Lattice Boltzmann method for simulating transport phenomena avoiding the use of lattice units

Abstract

In this paper, we propose a dimensional lattice Boltzmann method (LBM) that numerically solves the discrete lattice Boltzmann equation directly in physical units. This procedure facilitates the LBM application for simulating transport phenomena completely avoiding the use of lattice units and consequently of any particular unit conversion system. Several test problems related to different physical phenomena are simulated, such as heat diffusion, lid-driven cavity, forced convection in channels (both developed and under development) and two-phase liquid–gas systems, considering stationary and dynamic flows under very high density and viscosity ratios. We compare the numerical results with analytical or finite difference solutions, finding a good agreement between them. Similarly, we performed a stability analyses for three of the test cases. The traditional LBM was also considered for the sake of comparison, showing both the same accuracy and stability, as expected. Furthermore, we present solutions using the Allen–Cahn phase-field LBM model for high liquid/gas density and gas/liquid kinematic viscosity ratios, up to 43,300 and 470, respectively, commonly not found in open literature. The proposed methodology enhances the LBM use as a simulation tool for the wide transport phenomena where it finds application.

Appendices

Appendix 1: 1D advection–diffusion equation

In this example, we solve the advection–diffusion equation in a 1D domain. Air at an average temperature of 335.50 K is considered, with the following properties: $\rho = 1.052$ kg m$^{-3}$, $c_p = 1008.174$ J kg$^{-1}$K$^{-1}$ and $k = 0.029$ W m$^{-1}$K$^{-1}$. Initially, the domain is at $T_\textrm{ini} = 298.0$ K, and suddenly the right boundary is subjected to a temperature of $T_L = 373.0$ K, while the other end is maintained at $T_0 = 298.0$ K. The domain length is $L = 1.0$ m, and the air is moving with a constant speed $u = 0.001$ m s$^{-1}$.

The macroscopic equation that represents the physical problem is expressed by Eq. 53, and the corresponding analytical solution for the steady-state condition is given by Eq. 54.

$$\begin{aligned}{} & {} u\frac{\partial T}{\partial x} = \alpha \frac{\partial ^2 T}{\partial x^2} \end{aligned}$$

(53)

$$\begin{aligned}{} & {} T(x) = T_0 + (T_L - T_0)\left[ \frac{\text{ exp }\left( \frac{\rho c_p u x}{k} - 1 \right) }{\text{ exp }\left( \frac{\rho c_p u L}{k} - 1 \right) } \right] \end{aligned}$$

(54)

For this specific problem, given its simplicity in handling boundary conditions in the 1D case, we used the wet-node scheme for the boundaries instead of the link-wise scheme. This last scheme was used in the remainder of all simulations carried out in the paper. Then, using the D1Q3 velocity scheme, the BCs were implemented according to Eq. 55. Here, we considered the traditional BGK operator for the LBE, represented in Eq. 16.

$$\begin{aligned} {\left\{ \begin{array}{ll} g_1(0,t+\Delta t) = T_0 - g_0(0,t+\Delta t) - g_2(0,t+\Delta t), \text{ for } x=0.0;\\ g_2(L,t+\Delta t) = T_L - g_0(L,t+\Delta t) - g_1(L,t+\Delta t), \text{ for } x=L; \end{array}\right. } \end{aligned}$$

(55)

The LBM models are solved considering $\Delta x = 0.0125$ m and $\Delta t = 0.10$ s. The numerical solutions with LBM and the analytical solution are all shown in Fig. 17. Comparison of the solutions resulted in a global error of $E_2 = 0.030 \%$ for both LBM models relative to the theoretical solution.

Fig. 17

Steady-state temperature distribution for the one-dimensional advection–diffusion problem

The results provided verify the accuracy of the dimensional LBM.

Appendix 2: isothermal channel flow

We also applied both LBM models to simulate an isothermal Poiseuille flow between two parallel plates. The distance between the plates was assumed to be $H = 0.50$ mm. As the analytical solution does not depend on the channel length (given by Eq. 44), we used 10 computational cells ($10 \Delta x$) in the X-direction. We employed the D2Q9 velocity scheme with $\Delta x = 5.0\cdot 10^{-6}$ m and $\Delta t = 1.0\cdot 10^{-7}$ s.

