Lattice-Boltzmann method

The Lattice-Boltzmann method (also Lattice-Boltzmann method or grid-Boltzmann method ) is a method for numerical flow simulation developed in the late 1980s . As the name suggests, the phase space for the numerical solution of the Boltzmann equation is discretized by a grid . By incorporating further models, other physical processes in a continuum, such as thermodynamic processes in fluids or solids, can also be calculated using Lattice-Boltzmann methods.

The Lattice-Boltzmann method is based on the calculation of a greatly simplified particle microdynamics. This means that a simulation is carried out on the particle level. Due to the internal structure (low memory and computation requirements per cell), the method is particularly suitable. a. for the calculation of flows in complex geometries. The Lattice-Boltzmann method has its theoretical basis in statistical physics. The interaction of the microscopic particles is described by the Boltzmann equation.

Clear presentation of the algorithm

Schematic representation of a time step in a D2Q9 model

To solve the Boltzmann equation, it is discretized. The discretization takes place by introducing a grid in the spatial area, which also discretizes the velocity directions. Thus the whole phase space is discretized. A two-dimensional space can be discretized with the D2Q9 model shown here, for example. In the figure, the points represent points in spatial space, while the arrows represent how likely the speed of the particles associated with a point occurs in the direction of the arrow at the respective point. A fluid particle can stay in the same place per time step or move in the adjacent cells of the square grid. It therefore has nine possible speeds , with the index indicating the direction. ${\displaystyle {\vec {c}}_{i}}$ ${\displaystyle i=0,\ldots ,8}$

The algorithm can be divided into two sub-steps, the order of which is fixed but arbitrary:

1. Collision step
2. Flow step

The collision rules are applied in the collision step. These rules must preserve the mass as well as the momentum. A correspondingly calculated collision term is added to each phase space density at the location : ${\displaystyle f_{i}}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle \Delta _{i}}$

${\displaystyle f_{i}^{*}({\vec {x}},t)=f_{i}({\vec {x}},t)+\Delta _{i}}$

One possible collision term is the Bhatnagar – Gross – Krook (BGK) operator

${\displaystyle \Delta _{i}={\frac {1}{\tau }}(f_{i}^{\mathrm {eq} }-f_{i})}$.

The relaxation time determines how quickly the fluid approaches equilibrium and thus depends directly on the viscosity of the fluid. The value is the local equilibrium function which approximates the Boltzmann distribution. ${\displaystyle \tau }$ ${\displaystyle f_{i}^{\mathrm {eq} }}$

During the flow step, all arrows are moved (according to their direction) to the next grid point:

${\displaystyle f_{i}({\vec {x}}+{\vec {c}}_{i}\delta t,t+1)=f_{i}^{*}({\vec {x}},t)}$

The arrows shifted in this way form the starting point for the next collision step.

literature

• S. Chen and GD Doolen: Lattice Boltzmann method for fluid flows , Annu. Rev. Fluid Mech., 30: 329-364, (1998) doi : 10.1146 / annurev.fluid.30.1.329
• X. He and L.-S. Luo. Theory of the lattice boltzmann method: From the boltzmann equation to the lattice boltzmann equation . Phys. Rev. E, 56 (6): 6811-6817, (1997) doi : 10.1103 / PhysRevE.56.6811
• Dieter A. Wolf-Gladrow: Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction, Springer Verlag, 2000
• Sauro Succi : The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford University Press, 2001
• S. Scheiderer: Diploma thesis Efficient parallel Lattice-Boltzmann simulation for turbulent flows (PDF file; 8.18 MB), 2006
  (overview of theory and simulation of turbulent flows using the LB method. It will also use the multi-relaxation time (MRI) scheme.)
• Axel Reiser: Real-time solid-state simulation with the Lattice Boltzmann method , Bachelor thesis University of Stuttgart (explanation of the basic scheme in German)

Individual evidence

1. ^ AA Mohamad, Lattice Boltzmann method, Springer 2011, p. 62