A Novel Approach of Unit Conversion in the Lattice Boltzmann Method

Abstract

The lattice Boltzmann method (LBM) is an alternative method to the conventional computational fluid dynamic (CFD) methods. It gained popularity due to its simplicity in coding and dealing with a complex fluid flow such as the multiphase flow. The method is based on the kinetic theory, which is mesoscopic scale. Hence, applying the LBM method for macroscopic problems requires a proper conversion from the physical scale (conventional units) to the mesoscopic scale (lattice units) and vice versa. The Buckingham π theorem and the principle of corresponding states are the popular methods used for data reductions and unit conversion processes in the LBM. Nevertheless, those methods have some issues, such as difficulty in converting specific quantities, such as thermo-physical properties. The current work uses a novel dimensional analysis method systematically for mapping properties’ units between scales. Moreover, the approach has the flexibility in selecting parameters to ensure the stability of the method of solution. Several benchmark examples are used to evaluate the feasibility and accuracy of the proposed approach. In conclusion, the proposed approach showed the flexibility of the mapping between meso-scale to macro-scales and vice versa on solid bases rather than ad-hoc methods.

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 π 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 current methods used for mapping processes from the physical domain to the lattice domain and vice versa, especially for multi-phase problems.

The Buckingham π theorem is a well established method to reduce the number of experimental variables [5,6,7,8,9]. Numbers of π 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. Huang et al. [5] presented a few examples to show how the Buckingham π 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/m³ and a surface tension $\sigma$ = 72.13 × 10⁻³ N/m is overlain by air with a density ${\rho }_{air}=$ 1.23 kg/m³. They estimated the Bond number (Bo) to be 1/7.36. The Bo number is

$$Bo=\frac{r^2\left({\rho }_{water}-{\rho }_{air}\right)g}{\sigma },$$

(1)

where $g=$ 9.8 m/s².

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/lu³ 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 σ = 0.096 when τ = 1, where τ is the dimensionless relaxation time. Accordingly, they estimated the gravity to be $g$ = 3.84 × 10⁻⁶ lu/ts², 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 π theorem. Moreover, as seen in the above example, the Buckingham π theorem method depends on assuming some parameters’ values, which is increase the numerical effort. This effort appears in increasing the trials that needed to achieve a converge numerical solution, in other words, finding the appropriate values for assumed parameters. Therefore, the second method, i.e., the principle of corresponding states, comes to the picture to deal with those issues. The principle of corresponding states is also called the scaling method. In most numerical simulations the governing equations are known. Therefore, the non-dimensionalization process is to set a list of references for the variables (i.e., length, velocity, temperature, pressure, etc.). Those references must be known and constants. For instance in the multiphase problems, a critical state of density, volume, temperature, and pressure is taken as a reference to non-dimensionalize the property and reduce the number of parameters. For instance, the R-K EoS is as follows:

$$P=\frac{\rho RT}{1-b\rho }-\frac{a{\rho }^2}{\sqrt{T}\left(1+b\rho \right)}.$$

(2)

The equation includes six unknown parameters ($P$ is pressure, $\rho$ is density, $T$ is temperature, $R$ is gas constant, $a$ is a parameter characterizing the attraction of gas particles, and $b$ is effectively a minimum molar volume [8,9]). The variables $a$ and $b$ can be found by taking the first and second derivatives of the R-K EoS with respect to density at the critical state and equating them to zero. Solving the obtained system of equations yields:

$$a=\frac{0.42748R^2{T}_c{}^{2.5}}{P_c}=1.28157\frac{R{T}_c\sqrt{T_c}}{{\rho }_c},$$

(3)

and

$$b=0.08664\frac{R{T}_c}{P_c}=\frac{20}{77{\rho }_c},$$

(4)

where the subscript $c$ refers to the critical state. Using a critical state of each property yields the following dimensionless variables:

$$P_r=\frac{P}{P_c}, T_r=\frac{T}{T_c}, and {\rho }_r=\frac{\rho }{{\rho }_c},$$

(5)

where the subscript $r$ refers to the reduced state.

By substituting Equations (3)–(5) into Equation (2), the R-K EoS becomes:

$$P_r=\frac{T_r{\rho }_r}{0.33356-0.08664{\rho }_r}-\frac{{\rho }_r{}^2}{\sqrt{T_r}\left(0.26028+0.067604{\rho }_r\right)}.$$

(6)

The number of unknown parameters is reduced to half. It is worth mentioning that the R-K EoS is one of the most accurate EoS, which is adequate for the calculation of gas-phase properties when $\frac{P}{P_c}<\frac{T}{2T_c}$ [10].

A flow in a lid-driven cavity is another example, where the lid velocity ($u_{lid}$) is used to scale the velocity field; the height of the cavity (H) is used to scale the length, etc. The non-dimensional governing equation can be written as:

$$\frac{\partial U}{\partial t^{*}}+U.\nabla U=-\mathsf{\Delta }P+\frac{1}{Re}{\nabla }^{2}U,$$

(7)

where $t^{*}$ is non-dimensional time scaled by $H/u_{lid}$, and $Re=u_{lid}H/\nu$.

However, another example, utilizing natural convection in a differentially heated cavity, the velocity scale can be defined as either the kinematic viscosity ($\nu$) divided by the length scale or the thermal diffusion coefficient ($\alpha$) divided by the length scale. The open literature is full of examples of non-dimensionalized equations using the above-mentioned method. Note, the proper scaling is different than non-dimensionalizing. However, for numerical simulation, it does matter.

However, there are no reference states of some properties (e.g., specific heat capacity, viscosity, thermal diffusivity, coefficient of expansion, enthalpy, etc.). Usually, the researchers [5,11,12,13,14,15,16] couple the Buckingham π theorem with the principle of corresponding states (scaling) method to overcome the mentioned disadvantages. Huang et al. [17] mentioned that not every equation or variable could be easily converted by the Buckingham π theorem or scaling method. Consequently, they [17] tried to solve the whole problem by utilizing the Planck unit system. They tested their approach on a 2D convective heat transfer problem in tube banks. They used three unit systems for mapping properties; physical, Planck, and lattice unit systems. The approach is complicated and has some issues. The complication appears on their methodology to represent the variables on the unit systems. Each variable was represented seven times: once in a physical system, thrice in the Planck system, and thrice in a lattice system. For instance, the length was represented as lattice length in the physical, Planck, and lattice unit systems; Planck length in the physical, Planck, and lattice unit systems; and physical length in the physical unit system. In practice, it is preferred to solve some physical problems in the non-dimensional space. However, their approach does not provide this option, as the Planck system was used for scaling.

