# A Novel Approach of Unit Conversion in the Lattice Boltzmann Method
## 2 Unit Conversion from/to Physical-Scale to/from Lattice-Scale
### Boltzmann Transport Equation and Its Dimensionless Form  
The Bhatnagar-Gross-Krook (BGK) [18] Boltzmann transport equation can be written as [4; 19; 20; 21; 22]:  
\[\frac{\partial f}{\partial t}+\mathbf{e}\cdot\
abla\ f+\mathbf{a}\cdot\
abla_{e}\ f= \frac{1}{\lambda}\Big{(}f^{M}-f\Big{)}, \tag{8}\]  
where \(f=f(\mathbf{x},\mathbf{e},t)\) is the particle distribution function, \(\mathbf{e}\) is the microscopic velocity vector, \(\mathbf{a}\) is the acceleration, \(\lambda\) is the relaxation time, and \(f^{M}\) is the Maxwell-Boltzmann distribution function which is given as [4; 19; 20; 21; 22]  
\[f^{M}=\frac{\rho}{\left(2\pi RT\right)^{D/2}}\exp\left[\frac{-\{(\mathbf{e}-\mathbf{u} )\cdot(\mathbf{e}-\mathbf{u})\}}{2 RT}\right], \tag{9}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Primary Dimension** & **SI Unit** & **Lattice Unit** \\ \hline Mass [\(M\)] & kilogram (kg) & mass unit (mu) \\ Length [\(L\)] & meter (m) & length unit (lu) \\ Time [\(t\)] & second (s) & time unit (ts) \\ Temperature [\(\Theta\)] & Kelvin (K) & temperature unit (tu) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Primary dimensions in SI and lattice system units. where \(\rho\) is the density, \(\mathbf{u}\) is the macroscopic velocity, \(R\) is the gas constant, \(T\) is the temperature, and \(D\) is the number of spatial dimensions. The particle distribution function (\(f\)) has the same unit of \(f^{M}\). The exponential term in Equation (9) is non-dimensional. Therefore, the unit of the \(f\) is the unit of \(\frac{\rho}{(RT)^{D/2}}\). Using dimensional analysis yields:  
\[\{f\}=\Big{\{}M\cdot L^{-3}\cdot t^{\rm D}\cdot L^{-D}\Big{\}}. \tag{10}\]  
It is interesting to notice that the unit of \(f\) gives a physical insight about its meaning. The \(f\) is density in a hyperspace. The density can be found by using the multiple integrals of \(f\) over the velocity  
\[\rho=\int fd\mathbf{e}, \tag{11}\]  
where the integration is single, double, or triple for 1D, 2D, or 3D problems, respectively. [PERSON] [19] used the characteristic length scale, the reference speed, the reference density, and the time between particle collisions to non-dimensionalize the Boltzmann transport equation, Equation (8). The current work uses unknown reference basic parameters to non-dimensionalize Equation (8) as:  
\[\frac{\partial f^{*}}{\partial t^{*}}+\mathbf{e}^{*}\cdot\
abla^{*}\,f^{*}+\mathbf{a} ^{*}\cdot\
abla_{e}^{*}\,f^{*}=\frac{1}{\lambda^{*}}\left(f^{M,*}-f^{*}\right), \tag{12}\]  
where  
\[f^{*}=f\frac{x_{o}^{3}}{m_{o}}\frac{x_{o}^{D}}{t_{o}^{D}}\,, \tag{13}\]  
\[t^{*}=\frac{t}{t_{o}}, \tag{14}\]  
\[\mathbf{e}^{*}=\mathbf{e}\frac{t_{o}}{x_{o}}, \tag{15}\]  
\[\
abla^{*}=x_{o}\
abla, \tag{16}\]  
\[\mathbf{a}^{*}=\mathbf{a}\frac{t_{o}^{2}}{x_{o}}, \tag{17}\]  
\[\
abla_{e}^{*}=\frac{x_{o}}{t_{o}}\
abla_{e}, \tag{18}\]  
and  
\[\{f\}=\Big{\{}M\cdot L^{-3}\cdot t^{\rm D}\cdot L^{-D}\Big{\}}. \tag{10}\]0  
where \(m_{o}\), \(x_{o}\), and \(t_{o}\) are unknown references of mass, length, and time, respectively. Those basic parameters are chosen based on the problem to be solved, as will be presented in the upcoming section. Moreover, those basic parameters including temperature, \(\Theta_{o}\), can be used to non-dimensionalize the controlling parameters, such as density, velocity, specific heat capacity, viscosity, etc. The present work's primary key is to take advantage of transferring the units from one-unit scale to another using the basic reference parameters. Hence, we will have better control of the relaxation time and characteristic velocity of the LBM, which improves the accuracy and stability of the simulation of the given problem. In the following section, the methodology of converting the units of quantities from/to the physical-scale to/from the lattice-scale will be explained in detail.