Benchmark computations based on lattice-Boltzmann, finite element and finite volume methods for laminar flows

Abstract

The goal of this article is to contribute to the discussion of the efficiency of lattice-Boltzmann (LB) methods as CFD solvers. After a short review of the basic model and extensions, we compare the accuracy and computational efficiency of two research simulation codes based on the LB and the finite-element method (FEM) for two-dimensional incompressible laminar flow problems with complex geometries. We also study the influence of the Mach number on the solution, since LB methods are weakly compressible by nature, by comparing compressible and incompressible results obtained from the LB code and the commercial code CFX. Our results indicate, that for the quantities studied (lift, drag, pressure drop) our LB prototype is competitive for incompressible transient problems, but asymptotically slower for steady-state Stokes flow because the asymptotic algorithmic complexity of the classical LB-method is not optimal compared to the multigrid solvers incorporated in the FEM and CFX code. For the weakly compressible case, the LB approach has a significant wall clock time advantage as compared to CFX. In addition, we demonstrate that the influence of the finite Mach number in LB simulations of incompressible flow is easily underestimated.

Introduction

During the last decade much progress has been achieved in designing kinetic models of minimum complexity for fluid flow problems, especially in terms of the lattice-Boltzmann (LB) method. We refer to [1], [2], [3] and the literature cited therein for an introduction to the field. LB methods are used to simulate a variety of complex flows, but there is insufficient evidence to demonstrate the computational efficiency of a LB-based CFD solver is comparable to a state-of-the-art solver based on the direct discretization of the incompressible Navier–Stokes equations. Recently there have been several works comparing the results of simulations based on kinetic and macroscopic methods [11], [12], [13], [14], [15], [18], but either the details of the LB-models used are unknown (PowerFlow [19]) and any deviations in simulation results cannot be analyzed or the latest model extensions such as local grid refinement, improved second order boundary conditions, or multiple relaxation time models (see below) have not yet been incorporated into the kinetic simulation prototype. Although most engineering fluid flow applications have to deal with the issue of turbulence, we restrict ourself in this work to the incompressible laminar flow regime. The paper is organized as follows: In Section 2 a short summary of the LB basics and the potential advantages and disadvantages of LB methods for CFD simulations are discussed. Also in Section 2 we motivate the necessity of some extensions utilized for the benchmark. In Section 3 we sketch some properties of the finite element prototype and in Section 4 we define the benchmark problems. The results of the computations together with the meshes used for the different methods are presented and discussed in the last two sections, Sections 5 Results, 6 Discussion and outlook.