The literature review shows that the Buckingham π theorem and scaling are the common methods used for transferring quantities’ units between scales. However, those methods suffer some issues, such as unable to find a required number of dimensionless numbers for the mapping process and there are no reference states of some properties for the scaling process. The current article proposes a dimensional analysis approach to resolve those mentioned issues, where the scaling is performed based on the primary units.

Table 1. Primary dimensions in SI and lattice system units.

Primary Dimension	SI Unit	Lattice Unit
Mass {M}	kilogram (kg)	mass unit (mu)
Length {L}	meter (m)	length unit (lu)
Time {t}	second (s)	time unit (ts)
Temperature {Θ}	Kelvin (K)	temperature unit (tu)

summarizes the primary dimensions in SI and lattice system units. The proposed approach systematically transforms the quantities’ units from/to the physical-scale to/from the lattice-scale. It is known that the LBM is a pseudo-compressible method, which simulates incompressible flows by having a small Mach number to ensure the fluctuations of density in time are not that big. Therefore, the imposed velocity should be selected to ensure a low Mach number. In addition, the stability of the LBM is related to the relaxation time (τ), where $\nu =\left(\tau -0.5\right)C_s^2\Delta t$ ($C_s$ is the pseudo-speed of sound of the lattice). Therefore, τ is restricted to be higher than 0.5 to ensure the positivity of the viscosity ($\nu$). The proposed method has the flexibility to ensure the stability and accuracy of LBM by selecting proper mapping parameters.

In the following section, the proposed method of mapping quantities’ units is presented and discussed with supporting example. Section 3 presents and discusses the results of the current work. Six benchmark examples are presented to evaluate the proposed approach. Finally, Section 4 summarizes the conclusions.

2. Unit Conversion from/to Physical-Scale to/from Lattice-Scale

In the following, the proposed approach is presented and explained. Firstly, reference parameters that are used to non-dimensionalize the Boltzmann transport equation and any property are inferred. Then, a step-by-step example is performed to illustrate the concept using the inferred reference parameters.

2.1. 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]:

$$\frac{\partial f}{\partial t}+\mathit{e} · \nabla f+\mathit{a} · {\nabla }_e f=\frac{1}{\lambda }\left(f^M-f\right),$$

(8)

where $f=f\left(\mathit{x},\mathit{e},t\right)$ is the particle distribution function, $\mathit{e}$ is the microscopic velocity vector, $\mathit{a}$ 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]

$$f^M=\frac{\rho }{{\left(2\pi RT\right)}^{D/2}}\exp \left[\frac{-\left\{\left(\mathit{e}-\mathit{u}\right)·\left(\mathit{e}-\mathit{u}\right)\right\}}{2RT}\right],$$

(9)

where $\rho$ is the density, $\mathit{u}$ 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 $\frac{\rho }{{\left(RT\right)}^{D/2}}$. Using dimensional analysis yields:

$$\left\{f\right\}=\left\{M·L^{-3}·t^D·L^{-D}\right\}.$$

(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

$$\rho =\int fd\mathit{e},$$

(11)

where the integration is single, double, or triple for 1D, 2D, or 3D problems, respectively.

Wolf–Gladrow [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:

$$\frac{\partial f^{*}}{\partial t^{*}}+{\mathit{e}}^{*} · {\nabla }^{*} f^{*}+{\mathit{a}}^{*} · {\nabla }_e^{*} f^{*}=\frac{1}{{\lambda }^{*}}\left(f^{M,*}-f^{*}\right),$$

(12)

where

$$f^{*}=f\frac{x_o^3}{m_o}\frac{x_o^D}{t_o^D} ,$$

(13)

$$t^{*}=\frac{t}{t_o},$$

(14)

$${\mathit{e}}^{*}=\mathit{e}\frac{t_o}{x_o},$$

(15)

$${\nabla }^{*}=x_o\nabla ,$$

(16)

$${\mathit{a}}^{*}=\mathit{a}\frac{t_o^2}{x_o},$$

(17)

$${\nabla }_e^{*}=\frac{x_o}{t_o}{\nabla }_e,$$

(18)

and

$${\lambda }^{*}=\lambda /t_o,$$

(19)

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, ${\mathsf{\Theta }}_o$, 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.

In order to solve Equation (8) numerically, the discretized form is used:

$$f_i\left(\mathit{x}+{\mathit{e}}_i\Delta t,t+\Delta t\right)=f_i\left(\mathit{x},t\right)+\frac{1}{\tau }\left(f_i^{eq}\left(\mathit{x},t\right)-f_i\left(\mathit{x},t\right)\right)+\Delta t{S}_i\left(\mathit{x},t\right),$$

(20)

where ${\mathit{e}}_i$ is discrete lattice velocity, $\Delta t$ is the time step, $\tau =\frac{\lambda }{\Delta t}$ is the dimensionless relaxation time, $f_i^{eq}$ is the Maxwell–Boltzmann equilibrium distribution function, and $S_i$ is the lattice force term. Recently, we [11] reviewed and evaluated twelve force terms that proposed to implement the momentum forces in the LBM.

2.2. 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 ($x_{o\left(PS\right)}, t_{o\left(PS\right)}, m_{o\left(PS\right)}$, and ${\mathsf{\Theta }}_{o\left(PS\right)}$) 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 ($x_{o\left(LS\right)}, t_{o\left(LS\right)}, m_{o\left(LS\right)}$, and ${\mathsf{\Theta }}_{o\left(LS\right)}$) 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/m³ of density ($\rho$), 1.56 × 10⁻⁵ m²/s of viscosity ($\nu$), 2.14 × 10⁻⁵ m²/s of thermal diffusivity ($\alpha$), and 3.4 × 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.

