## 5 Lattice Boltzmann Methods (LBMs)  
Due to the limited capacity for theoretical analysis, huge cost, and long periods of experimental investigation, numerical simulation has emerged as an effective alternative tool for various industrial applications. Among the multitude of methods that have emerged, we find the LBM approach and its variants because of their significantly simple formulation compared to traditional methods dealing with Navier-Stokes equations (NSE). They are the starting point in the scale of molecular cluster-based discrete approaches (e.g., _MD for molecular dynamics, DSMC for direct simulation Monte Carlo, DPD for Dissipative Particle Dissipation_, etc.) as depicted in Figure 16[130]. Lattice Boltzmann methods (LBMs) have become an active subject of investigation in the fields of computational fluid dynamics and thermal science. Over the past decade, they have led to unprecedented results in the field of low-Mach, thermal, hydrodynamics, transport in porous media, groundwater flows, textile materials, fuel cells, aerodynamics, aeroacoustics, etc. The findings also include the appreciable stability of its streaming and collision algorithm in solving incompressible flows, its programming and parallelization flexibility, the treatment of nonlinear convective terms and pressure (_Poisson's equation_), the simplicity of dealing with complex boundary conditions and the ability to incorporate microscopic interactions. In practice, the CPU costs would be significantly lower than those of conventional Navier-Stokes solvers. In addition to being second-order accurate in time and space, the LBM method exhibits excellent dissipation properties for pressure waves, which are comparable to high-order numerical methods. It is based on a distribution of particles while aiming mainly to guess the macroscopic characteristics [131; 132]. It should be stressed that the low computational intensity (_number of iterations_, _computation time_, _CPU \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Heat Storage Density** & **Storage Period** & **Heat Transfer Time** & **Technical Maintenance** \\ \hline
**SHS** & \(\approx 50\) kWhm\({}^{-3}\) & limited & short & uncomplicated \\  & \(\approx 0.02-0.03\) kWhkg\({}^{-3}\) & & & \\ \hline
**LHS** & \(\approx 100\) kWhm\({}^{-3}\) & limited & short & complicate \\  & \(\approx 0.05-0.1\) kWhkg\({}^{-1}\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Technical parameters of SHS and LHS, reprinted/adapted with permission from Ref. [27], 2022, Elsevier. Figure 16: Hierarchy scale of modeling and simulation methods ranging from Euler to molecular dynamics, reprinted/adapted with permission from Ref. [130], 2022, World Scientific. use_, etc.) of this approach [133] is one of its most important advantages over traditional computational fluid dynamics (CFD) methods. In the case of 2D porous structures, [PERSON] et al. [133] reported a comparison between LBM and the finite difference method (FDM) for two different flow problems (_lid-driven cavity flow and buoyancy-driven cavity flow_). In the first case, they depicted the velocity distribution profiles at \(Re\leq 10^{3}\) and showed that the LBM results are more accurate than the FDM ones with a slight difference in the computation time. However, at [PERSON]Re>10^{3}\), the LBM simulations take longer (1.3 times) than simulations via FDM. In the second case, the authors found that the CPU time of LBM is generally lower, demonstrating the efficiency of LBM compared to FDM.Nevertheless, for high Rayleigh numbers (Ra), LBM requires more grid numbers than FDM to ensure its numerical stability. A summary statement of these findings [133] is depicted in Table 8.  
For the finite difference approach, the most influential parameters are the number of discretization points and the CFL number. Unlike the FDM, the equivalent to a CFL number for the LBM cannot be chosen explicitly. LB velocities, cell spacing and time step sizes are bound to the used lattice.