# 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  
Converting the quantities' units of a certain problem from/to physical-scale to/from lattice-scale is not difficult. However, it needs careful treatment that ensures a correct and consistent scaling of all quantities. Thus, the current approach suggests using basic reference parameters of the physical-scale and lattice-scale in the converting process. Figure 2 shows a flowchart of the unit conversion methodology of the present approach. The basic reference parameters of the physical-scale (\(x_{o(PS)},~{}t_{o(PS)},~{}m_{o(PS)}\), and \(\Theta_{o(PS)}\)) are applied to convert properties units to the dimensionless-scale. Those properties at the dimensionless-scale are converted to the lattice-scale using the basic reference parameters of the lattice-scale (\(x_{o(LS)},~{}t_{o(LS)},~{}m_{o(LS)}\), and \(\Theta_{o(LS)}\)) and vice versa. It is better to illustrate the methodology with an example. Assume natural convection in a cavity filled with air at room temperature (298 K) has 0.02 m and 0.04 m in high (H) and in length (L), respectively. The air properties at room temperature are 1.2 kg/m\({}^{3}\) of density (\(\rho\)), 1.56 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/s of viscosity (\(v\)), 2.14 \(\times\) 10\({}^{-5}\) m\({}^{2}\)/s of thermal diffusivity (\(a\)), and 3.4 \(\times\) 10\({}^{-3}\) K\({}^{-1}\) of the coefficient of expansion (\(\beta\)). The west wall is heated to a temperature of 500 K, while the east wall is kept at room temperature. The bottom and upper boundaries are subjected to adiabatic conditions. Figure 3 displays the schematic diagram of the problem. Figure 3: Schematic diagram of the differentially heated cavity. Figure 2: Flowchart of unit conversion approach. The dimensionless governing equations (Navier-Stokes equations) for natural convection can be written as:  
Continuity equation:  
\[\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0, \tag{21}\]  
\(\mathrm{x}\)-momentum equation:  
\[U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P }{\partial X}+\frac{1}{Re}\left(\frac{\partial^{2}U}{\partial X^{2}}+\frac{ \partial^{2}U}{\partial Y^{2}}\right), \tag{22}\]  
\(\mathrm{y}\)-momentum equation:  
\[U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{ \partial P}{\partial Y}+\frac{1}{Re}\left(\frac{\partial^{2}V}{\partial X^{2} }+\frac{\partial^{2}V}{\partial Y^{2}}\right)+\frac{Gr}{Re^{2}}\;\theta,\; \mathrm{and} \tag{23}\]  
Energy equation:  
\[U\frac{\partial\theta}{\partial X}+V\frac{\partial\theta}{\partial Y}=\frac{ 1}{Re\;Pr}\bigg{(}\frac{\partial^{2}\theta}{\partial X^{2}}+\frac{\partial^{ 2}\theta}{\partial Y^{2}}\bigg{)}, \tag{24}\]  
where  
\[X=\frac{x}{H},\;Y=\frac{y}{H}, \tag{25}\]  
\[U=\frac{u}{u_{o}},\;V=\frac{v}{u_{o}}, \tag{26}\]  
\[P=\frac{p}{\rho u_{o}^{2}},\;\mathrm{and}\;\theta=\frac{T-T_{\mathrm{\textit{ vest}}}}{T_{\mathrm{\textit{e}ast}}-T_{\mathrm{\textit{vest}}}}. \tag{27}\]  
The controlling parameters for the example are summarized in Table 2. As the LBM is a pseudo-compressible method, the magnitude of \(u\) should be ensured that the Mach number is within the limit of incompressible flow (\(Ma<0.3\)). The formulas and values of \(u\) and \(Ma\) are listed in Figure 2:0  
To convert the problem to the lattice-scale, the next steps should be followed:  
**Physical-scale:** In the data reduction process, there is a need to select the basic reference variables of the physical-scale of the problem as the characteristic scales. Therefore, \(x_{o(PS)}=0.02\) m and \(\Theta_{o(PS)}=500\) K are chosen as the reference variables for length and temperature, respectively. Furthermore, the reference of time (\(t_{o(PS)}\)) and mass (\(m_{o(PS)}\)) scales can be estimated by selecting the dimensionless parameters, \(\rho^{*}\) and \(u^{*}\), which are the dimensionless density and velocity, respectively. As free parameters, any values for \(\rho^{*}\) and \(u^{*}\) can be selected. Let us set \(\rho^{*}=u^{*}=1\) for simplicity.