The dimensionless governing equations (Navier–Stokes equations) for natural convection can be written as:

Continuity equation:

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0,$$

(21)

x-momentum equation:

$$U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{1}{Re} \left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right),$$

(22)

y-momentum equation:

$$U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial Y}+\frac{1}{Re} \left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)+\frac{Gr}{R{e}^2} \theta , \mathrm{and}$$

(23)

Energy equation:

$$U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{1}{Re Pr}\left(\frac{{\partial }^2\theta }{\partial X^2}+\frac{{\partial }^2\theta }{\partial Y^2}\right),$$

(24)

where

$$X=\frac{x}{H}, Y=\frac{y}{H},$$

(25)

$$U=\frac{u}{u_o}, V=\frac{v}{u_o},$$

(26)

$$P=\frac{p}{\rho u_o^2}, \mathrm{and} \theta =\frac{T-T_{west}}{T_{east}-T_{west}}.$$

(27)

The controlling parameters for the example are summarized in

Table 2. Parameters and dimensionless numbers of the example. $g=9.81 \mathrm{m}/{\mathrm{s}}^2$ is the gravity and $C=343 \mathrm{m}/\mathrm{s}$ is the speed of sound.

Parameter	Formula	Value
Prandtl number	$Pr=\frac{\nu }{\alpha }$	$0.73$
Grashof number	$Gr=\frac{g\beta \Delta T{H}^3}{{\nu }^2}$	$2.2145\times {10}^5$
Reynold number	$Re=\sqrt{Gr}$	$470.6196$
Velocity	$u=\sqrt{g\beta \Delta TH}$	$0.3671 \mathrm{m}/\mathrm{s}$
Mach number	$Ma=\frac{u}{C}$	$1.07\times {10}^{-3}$

. 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 Table 2.

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, $x_{o\left(PS\right)}=0.02 \mathrm{m}$ and ${\mathsf{\Theta }}_{o\left(PS\right)}=500 \mathrm{K}$ are chosen as the reference variables for length and temperature, respectively. Furthermore, the reference of time ($t_{o\left(PS\right)}$) and mass ($m_{o\left(PS\right)}$) 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. 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 ($m_{o\left(PS\right)}$ and $t_{o\left(PS\right)}$) can be calculated as:

$$m_{o\left(PS\right)}=\frac{\rho }{{\rho }^{*}}{x}_{o\left(PS\right)}^3=9.6\times {10}^{-6} \mathrm{kg},$$

(28)

and

$$t_{o\left(PS\right)}=\frac{u^{*}}{u}{x}_{o\left(PS\right)}=0.0545 \mathrm{s}.$$

(29)

Dimensionless-scale: Using the mentioned reference variables of the physical-scale, the non-dimensionalized parameters are as follows:

$$H^{*}=\frac{H_{\left(PS\right)}}{x_{o\left(PS\right)}}=1,$$

(30)

$$L^{*}=\frac{L_{\left(PS\right)}}{x_{o\left(PS\right)}}=2,$$

(31)

$$C^{*}=C\frac{t_{o\left(PS\right)}}{x_{o\left(PS\right)}}=934.3928,$$

(32)

$${\nu }^{*}={\nu }_{\left(PS\right)}\frac{t_{o\left(PS\right)}}{x_{o\left(PS\right)}^2}=2.12\times {10}^{-3},$$

(33)

$${\alpha }^{*}={\alpha }_{\left(PS\right)}\frac{t_{o\left(PS\right)}}{x_{o\left(PS\right)}^2}=2.91\times {10}^{-3},$$

(34)

$$g^{*}=g_{\left(PS\right)}\frac{t_{o\left(PS\right)}^2}{x_{o\left(PS\right)}}=1.456,$$

(35)

$${\beta }^{*}={\beta }_{\left(PS\right)}{\mathsf{\Theta }}_{o\left(PS\right)}=1.7,$$

(36)

$${\mathsf{\Theta }}_{west}^{*}=\frac{{\mathsf{\Theta }}_{west\left(PS\right)}}{{\mathsf{\Theta }}_{o\left(PS\right)}}=1,$$

(37)

and

$${\mathsf{\Theta }}_{east}^{*}={\mathsf{\Theta }}_{south}^{*}=\frac{{\mathsf{\Theta }}_{east\left(PS\right)}}{{\mathsf{\Theta }}_{o\left(PS\right)}}=0.596.$$

(38)

The dimensionless numbers will be

$$Pr=\frac{{\nu }^{*}}{{\alpha }^{*}}=0.73,$$

(39)

$$Gr=\frac{g^{*}{\beta }^{*}\Delta {\mathsf{\Theta }}^{*}{H}^{{*}^3}}{{\nu }^{{*}^2}}=2.2145\times {10}^5,$$

(40)

$$Re=\sqrt{Gr}=470.6196,$$

(41)

and

$$Ma=\frac{u^{*}}{C^{*}}=1.07\times {10}^{-3}.$$

(42)

Lattice-scale: The reference variables of the lattice-scale (${\mathsf{\Theta }}_{o\left(LS\right)}$, $m_{o\left(LS\right)}$, $x_{o\left(LS\right)}$, and $t_{o\left(LS\right)}$) 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,

$${\rho }^{*}={\rho }_{\left(LS\right)}\frac{x_{o\left(LS\right)}^3}{m_{o\left(LS\right)}}=1,$$

(43)

$$u^{*}=u_{\left(LS\right)}\frac{t_{o\left(LS\right)}}{x_{o\left(LS\right)}}=1,$$

(44)

$${\nu }^{*}={\nu }_{\left(LS\right)}\frac{t_{o\left(LS\right)}}{x_{o\left(LS\right)}^2}=2.12\times {10}^{-3},$$

(45)

$${\alpha }^{*}={\alpha }_{\left(LS\right)}\frac{t_{o\left(LS\right)}}{x_{o\left(LS\right)}^2}=2.91\times {10}^{-3},$$

(46)

and

$${\mathsf{\Theta }}_{west\left(LS\right)}^{*}=\frac{{\mathsf{\Theta }}_{west\left(LS\right)}}{{\mathsf{\Theta }}_{o\left(LS\right)}}=1.$$