The mean velocity of the channel was $u_m = 0.20 m\ s^{-1}$ and the driving force in the x direction was $F_x = 12u_m\zeta /H^2$. The fluid is water at a mean temperature of 301 K, with properties listed in Table 1. The results are depicted in Fig. 18, and the global errors for both LBM models compared to the analytical solution are $E_2^\textrm{dim} = E_2^\textrm{conv} = 0.011\%$, indicating very good accuracy.

Fig. 18

Steady-state velocity profiles of the isothermal Poiseuille flow between two parallel plates obtained for the dimensional and conventional LBM simulations and the analytical solution

Appendix 3: natural convection in a square enclosure

In this Appendix, we simulate natural convection in a square enclosure with length $L = 0.0130$ m containing air initially at $T_\textrm{ini} = 293.85$ K. The left wall of the domain is maintained at a higher constant temperature equal to $T_h = 373.15$ K, while the right wall remains at the initial temperature of $T_c = 293.85$ K. The upper (top) and lower (bottom) walls are considered insulated. Air properties are calculated at a reference temperature of $T_\textrm{ref} = 333.50$ K: $\rho = 1.059$ kg m$^{-3}$, $c_p = 1008.045$ J kg$^{-1}$K$^{-1}$, $k = 0.029$ W m$^{-1}$K$^{-1}$, $\alpha = 2.702 \cdot 10^{-5}$ m$^2$ s, $\nu = 1.90 \cdot 10^{-5}$ m$^2$ s and $\beta _\textrm{exp} = 3.004 \cdot 10^{-3} K^{-1}$ (thermal expansion coefficient).

The temperature difference between the walls causes a mass flux due to the difference in density between the hot and cold fluids. To consider this effect without changing the fluid density in the simulations, we assumed a buoyancy force given by Eq. 56 [57, 60, 77]. This is the so-called Boussinesq approximation. In this equation, ${\overline{\rho }}$ is the reference density, calculated at the reference temperature $T_\textrm{ref}$, and ${\textbf{g}} = (0,-\,9.81)$ m s$^{-2}$ is the gravitational acceleration.

$$\begin{aligned} \mathbf {F_b}({\textbf{x}},t) = -{\overline{\rho }} \beta _\textrm{exp} \left[ T({\textbf{x}},t) - T_\textrm{ref} \right] {\textbf{g}} \end{aligned}$$

(56)

We considered the D2Q9 velocity set, with the BGK and MRT collision operators for momentum and thermal LBEs, respectively. For stationary walls, we used the bounce-back BC for the momentum distribution function (Eq. 13). Furthermore, for the fixed temperature BCs (left and right walls), we employed the anti-bounce-back rule (Eq. 20), and the upper and lower walls were modeled as thermally insulated, just applying the BB rule (Eq. 21) with zero heat flux ($q'' = 0$).

The problem can be characterized by the Rayleigh number (Eq. 57), which will be considered as Ra $= 10^4$ for the first case and Ra $= 10^6$, for the second. In this last test, to obtain Ra $= 10^6$ without changing the average fluid temperature, we considered a new square cavity size, equal to $L = 0.60$ m, and the wall temperatures were changed to $T_h = 373.85$ K and $T_c = 293.15$ K, keeping $T_\textrm{ref} = 333.50$ K. Therefore, the air properties in both tests were maintained constant and unchanged.

$$\begin{aligned} \textrm{Ra} = \frac{|{\textbf{g}}|\beta _\textrm{exp}L^3(T_h - T_c)}{\nu \alpha } \end{aligned}$$

(57)

For both cases, discrete space and time intervals equal to $\Delta x = 2.0 \cdot 10^{-4}$ m and $\Delta t = 2.0 \cdot 10^{-4}$ s, were considered, respectively. Steady-state results for the temperature contours and streamlines are presented in Figs. 19 and 20. To evaluate the dimensional LBM, its solution is compared with the results of the conventional LBM, and both numerical solutions are validated through a comparison with the benchmark solutions found in the literature [14]. All these solutions are presented in Table 6. Both LBM models showed good agreement with the benchmark expected values, presenting very small global errors, $E_2$.

Table 6 Calculated average Nusselt numbers from the simulated results by both LBM models and the benchmark solution [14]

Fig. 19

Simulated temperature contours (a) and streamlines (b) for Ra $= 10^4$ with the dimensional LBM

Fig. 20

Simulated temperature contours (a) and streamlines (b) for Ra $= 10^6$ with the dimensional LBM