# A Novel Approach of Unit Conversion in the Lattice Boltzmann Method
## 2 Unit Conversion from/to Physical-Scale to/from Lattice-Scale
### Methodology of Quantities Converting
The subscript \(PS\) refers to the physical-scale and the superscript \(*\) refers to the dimensionless-scale. Using the dimensionless density and velocity, the other reference parameters (\(m_{o(PS)}\) and \(t_{o(PS)}\)) can be calculated as:  
Figure 2:1 \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Formula** & **Value** \\ \hline Prandtl number & \(Pr=\frac{v}{\theta}\) & 0.73 \\ \hline Grashof number & \(Gr=\frac{\theta\beta\Delta TH^{2}}{v^{2}}\) & \(2.2145\times 10^{5}\) \\ \hline Reynold number & \(Re=\sqrt{Gr}\) & 470.6196 \\ \hline Velocity & \(u=\sqrt{\beta\beta\Delta TH}\) & 0.3671 m/s \\ \hline Mach number & \(Ma=\frac{u}{\mathcal{C}}\) & \(1.07\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
and  
\[t_{o(PS)}=\frac{u^{*}}{u}x_{o(PS)}=0.0545\,\mathrm{s}. \tag{29}\]  
**Dimensionless-scale:** Using the mentioned reference variables of the physical-scale, the non-dimensionalized parameters are as follows:  
\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]  
\[L^{*}=\frac{L_{(PS)}}{x_{o(PS)}}=2, \tag{31}\]  
\[C^{*}=C\frac{t_{o(PS)}}{x_{o(PS)}}=934.3928, \tag{32}\]  
\[\
u^{*}=\
u_{(PS)}\frac{t_{o(PS)}}{x_{o(PS)}^{2}}=2.12\times 10^{-3}, \tag{33}\]  
\[\alpha^{*}=\alpha_{(PS)}\frac{t_{o(PS)}}{x_{o(PS)}^{2}}=2.91\times 10^{-3}, \tag{34}\]  
\[g^{*}=g_{(PS)}\frac{t_{o(PS)}^{2}}{x_{o(PS)}}=1.456, \tag{35}\]  
\[\beta^{*}=\beta_{(PS)}\Theta_{o(PS)}=1.7, \tag{36}\]  
\[\Theta_{\text{{\em west}}}^{*}=\frac{\Theta_{\text{{\em nest}}(PS)}}{\Theta_{o (PS)}}=1, \tag{37}\]  
and  
\[\Theta_{\text{{\em east}}}^{*}=\Theta_{\text{{\em south}}}^{*}=\frac{\Theta_{ \text{{\em east}}(PS)}}{\Theta_{o(PS)}}=0.596. \tag{38}\]  
The dimensionless numbers will be  
\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]0  
\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]1  
\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]2  
and  
\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]3  
**Lattice-scale:** The reference variables of the lattice-scale (\(\Theta_{o(LS)}\), \(m_{o(LS)}\), \(x_{o(LS)}\), and \(t_{o(LS)}\)) can be calculated. Notice that the formula of each non-dimensional parameter expressed on the physical-scale, i.e., Equations (28)-(38), should be similar to that expressed on the lattice-scale. Hence,  
\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]4  
\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]5  
\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]6\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]7  
and  
\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]8  
where the subscript \(LS\) refers to the lattice-scale. Again, we have two free parameters, \(\rho_{(LS)}\) and \(\Theta_{o(LS)}\). Any reasonable values can be selected, usually set to be one. Other scales, \(m_{o(LS)}\), \(x_{o(LS)}\), and \(t_{o(LS)}\) can be found by manipulating Equations (43)-(47):  
\[H^{*}=\frac{H_{(PS)}}{x_{o(PS)}}=1, \tag{30}\]9  
\[L^{*}=\frac{L_{(PS)}}{x_{o(PS)}}=2, \tag{31}\]0  
and  
\[L^{*}=\frac{L_{(PS)}}{x_{o(PS)}}=2, \tag{31}\]1  
The reference lattice velocity (\(u_{(LS)}\)), lattice viscosity (\(\
u_{(LS)}\)), and lattice thermal diffusivity (\(\alpha_{(LS)}\)) are free parameters. They should be selected to satisfy stability and accuracy. It is well known that the stability of the LBM is related to the relaxations time (\(\tau_{f}\) and \(\tau_{e}\)), where \(\
u_{(LS)}=\left(\tau_{f}-0.5\right)C_{s}^{2}\Delta t\) and \(\alpha_{(LS)}=(\tau_{e}-0.5)C_{s}^{2}\Delta t\). The subscripts \(f\) and \(e\) refer to the momentum and energy equations, respectively. \(C_{s}\) is the pseudo-speed of sound of the lattice. The pseudo-speed of sound is the maximum speed needed to transfer data in the lattice, which is a function of the lattice model, for D2Q9, is equal to \(\frac{1}{\sqrt{3}}\). Hence, the values of \(\tau_{f}\) and \(\tau_{e}\) should not be close to 0.5 to satisfy stability. The restriction imposed on the lattice velocity (\(u_{(LS)}\)) is that \(Ma_{(LS)}=u_{(LS)}/C_{s}<<1\). \(Ma_{(LS)}\) is the lattice Mach number in the lattice-scale, which is different from the physical Mach number.