(47)

where the subscript $LS$ refers to the lattice-scale. Again, we have two free parameters, ${\rho }_{\left(LS\right)}$ and ${\mathsf{\Theta }}_{o\left(LS\right)}$. Any reasonable values can be selected, usually set to be one. Other scales, $m_{o\left(LS\right)}$, $x_{o\left(LS\right)}$, and $t_{o\left(LS\right)}$ can be found by manipulating Equations (43)–(47):

$$m_{o\left(LS\right)}=\frac{x_{o\left(LS\right)}^3}{{\rho }^{*}}{\rho }_{\left(LS\right)}$$

(48)

$$t_{o\left(LS\right)}=\frac{{\nu }_{\left(LS\right)}}{{\nu }^{*}}{\left(\frac{u^{*}}{u_{\left(LS\right)}}\right)}^2=\frac{{\alpha }_{\left(LS\right)}}{{\alpha }^{*}}{\left(\frac{u^{*}}{u_{\left(LS\right)}}\right)}^2=\frac{u^{*}}{u_{\left(LS\right)}}{x}_{o\left(LS\right)},$$

(49)

and

$$x_{o\left(LS\right)}=\frac{{\nu }_{\left(LS\right)}}{{\nu }^{*}}\frac{u^{*}}{u_{\left(LS\right)}}=\frac{{\alpha }_{\left(LS\right)}}{{\alpha }^{*}}\frac{u^{*}}{u_{\left(LS\right)}}=\frac{u_{\left(LS\right)}}{u^{*}}{t}_{o\left(LS\right)}.$$

(50)

The reference lattice velocity ($u_{\left(LS\right)}$), lattice viscosity (${\nu }_{\left(LS\right)}$), and lattice thermal diffusivity (${\alpha }_{\left(LS\right)}$) 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 ${\nu }_{\left(LS\right)}=\left({\tau }_f-0.5\right)C_s^2\Delta t$ and ${\alpha }_{\left(LS\right)}=\left({\tau }_e-0.5\right)C_s^2\Delta t$. 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 ($u_{\left(LS\right)}$) is that $M{a}_{\left(LS\right)}=u_{\left(LS\right)}/C_s<<1$. $M{a}_{\left(LS\right)}$ is the lattice Mach number in the lattice-scale, which is different from the physical Mach number.

As mentioned before, the stability condition of LBM is that ${\tau }_f$ and ${\tau }_e$ should be greater than 0.5. Let ${\tau }_f=1$. For incompressibility condition the Ma number should be small. Let $u_{\left(LS\right)}=0.1$, which yields $M{a}_{\left(LS\right)}$ of $0.1732$, which is $<<1$. Hence, Equations (48)–(50) yield:

$${\nu }_{\left(LS\right)}=\frac{1-0.5}{3}=0.1667 \mathrm{l}{\mathrm{u}}^2/\mathrm{ts},$$

(51)

$$t_{o\left(LS\right)}=\frac{0.1667}{2.12\times {10}^{-3}}\times {\left(\frac{1}{0.1}\right)}^2=7.84\times {10}^3 \mathrm{ts},$$

(52)

and

$$x_{o\left(LS\right)}=\frac{0.1667}{2.12\times {10}^{-3}}\times \frac{1}{0.1}=784.3661 \mathrm{lu}.$$

(53)

The relaxation time related to the energy equation can be calculated as:

$${\alpha }_{\left(LS\right)}=784.3661\times 2.91\times {10}^{-3}\times \frac{0.1}{1}=0.2286 \mathrm{l}{\mathrm{u}}^2/\mathrm{ts},$$

(54)

then

$${\tau }_e=\frac{{\alpha }_{\left(LS\right)}}{C_s^2\Delta t}+0.5=1.1859.$$

(55)

Finally, the parameters of the problem in the lattice-scale will be:

$$H_{\left(LS\right)}=x_{o\left(LS\right)}{H}^{*}=784.3661 \mathrm{lu},$$

(56)

$$L_{\left(LS\right)}=x_{o\left(LS\right)}{L}^{*}=1568.7321 \mathrm{lu},$$

(57)

$$C_{\left(LS\right)}=C\frac{x_{o\left(LS\right)}}{t_{o\left(LS\right)}}=93.4393 \mathrm{lu}/\mathrm{ts},$$

(58)

$$g_{\left(LS\right)}=g^{*}\frac{x_{o\left(LS\right)}}{t_{o\left(LS\right)}^2}=1.86\times {10}^{-5} \mathrm{lu}/{\mathrm{ts}}^2,$$

(59)

$${\beta }_{\left(LS\right)}=\frac{{\beta }^{*}}{{\mathsf{\Theta }}_{o\left(LS\right)}}=1.7 {\mathrm{tu}}^{-1},$$

(60)

$${\mathsf{\Theta }}_{west\left(LS\right)}={\mathsf{\Theta }}_{west}^{*}{\mathsf{\Theta }}_{o\left(LS\right)}=1 \mathrm{tu},$$

(61)

and

$${\mathsf{\Theta }}_{east\left(LS\right)}={\mathsf{\Theta }}_{south\left(LS\right)}={\mathsf{\Theta }}_{south}^{*}{\mathsf{\Theta }}_{o\left(LS\right)}=0.596 \mathrm{tu}.$$

(62)

To check the similarity, the dimensionless numbers can be calculated as:

$$Pr=\frac{{\nu }_{\left(LS\right)}}{{\alpha }_{\left(LS\right)}}=0.73,$$

(63)

$$Gr=\frac{g_{\left(LS\right)}{\beta }_{\left(LS\right)}{H}_{\left(LS\right)}^3\Delta {\mathsf{\Theta }}_{\left(LS\right)}}{{\nu }_{\left(LS\right)}^2}=2.2145\times {10}^5,$$

(64)

$$Re=\sqrt{Gr}=470.6196,$$

(65)

and

$$Ma=\frac{u_{\left(LS\right)}}{C_{\left(LS\right)}}=1.07\times {10}^{-3},$$

(66)

$$M{a}_{\left(LS\right)}=\frac{u_{\left(LS\right)}}{C_s}=0.1732,$$

(67)

One advantage of the method is freedom of selection ${\tau }_f$ or ${\tau }_e$ to ensure the stability of the solution and in selecting $u_{\left(LS\right)}$ to ensure the incompressibility conditions.

