Lattice Boltzmann Methods: High Performance Computing and Engineering Applications

G. Brennera; Th. Zeisera; K. Beronova; P. Lammersa; J. Bernsdorfb    a Institute of Fluid Mechanics, University of Erłangen-Nuremberg, Cauerstraße 4, 91058 Erlangen, Germany
b C&C Research Laboratories, Rathausallee 10, 53757 Sankt Augustin, Germany

The development of novel numerical methods for applications in computational fluid dynamics has made rapid progress in recent years. These new techniques include the lattice gas and lattice Boltzmann methods. Compared to the traditional CFD methods, the lattice Boltzmann methods are based on a more rigorous physical modelling, the Boltzmann equation. This allows to circumvent many deficiencies inherent in existing Navier-Stokes based approaches. Thus, the lattice Boltzmann methods have attracted a lot of attention in the fluid dynamics community and emerged as an attractive alternative in many application areas. In the present paper, we discuss some perspectives of the lattice Boltzmann methods, in particular for industrial applications and present some successful examples from projects related to aerodynamics, chemical and process engineering.

1 Introduction

In the past years, the methods of lattice gas cellular automata (LGCA) and the lattice Boltzmann methods (LBM) have attained a certain maturity and subsequently challenged the traditional methods of computational fluids dynamics (CFD) in many areas. In that context, traditional methods of CFD are understood to include all numerical schemes, that aim to solve the Navier-Stokes equations by some direct discretisation. In contrast to that, the LBM is based on a more rigorous description of the transport phenomena, the Boltzmann equation. Compared to other attempts, that have been made to solve this equation in the past, the LBM makes use of several significant, physically motivated simplifications that allow to construct efficient and competitive or even superior computational codes as compared to the classical approaches.

Lattice gas cellular automata and even more lattice Boltzmann methods are relatively new. Just about 15 years ago, the field of LGCA started almost out of the blue with the now famous paper of Frisch, Hasslacher and Pomeau [1], who showed that some simplified kind of ”billiard game” representing the propagation and collision of fluid particles leads to the Navier-Stokes equations in a suitable macroscopic limit. In particular, the authors showed how the propagation and collisions of particles have to be abstracted in order to conserve mass and momentum and how the underlaying lattice has to be designed in order t,o provide sufficient symmetries to obtain Navier-Stokes like behaviour. Each month, several papers appear to present new models or to investigate existing models, to demonstrate and assess the use of LBM in application fields or to evaluate high performance computing (HPC) aspects. Summerschools, special conferences and LBM sessions in existing conferences have been organised to satisfy also the growing interest of developers and potential users in this technique. Besides that, commercial products are available with remarkable success (see e. g [2])

The goal of the present paper is to show the potential of the lattice Boltzmann method in CFD and in related areas. Besides the classical application fields, such as aerodynamics, these are in particular problems related to chemical and process engineering. Due to the complexity of the relevant transport and chemical conversion mechanisms, that have to be modelled, these areas open new challenges also for the LBM.

In the present paper, after a short summary of the basic principles of the LBM, examples related to turbulent flows, reacting flows and the respective application fields are discussed.

2 Lattice Gas and Boltzmann Method

From a gaskinetical, i.e. microscopic, point of view, the movement of a fluid may be considered as the propagation and collision of molecular particles governed by fundamental laws of physics. The modelling of this motion may be carried out on several levels, starting with the Hamilton equation of a set of discrete particles. Since this approach prohibits itself because of the large number of freedoms to be considered, several attempts have been made to simplify this picture by extracting only the essential criteria required to model e. g. the motion of a Newtonian fluid. In that context, the lattice gas automata may be seen as an abstraction of the fluid making use of the fact, that the statistics of the gas may be correctly described by a significantly reduced number of molecules and by applying simplyfied dynamics of the particles. This can be explained by the fact, that the conservation principles as well as associated symmetries are the basic building blocks for the continuum equations of fluids. Thus, in oder to simulate a continuum flow, the approximation of the computer gas has to recover only these principles to a certain extend. The FHP automata, named after [1], was a first successful attempt to construct a discrete model to compute the motion of a Newtonain fluid. Although this discrete particle approach seems promising, there are problems due to spurious invariants and random noise in the solutions. These deficiencies can be overcome by applying the idea of McNamara and Zanetti [3], who considered the discrete Boltzmann equation as a base for the numerical algorithm. This approach may be briefly explained as follows: The Boltzmann equation is an integro-đifïerential equation for the single particle distribution function tx→υ→, which describes the propability to find a particle in a volume →,x→+dx→ and with a velocity in the range. →,υ→+dυ→ Neglecting body forces one has:

tf+υ→∇f=Qf

A suitable simplification of the complicated collision integral Q(f) is the BGK approximation,

f≈1τfeq−f

which preserves the lower moments an satisfies an H-Theorem like the original equations 1. Here feq is the Maxwell equilibrium distribution. The discretisation of this equation requires a finite representation of the distribution function in the velocity space. One way to realise this is to introduce a finite set of velocities →i and associated distribution functions itx→υ→i, which are governed by the discrete velocity Boltzmann (BGK) equation:

tfi+υ→i∇fi=1τfieq−fi

Next, the discretisation in space and time is accomplished by an explicit finite difference approximation. With a scaling of the lattice spacing, the time step and the discrete velocities according to →i=Δx→i/Δt, the discretised equation takes the following form:

ix+υ→iΔt,t+Δt−fix,t=1τfieq−fix,t.

The discrete values of the equilibrium functions are chosen of Maxwell type, in the sense that their velocity moments up to fourth order are identical with the velocity moments over the Maxwell distribution. The following definition satisfies this requirement:

ieq=tpρ1+υiαuαcs2+uαuβ2cs2υiαυiβcs2−δαβ.

The discrete equilibrium functions may be computed efficiently at each time step for each node, from the components of the local macroscopic flow velocity uα, the fluid density ρ, the ”speed of sound cs” and a direction-dependent lattice geometry weighting factor tp. The viscosity ν of the simulated fluid can be controlled by the relaxation time τ, according to

=13τ−12

Further technical details of this method, in particular concerning the formulation of boundary conditions, may be found in [4–6].

From the computational point of view the above approach is interesting as it resembles a simple finite difference scheme applied to a first oder (in time and space) hyperbolic system of equations in diagonal form. This extremely simplifies the design of a numerical scheme. However, finally the solution of the Navier-Stokes equations with second order accuracy in the limit of low Mach numbers s2>>υ→2 is recovered, as can be shown rigorously in [7] by applying the Chapman-Enskog procedure to eq. (4).

The approach presented above leads to the basic version of LBM. Many improvements have been designed in order to broaden the methods range of applicability, see e.g. the review article of Chen and Doolen [8]. Current issues are the multi-time relaxation methods to enhance the stability of the method [9,10], implicit methods [11] or the application of nonuniform and...