tions. Here, a three-dimensional model with 19 velocities (commonly denoted as D3Q19 ) will be used. Alternatives are models with 15 or 27 velocities. However, the latter one has no clear advantages over the 19 velocity model, while the model with 15 velocities has a decreased stability. The D3Q19 model is thus usually preferable as it requires less 2 The Boltzmann equation itself has been known since 1872. It is named after the Austrian scientist Ludwig Boltzmann, and is part of the classical statistical physics that describe the behavior of a gas on the microscopic scale. The LBM follows the approach of cellular automata to model even complex systems with a set of simple and local rules for each cell [Wol02]. As the LBM computes macroscopic behavior, such as the motion of a fluid, with equations describing microscopic scales, it operates on a so-called ”mesoscopic” level in between those two extremes. Historically, the LBM evolved from methods for the simulation of gases that computed the motion of each molecule in the gas purely with integer operations. In [HYP76], there was a first attempt to perform fluid simulations with this approach. It took ten years to discover that the isotropy of the lattice vectors is crucial for a correct approximation of the NS equations [FdH + 87]. Motivated by this improvement, [MZ88] developed the first algorithm that was actually called LBM by performing simulations with averaged floating point values instead of single fluid molecules. The third important contribution to the basic LBM was the simplified collision operator with a single time relaxation parameter [BGK54, CCM92, QdL92].
60 Real Time Physics Class Notes D2Q9 D3Q19 DFs of length 1 DFs of length 0 DFs of length 2 Cell boundary 2 Dimensions 9 Velocities 3 Dimensions 19 Velocities Figure 10.1: The most commonly used LBM models in two and three dimensions. memory than the 27 velocity model. For two dimensions the D2Q9 model with nine veloc- ities is the most common one. The D3Q19 model with its lattice velocity vectors e 1 .. 19 is shown in Fig. 10.1 (together with the D2Q9 model). The velocity vectors take the fol- lowing values: e 1 = ( 0 , 0 , 0 ) T , e 2 , 3 = ( ± 1 , 0 , 0 ) T , e 4 , 5 = ( 0 , ± 1 , 0 ) T , e 6 , 7 = ( 0 , 0 , ± 1 ) T , e 8 .. 11 = ( ± 1 , ± 1 , 0 ) T , e 12 .. 15 = ( 0 , ± 1 , ± 1 ) T , and e 16 .. 19 = ( ± 1 , 0 , ± 1 ) T . As all formulas for the LBM usually only depend on the so-called particle distribution functions ( DFs ), all of these two-dimensional and three-dimensional models can be used with the method presented here. To increase clarity, the following illustrations will all use the D2Q9 model. For each of the velocities, a floating point number f 1 .. 19 , representing a blob of fluid moving with this velocity, needs to be stored. As the LBM originates from statistical physics, this blob is thought of as a collection of molecules or particles. Thus, in the D3Q19 model there are particles not moving at all ( f 1 ), moving with speed 1 ( f 2 .. 7 ) and moving with speed √ 2 ( f 8 .. 19 ). In the following, a subscript of ˜ i will denote the value from the inverse direction of a value with subscript i . Thus, f i and f ˜ i are opposite DFs with in-
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