60 \({}^{\circ}\)C & \(\pm\)0.5 \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of instruments. Figure 2: Layout of measuring points. (**a**) Temperature measurement point distribution for rack rows (exemplified by Row AA). (**b**) Temperature measurement point distribution for single rack. (**c**) Velocity measurement points for perforated tile. \[\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{ \partial v}{\partial y}+\omega\frac{\partial v}{\partial z} =-\frac{1}{\rho}\frac{\partial p}{\partial y}+\mu(\frac{\partial^{2 }v}{\partial x^{2}}+\frac{\partial^{2}v}{\partial y^{2}}+\frac{\partial^{2}v}{ \partial z^{2}})+S_{v} \tag{3}\] \[\frac{\partial\omega}{\partial t}+u\frac{\partial\omega}{\partial x }+v\frac{\partial\omega}{\partial y}+\omega\frac{\partial\omega}{\partial z} =-\frac{1}{\rho}\frac{\partial p}{\partial z}+\mu(\frac{\partial^{2} \omega}{\partial x^{2}}+\frac{\partial^{2}\omega}{\partial y^{2}}+\frac{ \partial^{2}\omega}{\partial z^{2}})+S_{\omega} \tag{4}\]  
The energy conservation equation is expressed as follows [10]:  
\[\rho(\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{ \partial T}{\partial y}+\omega\frac{\partial T}{\partial z}=\frac{\lambda}{c _{p}}(\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2 }}+\frac{\partial^{2}T}{\partial z^{2}})+S_{T} \tag{5}\]  
where \(\rho\) is the fluid density, kg/m\({}^{3}\); \(u\), \(v\), and \(\omega\) represent the velocity vector components in the Cartesian coordinate system's x, y, and z directions, respectively (m/s); \(p\) is the time-averaged pressure, Pa; and \(S_{u}\), \(S_{v}\), and \(S_{\omega}\) signify the generalized source terms in the Cartesian coordinate system. For incompressible homogeneous fluids, these terms can be neglected; \(T\) is temperature, K; \(\lambda\) is transfer coefficient, W/(m\({}^{2}\cdot\)K); \(c_{p}\) is specific heat capacity of air, J/(kg\(\cdot\)K); and \(S_{T}\) is the viscous dissipation term. Due to the complex geometry, equipment layout, and boundary conditions in data centers, the airflow organization within server rooms typically exhibits turbulent flow characteristics. In numerical simulations of data center environments, the standard k-\(\varepsilon\) turbulence model demonstrates good applicability and has been widely adopted, as represented by Equations (6)-(8).