The scaling units between SI and lattice unit systems can be summarized in

Table 3. Scaling units between SI and lattice systems.

Primary Dimension	SI Unit	Lattice Unit
Mass	1 kg/kg	$5.027\times {10}^{13}$mu/kg
Length	1 m/m	$3.9218\times {10}^4$lu/m
Time	1 s/s	$1.4396\times {10}^5$ ts/s
Temperature	1 K/K	$2.0\times {10}^{-3}$ tu/K

. The values listed in Table 3 are calculated by dividing each reference variable by its corresponding value in physical-scale (SI unit). The Appendix A gives the factors to convert quantities from/to physical-scale to/from dimensionless-scale from/to lattice-scale.

3. Results and Discussions

To evaluate the feasibility and accuracy of the proposed approach, one- and two- dimensional benchmark examples whose analytical or numerical solutions available are performed. The D1Q3 lattice model is used to solve one-dimensional problems, while the D2Q5 and D2Q9 for two-dimensional. Figure 4 shows the index and direction of each microscopic velocity vector in the used lattice models.

3.1. One-Dimensional Examples

The solution of 1D diffusion, 1D advection-diffusion, and 1D phase change problems are provided to exemplify the capacity of the proposed approach. The D1Q3 lattice model is used to solve those examples, where it has the following equilibrium distribution function ($f_i^{eq}$), lattice weights ($w_i$), microscopic speeds (${\mathit{e}}_i)$, and speed of sound ($C_s$):

$$f_i^{eq}=w_i\varphi \left[1+\frac{{\mathit{e}}_i · {\mathit{u}}^{eq}}{C_s^2}\right],$$

(68)

$$w_i=\left[\begin{array}{ccc}4/6& 1/6& 1/6\end{array}\right],$$

(69)

$${\mathit{e}}_i=\left[\begin{array}{ccc}0& 1& -1\end{array}\right],$$

(70)

and

$$C_s^2=\frac{1}{3}.$$

(71)

3.1.1. 1D Diffusion

In this example, an insulated rod of copper whose left and right ends are maintained at constant temperatures of 298 K and 373 K, respectively. The rod has length L = 2 m and uniform heat generation q = 100 kW/m³. The properties of the copper in the SI unit system are listed in

Table 4. Converting data for diffusion problem. The values with * represent assumed values.

Reference Variables in Physical-Scale and Lattice-Scale
Reference Variables	Physical-Scale		Lattice-Scale
$m_o$	71,520 kg		1000 mu
$x_o$	2 m		10 lu
$t_o$	3.4635 × 104 s		600 ts
${\mathsf{\Theta }}_o$	373 K		1 * tu
Scaling units between SI and lattice systems
Primary dimension	Physical-scale		Lattice-scale
Mass	1 kg/kg		0.014 mu/kg
Length	1 m/m		5 lu/m
Time	1 s/s		0.017 ts/s
Temperature	1 K/K		2.68 × 10−3 tu/K
Converting properties between scales
Property	Physical-scale	Dimensionless-scale	Lattice-scale
Length (L)	2 m	1	10 * lu
$\mathrm{Cold} \mathrm{temperature} (T_c$)	298 K	0.799	0.799 tu
$\mathrm{Hot} \mathrm{temperature} (T_h$)	373 K	1	$1$ tu
$\mathrm{Density} \left(\rho \right)$	8940 kg/m3	1 *	1 * mu.lu−3
$\mathrm{Specific} \mathrm{heat} (C_p$)	385 J/kg.K	4.3068 × 1013	1.19 × 1010 lu2.ts−2.tu−1
Thermal conductivity (k)	397.5 W/m.K	4.3068 × 1013	1.99 × 109 mu.lu.ts−3.tu−1
Thermal diffusivity (α)	1.1549 × 10−4 m2/s	1 *	0.1667 lu2.ts−1
Uniform heat generation (q)	1 × 105 W/m3	1.1619×1014	5.3791 × 107 mu.lu−1.ts−3

. The problem is governed by the following equation

$$\rho C_p\frac{\partial T}{\partial t}=k\frac{{\partial }^{2}T}{\partial x^2}+q.$$

(72)

where $C_p$ is the specific heat at constant pressure and $k$ is the thermal conductivity.

The analytical solution of the steady-state temperature distribution is given as Ref. [23]:

$$T=\left[\frac{T_c-T_h}{L}+\frac{q}{2k}\left(L-x\right)\right]x+T_h.$$

(73)

where $T_c$ and $T_h$ are the cold and hot temperatures, respectively.

In LBM simulation, the following data are used:

$${\tau }_e=1,$$

(74)

and

$$S_i=w_i\frac{q}{\rho C_p}.$$

(75)

Using the previously mentioned steps in Section 2.2, each quantity can be mapped from the physical-scale to the dimensionless-scale after the reference variables of the physical-scale are defined. The current example is characterized by stability only. This gives more flexibility in choosing the length reference variable for the lattice-scale. Based on that ${\tau }_e=1$ and $x_{o\left(LS\right)}=10 lu$ are selected. By setting ${\rho }_{\left(LS\right)}={\mathsf{\Theta }}_{o\left(LS\right)}=1$ and manipulating Equations (48)–(50), all of the reference variables of the lattice-scale are defined. Follow Equations (56)–(62), each quantity is mapped from a dimensionless-scale to the lattice-scale. Table 4 shows that all quantities of the current example are mapped from the physical-scale to the lattice-scale using the present approach. Table 4 lists the reference variables of the physical-scale and lattice-scale, the scaling units between SI and lattice systems, and the transforming properties between scales. The values with * represent those values are assumed.

Figure 5 shows the temperature distribution along the rod. The steady-state condition is achieved when the difference between two successive iterations of temperature is less than 10⁻⁹. Figure 5a compares the LBM predictions with analytical solution (i.e., Equation (73)) at the lattice-scale. Figure 5b compares the LBM predictions with the analytical solution at the physical-scale. It can be concluded that excellent agreements between the numerical and analytical solutions at both scales can be achieved.

