# A Novel Approach of Unit Conversion in the Lattice Boltzmann Method
## 1 Introduction  
The problems encountered in engineering and sciences are either macroscopic, mesoscopic, microscopic in nature, or a combination of them, see Figure 1. At the macroscopic level, the problems are a continuum. Whereas, at the microscopic level, the material is treated as soft or solid particles. The lattice Boltzmann method (LBM) is based on meso-scales. The LBM became a very popular method since its birthday in late 1989. One of the main advantages of the LBM is its ability to incorporate the microscopic or mesoscopic physics while recovering the macroscopic laws at an affordable computational cost [1, 2, 3, 4]. Therefore, converting or mapping quantities' units from/to the macroscopic scale (physical-scale) to/from the mesoscopic scale (lattice-scale) should correctly be performed. The geometric and dynamics similarity conditions must be conserved between the physical-scale and lattice-scale. The most popular method is to match the dimensionless parameters (such as Reynolds number, Rayleigh number, etc.) besides the geometrical similarity, such as aspect ratio, i.e., the Buckingham \(\pi\) theorem. For single-phase and single component flows and heat and mass transfer, the controlling dimensionless parameters are few and easy to match. However, the problem of using similarity transformation for multi-phase flows and transports is not straightforward. In addition, the same argument equally applies to problems with variable thermo-physical properties and non-Newtonian flow. For completeness of the topic, a short introduction will be discussed on the currentmethods used for mapping processes from the physical domain to the lattice domain and vice versa, especially for multi-phase problems. The Buckingham \(\pi\) theorem is a well established method to reduce the number of experimental variables [5; 6; 7; 8; 9]. Numbers of \(\pi\) groups (i.e., dimensionless numbers, e.g., Reynolds number (Re), Nusselt number (Nu), Weber number (We)) are based on the degrees of freedom between the problem's quantities and primary units. [PERSON] et al. [5] presented a few examples to show how the Buckingham \(\pi\) theorem can be used to match the physical-scale with the lattice-scale. In a capillary rise example, they set a capillary slit's width of 0.002 m. Water with a density \(\rho_{water}=1000\) kg/m\({}^{3}\) and a surface tension \(\sigma=72.13\times 10^{-3}\) N/m is overlain by air with a density \(\rho_{air}=1.23\) kg/m\({}^{3}\). They estimated the Bond number (\(Bo\)) to be 1/7.36. The Bo number is  
\[Bo=\frac{r^{2}(\rho_{water}-\rho_{air})g}{\sigma}, \tag{1}\]  
where \(g=9.8\) m/s\({}^{2}\). In their LBM simulation, they assumed that the system is at a temperature of 0.177 tu, where tu is the unit of temperature in the lattice-scale. The Redlich-Kwong (R-K) Equation of State (EoS) was used and they found that the corresponding coexisting densities of that temperature are 5.44 and 0.81 mu/lu\({}^{3}\) of water and air, respectively. mu and lu are the units of mass and length in the lattice-scale, respectively. Using the Shan-Chen (SC) [6; 7] model, the corresponding surface tension is estimated to be \(\sigma=0.096\) when \(\tau=1\), where \(\tau\) is the dimensionless relaxation time. Accordingly, they estimated the gravity to be \(g=3.84\times 10^{-6}\) lu/ts\({}^{2}\), where ts is the unit of time in the lattice-scale. They assumed that the maximum capillary rise is 200 lu, while the LBM simulation predicts 218.4 lu. They stated that the discrepancy is due to the compressibility in the SC [6; 7] model. It is not easy to find a required number of dimensionless numbers to map the quantities' units from/to the physical-scale to/from the lattice-scale for some problems by the Buckingham \(\pi\) theorem. Moreover, as seen in the above example, the Buckingham \(\pi\) theorem method depends on assuming some parameters' values, which is increase the numerical effort.