B. Ivancic CFD2k-Software Development and Engineering Corporation (CFD2k-SDEC) 5210 Windisch ; Switzerland Email: info@cfd2k.eu (Dated: February 25, 2008)

Abstract There are several approaches to describe flows with particles e.g. Lattice-Gas Automata (LGA), Lattice-Boltzmann method (LBM) or smoothed particle hydrodynamics (SPH). These approaches do not use fixed grids on which the Navier- Stokes equations are solved via e.g. finite volume method. The flow is simulated using a multitude of particles or particle density distributions, which interacts and due to statistical laws and an even more fundamental approach than the Navier-Stokes equation, the averaged flow variables can be derived. After a short summary of the most popular particle methods the new DMPC (Dissipative Multiple Particles Collision) approach will be presented. The DMPC-model eliminates some of the weak points of the established particle methods and shows high potential for more accurate CFD solution especially in areas where standard CFD tools still have problems (e.g. aero-acoustics). The DMPC-model deals with discrete circular particles and calculates the detailed collision process (micro scale) of several overlapping particles. With thermodynamic, statistical and similarity laws global (large scale) flow variables can be derived. The model is so far 2d and the particles can move in every direction in the 2d plane depending on the forces acting on it. The possible overlap between neighbouring particles and multi-particle interactions are important features of this model. A freeware software is developed and published under www.cfd2k.eu . There the executable, the user guide and several exemplary cases can be downloaded.

Keywords: Particles, CFD, computational fluid dynamics, compressible, ideal gas, turbulence, Lagrange-method, particle-in-cell method, Lattice-Boltzmann, mesh free method, Voronoi-cell, Voronoi-diagram

Introduction Standard CFD solvers work with fixed grids (Euler approach). On this grid the governing equation are solved numerically with a variety of approximate schemes, which are available nowadays [1]. The most general governing equations are the Navier-Stokes equations but models like Euler-equations for inviscid flows are preferably used if this simplification is possible. In fluid dynamics for Euler approaches most popular solver methods are the finite volume methods, finite difference methods and finite element methods. Particle methods work principally different. They use no fixed grid (mesh free method). The particles are distributed over the whole domain and interaction between the particles combined with statistical approaches enables a derivation of averaged flow field variables. Particle methods do not solve the Navier-Stokes (N-S) equations directly because the main difference is that N-S approaches start with a mathematical description of the flow at a continuum level. Particle methods work on a more fundamental i.e. kinetic level. The most popular particle methods are the Lattice- Boltzmann and the SPH (smoothed particle hydrodynamic) methods. These two methods will be now shortly summarised together with their advantages and disadvantages before the new DMPC method with its benefits is presented. The Lattice-Boltzmann method [2, 3] was developed at the end of the 80’s. It is based on the simulation of strongly simplified particle micro dynamics. The simulation on particle level needs a very small amount of computational resources per particle due to the simple inner structure. For this reason this method is appropriate for the simulation of very complex geometries like porous media etc. The Lattice-Boltzmann method has its origin in statistical physics and the governing equations are the Boltzmann equations:

Ω

∂ ∂ + ∂ ∂ + ∂ ∂ v f F x f v t f

(1)

The Boltzmann equation is an evolution equation for a single particle probability distribution function f(x,v,t) where x is the position vector, v the particle velocity vector, F is an external force and Ω is a collision integral. The lattice Boltzmann method discretizes equation (1) by limiting space to a lattice and the velocity space to a set of discrete particle velocities vi. The discretised Boltzmann equation, which is the Lattice Boltzmann equation, can be transformed in the N-S equations. The Lattice Boltzmann method considers particle distributions that are located on lattice nodes and not individual particles. The general form of the lattice Boltzmann equation is:

( ) ( ) i i i i t x f t t t v x f Ω +

Δ + Δ ⋅ + , ,

(2)

Where fi can be interpreted as the concentration of particles that flows with the velocity vi. With this discrete velocity the particle distribution travels to the next lattice node, which is reached in one time step Δt. There are several approaches

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