This work focuses on the numerical solution of the Stokes-Brinkman equation for a voxel-type porous-media grid, resolved by one to eight spacings per permeability contrast of 1 to 10 orders in magnitude. It is first analytically demonstrated that the lattice Boltzmann method (LBM) and the linear-finite-element method (FEM) both suffer from the viscosity correction induced by the linear variation of the resistance with the velocity. This numerical artefact may lead to an apparent negative viscosity in low-permeable blocks, inducing spurious velocity oscillations. The two-relaxation-times (TRT) LBM may control this effect thanks to free-tunable two-rates combination $\Lambda$. Moreover, the Brinkman-force-based BF-TRT schemes may maintain the nondimensional Darcy group and produce viscosity-independent permeability provided that the spatial distribution of $\Lambda$ is fixed independently of the kinematic viscosity. Such a property is lost not only in the BF-BGK scheme but also by “partial bounce-back” TRT gray models, as shown in this work. Further, we propose a consistent and improved IBF-TRT model which vanishes viscosity correction via simple specific adjusting of the viscous-mode relaxation rate to local permeability value. This prevents the model from velocity fluctuations and, in parallel, improves for effective permeability measurements, from porous channel to multidimensions. The framework of our exact analysis employs a symbolic approach developed for both LBM and FEM in single and stratified, unconfined, and bounded channels. It shows that even with similar bulk discretization, BF, IBF, and FEM may manifest quite different velocity profiles on the coarse grids due to their intrinsic contrasts in the setting of interface continuity and no-slip conditions. While FEM enforces them on the grid vertexes, the LBM prescribes them implicitly. We derive effective LBM continuity conditions and show that the heterogeneous viscosity correction impacts them, a property also shared by FEM for shear stress. But, in contrast with FEM, effective velocity conditions in LBM give rise to slip velocity jumps which depend on (i) neighbor permeability values, (ii) resolution, and (iii) control parameter $\Lambda$, ranging its reliable values from Poiseuille bounce-back solution in open flow to zero in Darcy's limit. We suggest an “upscaling” algorithm for $\Lambda$, from multilayers to multidimensions in random extremely dispersive samples. Finally, on the positive side for LBM besides its overall versatility, the implicit boundary layers allow for smooth accommodation of the flat discontinuous Darcy profiles, quite deficient in FEM.

The isothermal Navier–Stokes equations are determined by the leading three velocity moments of the lattice Boltzmann method (LBM). Necessary conditions establishing the hydrodynamic consistency of these moments are provided by multiscale asymptotic techniques, such as the second-order Chapman–Enskog expansion. However, for simulating incompressible hydrodynamics the structure of the forcing term in the LBM is still a discordant issue as far as its correct velocity expansion order is concerned. This work uses the traditional second-order Chapman–Enskog expansion analysis to demonstrate that the truncation order of the forcing term may depend on the time regime in this case. This is due to the fact that LBM does not reproduce exactly the incompressibility condition. It rather approximates it through a weakly compressible or an artificial compressible system. The present study shows that for the artificial compressible setup, as the incompressibility flow condition is singularly perturbed by the time variable, such a connection will also affect the LBM forcing formulation. As a result, for time-independent incompressible flows the LBM forcing must be truncated to first order whereas for a time-dependent case it is convenient to include the second-order term. The theoretical findings are confirmed by numerical tests carried out in several distinct benchmark flows driven by space- and/or time-varying body forces and possessing known analytical solutions. These results are verified for the single relaxation time, the multiple relaxation time and the regularized collision models.

The first nonequilibrium effect experienced by gaseous flows in contact with solid surfaces is the slip-flow regime. While the classical hydrodynamic description holds valid in bulk, at boundaries the fluid-wall interactions must consider slip. In comparison to the standard no-slip Dirichlet condition, the case of slip formulates as a Robin-type condition for the fluid tangential velocity. This makes its numerical modeling a challenging task, particularly in complex geometries. In this work, this issue is handled with the lattice Boltzmann method (LBM), motivated by the similarities between the closure relations of the reflection-type boundary schemes equipping the LBM equation and the slip velocity condition established by slip-flow theory. Based on this analogy, we derive, as central result, the structure of the LBM boundary closure relation that is consistent with the second-order slip velocity condition, applicable to planar walls. Subsequently, three tasks are performed. First, we clarify the limitations of existing slip velocity LBM schemes, based on discrete analogs of kinetic theory fluid-wall interaction models. Second, we present improved slip velocity LBM boundary schemes, constructed directly at discrete level, by extending the multireflection framework to the slip-flow regime. Here, two classes of slip velocity LBM boundary schemes are considered: (i) linear slip schemes, which are local but retain some calibration requirements and/or operation limitations, (ii) parabolic slip schemes, which use a two-point implementation but guarantee the consistent prescription of the intended slip velocity condition, at arbitrary plane wall discretizations, further dispensing any numerical calibration procedure. Third and final, we verify the improvements of our proposed slip velocity LBM boundary schemes against existing ones. The numerical tests evaluate the ability of the slip schemes to exactly accommodate the steady Poiseuille channel flow solution, over distinct wall slippage conditions, namely, no-slip, first-order slip, and second-order slip. The modeling of channel walls is discussed at both lattice-aligned and non-mesh-aligned configurations: the first case illustrates the numerical slip due to the incorrect modeling of slippage coefficients, whereas the second case adds the effect of spurious boundary layers created by the deficient accommodation of bulk solution. Finally, the slip-flow solutions predicted by LBM schemes are further evaluated for the Knudsen's paradox problem. As conclusion, this work establishes the parabolic accuracy of slip velocity schemes as the necessary condition for the consistent LBM modeling of the slip-flow regime.