Finite-volume lattice Boltzmann schemes in two and three dimensions

Integrating a Stabilized Radial Basis Function Method with Lattice Boltzmann Method

The lattice Boltzmann method (LBM) has two key steps: collision and streaming. In a conventional LBM, the streaming is exact, where each distribution function is perfectly shifted to the neighbor node on the uniform mesh arrangement. This advantage may curtail the applicability of the method to problems with complex geometries. To overcome this issue, a high-order meshless interpolation-based approach is proposed to handle the streaming step. Owing to its high accuracy, the radial basis function (RBF) is one of the popular methods used for interpolation. In general, RBF-based approaches suffer from some stability issues, where their stability strongly depends on the shape parameter of the RBF. In the current work, a stabilized RBF approach is used to handle the streaming. The stabilized RBF approach has a weak dependency on the shape parameter, which improves the stability of the method and reduces the dependency of the shape parameter. Both the stabilized RBF method and the streaming of the LBM are used for solving three benchmark problems. The results of the stabilized method and the perfect streaming LBM are compared with analytical solutions or published results. Excellent agreements are observed, with a little advantage for the stabilized approach. Additionally, the computational cost is compared, where a marginal difference is observed in the favor of the streaming of the LBM. In conclusion, one could report that the stabilized method is a viable alternative to the streaming of the LBM in handling both simple and complex geometries.

Impact of collision models on the physical properties and the stability of lattice Boltzmann methods

The lattice Boltzmann method (LBM) is known to suffer from stability issues when the collision model relies on the BGK approximation, especially in the zero viscosity limit and for non-vanishing Mach numbers. To tackle this problem, two kinds of solutions were proposed in the literature. They consist in changing either the numerical discretization (finite-volume, finite-difference, spectral-element, etc.) of the discrete velocity Boltzmann equation (DVBE), or the collision model. In this work, the latter solution is investigated in detail. More precisely, we propose a comprehensive comparison of (static relaxation time based) collision models, in terms of stability, and with preliminary results on their accuracy, for the simulation of isothermal high-Reynolds number flows in the (weakly) compressible regime. It starts by investigating the possible impact of collision models on the macroscopic behaviour of stream-and-collide based D2Q9-LBMs, which clarifies the exact physical properties of collision models on LBMs. It is followed by extensive linear and numerical stability analyses, supplemented with an accuracy study based on the transport of vortical structures over long distances. In order to draw conclusions as generally as possible, the most common moment spaces (raw, central, Hermite, central Hermite and cumulant), as well as regularized approaches, are considered for the comparative studies. LBMs based on dynamic collision mechanisms (entropic collision, subgrid-scale models, explicit filtering, etc.) are also briefly discussed. This article is part of the theme issue 'Fluid dynamics, soft matter and complex systems: recent results and new methods'.

Lattice Boltzmann method on unstructured grids: further developments

We discuss further developments of the finite-volume lattice Boltzmann formulation on unstructured grids. It is shown that the method tolerates significant grid distortions without showing any appreciable numerical viscosity effects at second order in the mesh size. A theoretical argument of plausibility for such a property is presented. In addition, a set of boundary conditions which permit to handle flows with open boundaries is also introduced and numerically demonstrated for the case of channel flows and driven cavity flows.

Numerical simulation of natural convection in a concentric annulus between a square outer cylinder and a circular inner cylinder using the Taylor-series-expansion and least-squares-based lattice Boltzmann method

In this paper, natural convective heat transfer in a horizontal concentric annulus between a square outer cylinder and a heated circular inner cylinder is numerically studied using the Taylor-series-expansion and least-squares-based lattice Boltzmann method (TLLBM). The TLLBM is used to extend the current thermal model to more practical applications. Since the TLLBM is basically a meshless approach and can be applied to any complex geometry, we can easily use it to solve the complex thermal problem accurately and effectively. The present method is validated by comparing its numerical results with available data in the literature, and very good agreement has been achieved.

Accuracy and stability of a lattice-Boltzmann model with subgrid scale boundary conditions

A lattice-Boltzmann method has recently been developed to incorporate solid-fluid boundary conditions on length scales less than the grid spacing. By introducing a real numbered parameter, specified at each node and representing the fluid volume associated with that node, we were able to accurately simulate arbitrary geometries without the need to specify surface normals. In this paper a detailed description of the rules is presented and the accuracy and stability of the method is discussed, based on numerical results for flow in systems with planar surfaces and for flow through periodic arrays of disks and spheres.

Lattice boltzmann method on composite grids

A composite block-structured lattice Boltzmann method is proposed for the simulation of two-dimensional incompressible fluid flows. The grid structure is composed of a coarse base grid and one or several fine grid(s). The former covers the entire physical domain; the latter are placed at regions where local grid refinement is desirable. The simulation is first carried out on the base grid level at a smaller relaxation time, allowing a rapid propagation of boundary information throughout the entire domain. Thus large-scale flow features can be resolved efficiently at a relatively low cost. At a later time, fine grid variables are initiated. The dependent variables on both grid levels are, then, advanced in time simultaneously with the fine grid boundary conditions obtained from the base grid solution at the grid interface. As a demonstration, the lid-driven cavity flow is selected for study. The results show good agreement with benchmark numerical data and those calculated from the finite-volume U2RANS model. The proposed method is able to produce accurate solutions on fine grids, with a considerable saving in CPU time.