3.1.2. 1D Advection-Diffusion

In this example, a property $\varphi$ is transported by a constant velocity of 0.001 m/s by advection-diffusion through the 1D domain. The boundary conditions are $\varphi$ x = 0 = 373 K and $\varphi$ x = L = 298 K, where L = 1 m. The property has 1.2 kg/m³ of density, 1.005 kJ/kg.K of specific heat, and 0.026 W/m.K of thermal conductivity. Due to the advection velocity of the current example is constant, the accuracy criterion does not come to the picture. The 1D advection-diffusion equation can be expressed as follows:

$$\rho C_p\frac{\partial \varphi }{\partial t}+\rho C_{p}u\frac{\partial \varphi }{\partial x}=k\frac{{\partial }^2\varphi }{\partial x^2}.$$

(76)

The analytical solution of the steady-state condition is given by [23]:

$$\varphi ={\varphi }_{x=0}+\left({\varphi }_{x=L}-{\varphi }_{x=0}\right)\left[\frac{exp\left(\rho C_{p}ux/k\right)-1}{exp\left(\rho C_{p}uL/k\right)-1}\right].$$

(77)

The following data are used in the LBM modeling:

$${\tau }_e=1,$$

(78)

$${\mathit{u}}^{eq}={\mathit{u}}_{\left(LS\right)},$$

(79)

and

$$S_i=0,$$

(80)

where ${\mathit{u}}_{\left(LS\right)}$ is the advection velocity at the lattice-scale, which is constant.

Figure 6 shows the simulation results for the given example using the listed data in

Table 5. Converting data for the advection-diffusion problem. The values with * represent assumed values.

Reference Variables in Physical-Scale and Lattice-Scale
Reference Variables	Physical-Scale		Lattice-Scale
$m_o$	1.2 kg		6.4 × 104 mu
$x_o$	1 m		40 lu
$t_o$	1.0 × 103 s		206.97 ts
${\mathsf{\Theta }}_o$	373 K		1 * tu
Converting properties between scales
Property	Physical-scale	Dimensionless-scale	Lattice-scale
Length (L)	1 m	1	40 * lu
Advection velocity (u)	0.001 m/s	1 *	0.1933 lu.ts−1
$\varphi$x = L	298 K	0.799	0.799 tu
$\varphi$x = 0	373 K	1	$1$ tu
$\mathrm{Density} (\rho$)	1.2 kg/m3	1 *	1 * mu.lu−3
$\mathrm{Specific} \mathrm{heat} (C_p$)	1005 J/kg.K	3.75 × 1011	1.4 × 1010 lu2.ts−2.tu−1
Thermal conductivity (k)	0.026 W/m.K	8.1 × 109	2.33 × 109 mu.lu.ts−3.tu−1
Thermal diffusivity (α)	2.156 × 10−5 m2/s	0.0216	0.1667 lu2.ts−1

. Excellent agreements with the analytical solutions at the lattice-scale (Figure 6a) and the physical-scale (Figure 6b) are shown. As mentioned, in such problems, the LBM results are not affected by the advection velocity. Therefore, the lattice Mach number ($M{a}_{\left(LS\right)}$) does not affect the results. However, the results of the conventional computational fluid dynamic (CFD) methods are significantly affected by the value of the advection velocity, which causes dispersion error [23]. Figure 7 shows a comparison between the LBM and finite difference method (FDM) at the physical-scale for different mesh sizes. The LBM solution is achieved at the lattice-scale then converted to the physical-scale. As shown, the error between the analytical and FDM solutions is significantly affected by mesh size, unlike the LBM.

3.1.3. 1D Melting in a Half-Scale (Phase Change)

A semi-infinite slab of pure aluminum initially is at the uniform melting temperature (660 °C). Suddenly, the left side of the slab (at x = 0) is subjected to a constant source of a temperature of 750 °C at times t ≥ 0. As a result, a melting begins from the left side, where the liquid/solid interface moves in the positive x-direction as time progresses. Since the solid phase is at a constant melting temperature, the problem is considered a one-region problem. The temperature distribution in the liquid phase for a given time can be written as Refs. [24,25]:

$$T=T_{x=0}+\left(T_{x=0}-T_{x=L}\right)\frac{erf\left(\frac{x}{2\sqrt{\alpha t}}\right)}{erf\left(\lambda \right)},$$

(81)

where $\lambda$ are solutions to the transcendental equation $\lambda e^{{\lambda }^2}erf\left(\lambda \right)=\frac{St}{\sqrt{\pi }}$. $St=\frac{C_p\left(T_{x=0}-T_{x=L}\right)}{h_{fg}}$ is the Stefan number, where $h_{fg}$ is the latent heat of aluminum.

Table 6. Converting data for phase change problem. The values with * represent assumed values.

Reference Variables in Physical-Scale and Lattice-Scale
Reference Variables	Physical-Scale		Lattice-Scale
$m_o$	21.59 kg		1 × 106 mu
$x_o$	0.2 m		100 lu
$t_o$	4.6267 × 102 s		6 × 104 ts
${\mathsf{\Theta }}_o$	1023 K		1 * tu
Converting properties between scales
Property	Physical-scale	Dimensionless-scale	Lattice-scale
Length (L)	0.2 m	1	100 * lu
$\mathrm{Latent} \mathrm{heat} (h_{fg}$)	3.869 × 105 J/kg	2.0705 × 1012	5.7514 × 106 lu2.ts−2
$\mathrm{Melting} \mathrm{temperature} (T$x = L)	933 K	0.912	0.912 tu
$\mathrm{Subjected} \mathrm{temperature} (T$x =0)	1023 K	1	$1$ tu
$\mathrm{Density} (\rho$)	2698.9 kg/m3	1*	1 * mu.lu−3
$\mathrm{Specific} \mathrm{heat} (C_p$)	900 J/kg.K	4.9272 × 1012	1.3687 × 107 lu2.ts−2.tu−1
Thermal conductivity (k)	210 W/m.K	4.9272×1012	2.2811×106 mu.lu.ts−3.tu−1
Thermal diffusivity (α)	8.6455 × 10−5 m2/s	1 *	0.1667 lu2.ts−1

summarizes the thermo-physical properties of pure aluminum in the SI unit system [15].

