High-order lattice Boltzmann models for wall-bounded flows at finite Knudsen numbers

Discrete Velocity Boltzmann Model for Quasi-Incompressible Hydrodynamics

In this paper, we consider the development of the two-dimensional discrete velocity Boltzmann model on a nine-velocity lattice. Compared to the conventional lattice Boltzmann approach for the present model, the collision rules for the interacting particles are formulated explicitly. The collisions are tailored in such a way that mass, momentum and energy are conserved and the H-theorem is fulfilled. By applying the Chapman–Enskog expansion, we show that the model recovers quasi-incompressible hydrodynamic equations for small Mach number limit and we derive the closed expression for the viscosity, depending on the collision cross-sections. In addition, the numerical implementation of the model with the on-lattice streaming and local collision step is proposed. As test problems, the shear wave decay and Taylor–Green vortex are considered, and a comparison of the numerical simulations with the analytical solutions is presented.

Consistent lattice Boltzmann modeling of low-speed isothermal flows at finite Knudsen numbers in slip-flow regime. II. Application to curved boundaries

Gaseous flows inside microfluidic devices often fall in the slip-flow regime. According to this theoretical description, the Navier-Stokes model remains applicable in bulk, while at solid walls a slip velocity boundary model shall be considered. Physically, it is well established that, to properly account for the wall curvature, the wall slip velocity must be determined by the shear stress, rather than the normal component of the velocity derivative alone, as commonly applied to planar surfaces. It follows that the numerical transcription of this type of boundary condition is generally a challenging task for standard computational fluid dynamics (CFD) techniques. This paper aims to show that the realization of the slip velocity condition on arbitrarily shaped boundaries can be accomplished in a natural way with the lattice Boltzmann method (LBM). To substantiate this conclusion, this work undertakes the following three studies. First, we examine the conditions under which the generic reflection-type boundary rules used by LBM become consistent models for the slip velocity boundary condition. This effort makes use of the second-order Chapman-Enskog expansion method, where we address both planar and curved boundaries. The analysis also clarifies the capabilities and limitations behind the considered reflection-type slip schemes. Second, we revisit the family of parabolic accurate LBM slip boundary schemes, originally formulated in [Phys. Rev. E 96, 013311 (2017)2470-004510.1103/PhysRevE.96.013311] on the basis of the multireflection framework, and discuss their characteristics when operating on curved boundaries as well as the limitations of other less accurate LBM slip boundary formulations, such as the linearly accurate slip schemes and the widely popular "kinetic-based" boundary schemes. In addition, we also discuss the numerical stability of the parabolic slip schemes previously developed, providing an heuristic strategy to improve their stable range of operation. Third, we evaluate the performance of the several slip boundary schemes debated in this paper. The numerical tests correspond to two classical 2D benchmark flow problems of slip over non-planar solid surfaces, namely: (i) the velocity profile of the cylindrical Couette flow, and (ii) the permeability of a slow rarefied gas over a periodic array of circular cylindrical obstacles. The obtained numerical results confirm the competitiveness of the LBM when equipped with slip boundary schemes of parabolic accuracy as CFD tool to simulate slippage phenomena over arbitrarily non-planar surfaces. Indeed, although operating on a simple uniform mesh discretization, the LBM yields a similar, or even superior, level of accuracy compared to state-of-the-art FEM simulations conducted on hardworking body-fitted meshes. This conclusion establishes the LBM as a very appealing CFD technique for simulating microfluidic flows in the slip-flow regime, a result that deserves further exploration in future studies.

Consistent lattice Boltzmann modeling of low-speed isothermal flows at finite Knudsen numbers in slip-flow regime: Application to plane boundaries

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.