# A Novel Approach of Unit Conversion in the Lattice Boltzmann Method
## 3 Results and Discussions
### Two-Dimensional Examples
#### 3.2.2 2D Differentially Heated Square Cavity  
A square cavity filled with air at a room temperature (298.15 K) is driven by natural convection. The cavity has 0.02 m in length. The air properties at room temperature are summarized in Table 8. The west wall of the cavity is heated to a temperature of 1549.27 K. This value is selected to ensure \(\text{Ra}=10^{6}\) for comparison with the benchmark solution. The east wall is kept at room temperature. The other boundaries are adiabatic. In such problems, the buoyancy force generated by the heated wall drives the flow inside the cavity [4; 11; 29; 30; 31]:  
\[\mathbf{F}=\rho\mathbf{g}\beta\left(T-T_{ref}\right), \tag{96}\] \begin{table}
\begin{tabular}{c c c} \hline \multicolumn{3}{c}{**Reference Variables in Physical-Scale and Lattice-Scale**} \\ \hline
**Reference Variables** & **Physical-Scale** & **Lattice-Scale** \\ \hline \(m_{o}\) & \(9.6\times 10^{-3}\) kg & \(2.49\times 10^{6}\) mu \\ \hline \(x_{o}\) & 0.2 m & 135.45 lu \\ \hline \(t_{o}\) & 25 s & 2708.99 ts \\ \hline \multicolumn{3}{c}{Converting properties between scales} \\ \hline Property & Physical-scale & Dimensionless-scale & Lattice-scale \\ \hline Gap (\(H\)) & 0.2 m & 1 & 135.45 lu \\ \hline Length (\(L\)) & 1.5 m & 7.5 & 1015.87 lu \\ \hline Velocity (\(V\)) & 0.008 m/s & 1 * & 0.05 * lu.ts\({}^{-1}\) \\ \hline Pressure drop (\(\partial P/\partial x\)) & \(-4.54\times 10^{-5}\) Pa/m & \(-0.1181\) & \(-2.18\times 10^{-6}\) mu.lu\({}^{-2}\).ts\({}^{-2}\) \\ \hline Density (\(\rho\)) & 1.2 kg/m\({}^{3}\) & 1 * & 1 * m.lu\({}^{-3}\) \\ \hline Dynamic viscosity (\(\mu\)) & \(1.89\times 10^{-5}\) kg/m.s & \(9.844\times 10^{-3}\) & 0.0667 mu.lu\({}^{-1}\).ts\({}^{-1}\) \\ \hline Kinematic viscosity (\(\
u\)) & \(1.58\times 10^{-5}\) m\({}^{2}\)/s & \(9.844\times 10^{-3}\) & 0.0667 lu\({}^{2}\).ts\({}^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 7: Converting data for flow in a channel problem. The values with * represent assumed values. Figure 9: Fully developed velocity profile; (**a**) solutions at the lattice-scale, and (**b**) solutions at the physical-scale. where \(T_{ref}\) is the reference temperature, which equals the average temperature between west and east walls. In the LBM simulation, the following relaxation times related to momentum and temperature equations are applied:  
\[\tau_{f}=0.7, \tag{97}\]  
Figure 10: Effect of the lattice Mach number (\(Ma_{(LS)}\)) on the accuracy of the LBM. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Reference Variables in Physical-Scale and Lattice-Scale**} \\ \hline
**Reference Variables** & **Physical-Scale** & **Lattice-Scale** \\ \hline \(m_{o}\) & \(9.6\times 10^{-6}\) kg & \(6.4\times 10^{7}\) mu \\ \hline \(x_{o}\) & 0.02 m & 400 lu \\ \hline \(t_{o}\) & 18.6916 s & \(1.7495\times 10^{6}\) ts \\ \hline \(\Theta_{o}\) & 1549.27 K & 1 * tu \\ \hline \multicolumn{3}{c}{Converting properties between scales} \\ \hline Property & Physical-scale & Dimensionless-scale & Lattice-scale \\ \hline Length (\(l\)) & 0.02 m & 1 & 400 * lu \\ \hline Room temperature (\(T_{room}\)) & 298.15 K & 0.1924 & 0.1924 tu \\ \hline West wall temperature (\(T_{west}\)) & 1549.27 K & 1 & 1 tu \\ \hline Density (\(\rho\)) & 1.2 kg/m\({}^{3}\) & 1 * & 1 * mu.lu\({}^{-3}\) \\ \hline viscosity (\(\upsilon\)) & 1.56 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/s & 7.2897\(\times\)10\({}^{-1}\) & \(6.6667\times 10^{-2}\) lu\({}^{2}\).ts\({}^{-1}\) \\ \hline Thermal diffusivity (\(\alpha\)) & 2.14 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/s & 1 * & \(9.1453\times 10^{-2}\) lu\({}^{2}\).ts\({}^{-1}\) \\ \hline Coefficient of expansion (\(\beta\)) & 3.4 \(\times\) 10\({}^{-3}\) K\({}^{-1}\) & 5.2675 & 5.2675 tu\({}^{-1}\) \\ \hline Gravity (\(g\)) & 9.81 m/s\({}^{2}\) & \(1.7137\times 10^{5}\) & \(2.2395\times 10^{-5}\) lu.ts\({}^{-2}\) \\ \hline Magnitude velocity (\(u\)) & 0.9136 m/s & 853.7985 & 0.1952 lu.ts\({}^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Converting data for the differentially heated square cavity example.