For the LBM simulation, Jiaung et al.’s [26] phase change model is used where the following data are used:

$${\tau }_e=1, \mathrm{and}$$

(82)

$$S_i=-w_i\frac{h_{fg}}{C_p}\frac{\partial f_l}{\partial t} ,$$

(83)

where $f_l$ is the volume-phase fraction of the liquid phase and is zero for the solid region and unity for the liquid region.

In the present example, the non-dimensional LBM form is used. Therefore, the dimensionless-scale data listed in Table 6 are applied. Figure 8 displays the temperature distribution along the slab for dimensionless time $t^{*}=1$ (i.e., for 4.6267 × 10² s). The good agreements between the results can be shown in Figure 8.

3.2. Two-Dimensional Examples

The D2Q9 lattice model is used to solve the Navier-Stokes equations, while the D2Q5 for the energy equation. $f_i$ is a distribution function used for the D2Q9, while $h_i$ for the D2Q5. The models have the following equilibrium distribution function, lattice weights, lattice velocities, and speed of sound.

D2Q9:

$$f_i^{eq}=w_i\rho \left[1+\frac{{\mathit{e}}_i·{\mathit{u}}^{eq}}{C_s^2}+\frac{{\left({\mathit{e}}_i·{\mathit{u}}^{eq}\right)}^2}{2C_s^4}-\frac{{\left({\mathit{u}}^{eq}\right)}^2}{2C_s^2}\right],$$

(84)

$$w_i=\left\{\begin{array}{c}4/9 i=0\\ 1/9 i=1,2,3,4\\ 1/36 i=5,6,7,8\end{array}\right.,$$

(85)

$${\mathit{e}}_i=\left[\begin{array}{ccc}0& 1& 0 \\ 0& 0& 1\end{array} \begin{array}{ccc}-1& 0& 1\\ 0& -1& 1\end{array}\begin{array}{ccc}-1& -1& 1\\ 1& -1& -1\end{array}\right],$$

(86)

and

$$C_s^2=\frac{1}{3}.$$

(87)

D2Q5:

$$h_i^{eq}=w_i\rho \left[1+\frac{{\mathit{e}}_i·{\mathit{u}}^{eq}}{C_s^2}\right],$$

(88)

$$w_i=\left\{\begin{array}{c}2/6 i=0\\ \\ 1/6 i=1,2,3,4\end{array}\right.,$$

(89)

$${\mathit{e}}_i=c\left[\begin{array}{ccc}0& 1& 0 \\ 0& 0& 1\end{array} \begin{array}{cc}-1& 0\\ 0& -1\end{array}\right],$$

(90)

and

$$C_s^2=\frac{1}{3}.$$

(91)

3.2.1. 2D Flow in a Channel

This example is characterized by the stability and accuracy criteria. Air flows in a channel with 8 mm/s of an average velocity (V) at the inlet. The channel has 1.5 m of a length (L) and 20 cm of gap (H). The pressure drop ($\partial P/\partial x$) inside the channel is −4.54 × 10⁻⁵ Pa/m. The density (ρ) and dynamic viscosity (µ) of air are 1.2 kg/m³ and 1.89 × 10⁻⁵ kg/m.s, respectively. The analytical solution of the velocity distribution at a full development regime is [27,28]:

$$u=\frac{H^2}{2\mathsf{\mu }}\left(\frac{\partial P}{\partial x}\right)\left[{\left(\frac{y}{H}\right)}^2-\left(\frac{y}{H}\right)\right].$$

(92)

The following data are used in the LBM simulation:

$${\tau }_f=0.7,$$

(93)

$$\rho {\mathit{u}}^{eq}=\sum f_i{e}_i,$$

(94)

and

$$S_i=0.$$

(95)

The LBM simulation of flow in a channel is performed using the tabled data,

Table 7. Converting data for flow in a channel problem. The values with * represent assumed values.

Reference Variables in Physical-Scale and Lattice-Scale
Reference Variables	Physical-Scale		Lattice-Scale
$m_o$	9.6 × 10−3 kg		2.49 × 106 mu
$x_o$	0.2 m		135.45 lu
$t_o$	25 s		2708.99 ts
Converting properties between scales
Property	Physical-scale	Dimensionless-scale	Lattice-scale
Gap (H)	0.2 m	1	135.45 lu
Length (L)	1.5 m	7.5	1015.87 lu
Velocity (V)	0.008 m/s	1 *	0.05 * lu.ts−1
$\mathrm{Pressure} \mathrm{drop} (\partial P/\partial x$)	−4.54 × 10−5 Pa/m	−0.1181	−2.18 × 10−6 mu.lu−2.ts−2
$\mathrm{Density} (\rho$)	1.2 kg/m3	1 *	1 * mu.lu−3
Dynamic viscosity (µ)	1.89 × 10−5 kg/m.s	9.844 × 10−3	0.0667 mu.lu−1.ts−1
$\mathrm{Kinematic} \mathrm{viscosity} (\nu$)	1.58 × 10−5 m2/s	9.844 × 10−3	0.0667 lu2.ts−1

. Figure 9 demonstrates the fully developed velocity profile. As shown, good agreements between the LBM and analytical solutions are achieved. To show the effect of the lattice Mach number ($M{a}_{\left(LS\right)}$) on the accuracy of the LBM, Figure 10 is plotted. As seen in Figure 10, as the $M{a}_{\left(LS\right)}$ increases, the error between the LBM and analytical results increases too. The error is due to the pseudo-incompressibility of the LBM. Based on the numerical experimentations, the stability limit of the LBM is at $M{a}_{\left(LS\right)}$ equals 0.49, where the scheme becomes unstable beyond this number.

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. Converting data for the differentially heated square cavity example. The values with * represent assumed values.

Reference Variables in Physical-Scale and Lattice-Scale
Reference Variables	Physical-Scale		Lattice-Scale
$m_o$	9.6 × 10−6 kg		6.4 × 107 mu
$x_o$	0.02 m		400 lu
$t_o$	18.6916 s		1.7495 × 106 ts
${\mathsf{\Theta }}_o$	1549.27 K		1 * tu
Converting properties between scales
Property	Physical-scale	Dimensionless-scale	Lattice-scale
Length (L)	0.02 m	1	400 * lu
$\mathrm{Room} \mathrm{temperature} (T_{room}$)	298.15 K	0.1924	0.1924 tu
$\mathrm{West} \mathrm{wall} \mathrm{temperature} (T_{west}$)	1549.27 K	1	$1$ tu
$\mathrm{Density} (\rho$)	1.2 kg/m3	1 *	1 * mu.lu−3
$\mathrm{viscosity} (\nu$)	1.56 × 10−5 m2/s	7.2897×10−1	6.6667 × 10−2 lu2.ts−1
Thermal diffusivity (α)	2.14 × 10−5 m2/s	1 *	9.1453 × 10−2 lu2.ts−1
$\mathrm{Coefficient} \mathrm{of} \mathrm{expansion} (\beta$)	3.4 × 10−3 K−1	5.2675	5.2675 tu−1
$\mathrm{Gravity} (g$)	9.81 m/s2	1.7137 × 105	2.2395 × 10−5 lu.ts−2
$\mathrm{Magnitude} \mathrm{velocity} (u$)	0.9136 m/s	853.7985	0.1952 lu.ts−1

. The west wall of the cavity is heated to a temperature of 1549.27 K. This value is selected to ensure Ra = 10⁶ 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]:

$$\mathit{F}=\rho \mathit{g}\beta \left(T-T_{ref}\right),$$

(96)

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:

$${\tau }_f=0.7,$$

(97)

and

$${\tau }_e=0.7744.$$

(98)

Luo’s [32,33] forcing scheme is used in the example, which has:

$$S_i=\frac{w_i}{C_s^2}{\mathit{e}}_i·\mathit{F},$$

(99)

and

$$\rho {\mathit{u}}^{eq}=\sum f_i{e}_i.$$

(100)

The obtained LBM results are compared with those of Quere and De Roquefortt [34].

Table 9. Dimensionless results for $Ra={10}^6$.

	$\mathit{N}{\mathit{u}}_{\mathit{a}\mathit{v}\mathit{e}}$	${\mathit{U}}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{Y}}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{V}}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{X}}_{\mathit{m}\mathit{a}\mathit{x}}$
Ref. [33]	8.814	64.81	0.850	220.57	0.0378
Present study	8.6915	63.9960	0.8529	216.8242	0.0399

shows a comparison between the LBM and Quere and De Roquefortt [34] results. In the table, $X_{max}$ and $Y_{max}$ are the locations of maximum $U_{max}$ and $V_{max}$, respectively. The average Nusselt number ($N{u}_{ave}$) is estimated as:

$$N{u}_{ave}=\frac{1}{\left(T_{west}-T_{east}\right)}{\displaystyle \sum }_{k=1}^{k=N}\frac{3T_{w,k}-4T_{w+1,k}+T_{w+2,k}}{2\Delta x},$$

(101)

where the subscripts $w$, $w+1$, and $w+2$ represent the lattice of the heated wall, the lattice adjacent to the heated wall, and the lattice after adjacent to the heated wall in the x-direction, respectively. Table 9 shows a good agreement between the results of the current study and Quere and De Roquefortt.

3.2.3. 2D Stationary Droplet

In the following example, we considered a stationary droplet placed at the center of the vapor phase domain. Such a problem is usually used to evaluate the stability and accuracy of the multiphase model, as well as to estimate the surface tension using Laplace’s law. In the current example, the surface tensions of water at different temperatures are estimated. Therefore, the inter-particle force between liquid and vapor phases can be presented as [35]:

$${\mathit{F}}_{int}\left(\mathit{x},t\right)=-G\psi \left(\mathit{x},t\right){\sum }_i{\omega }_i\psi \left(\mathit{x}+{\mathit{e}}_i\Delta t,t\right){\mathit{e}}_i,$$

(102)

where $G$ is a parameter that controls the strength of the inter-particle force and $\psi$ is the mean-field potential function. Yuan and Schaefer [36] modified the SC [6,7] model and defined $\psi$ as:

$$\psi =\sqrt{\frac{2\left\{P_{\mathrm{EoS}}-C_s^2\rho \right\}}{C_s^{2}G}},$$

(103)

where $P_{\mathrm{EoS}}$ is the EoS pressure. The R-K EoS is used in the present example. The parameters $a$ and $b$ of the R-K EoS can be defined as [37]:

$$a=1.282\frac{R{T}_c\sqrt{T_c}}{{\rho }_c},$$

(104)

and

$$b=\frac{0.2597}{{\rho }_c},$$

(105)

where

$$R{T}_c=0.27847,$$

(106)

and

$${\rho }_c=\ln \left(2\right){\rho }_o,$$

(107)

where ${\rho }_o$ is an arbitrary constant. The values of $R$ and ${\rho }_o$ are set to be 1 and 20 for water, respectively.

Table 10. Scaling units between SI and lattice systems.

Primary Dimension	Physical-Scale	Lattice-Scale
Mass	1 kg/kg	2.0972 × 103 mu/kg
Length	1 m/m	128 lu/m
Time	1 s/s	4.4184 × 10−3 ts/s
Temperature	1 K/K	4.3038 × 10−4 tu/K

shows the scaling parameters that are used to map the data from the lattice-scale to the physical-scale. The LBM predictions are compared with Jain et al. [38] LBM simulations and with the DDBST GmbH database [39], as shown in Figure 11. It is clear that there is a good agreement between the current predictions with published results.

4. Conclusions

The article deals with converting units of the quantities from/to the physical-space to/from the lattice-space. A novel dimensional analysis method is proposed. The proposed approach is characterized by high flexibility and simplicity in controlling the stability and accuracy of LBM. The accuracy of LBM is strongly dependent on the lattice Mach number ($M{a}_{\left(LS\right)}$), while its stability depends on the relaxation time. Increasing the lattice Mach number reduces the accuracy. Six benchmark examples are used to test the ability of the approach. The stability and accuracy of LBM are also evaluated. In conclusion, the proposed approach is a powerful and reliable tool to transform properties’ units from/to the physical-scale to/from the lattice-scale.