Bhatnagar-Gross-Krook relaxation

In this study, the Boltzmann equation is solved with the help of a single relaxation time collision operator approximated by the Bhatnagar-Gross-Krook (BGK) approach [1], Here, the relaxation of the distribution function to an equilibrium distribution is supposed to occur at a constant relaxation parameter r. The substitution of the continuous velocities in the Boltzmann equation by discrete ones leads to the discrete Boltzmann equation, where fai = fm(x, t). The number of available discrete velocity directions ai that connect the lattice nodes with each other depends on the applied model. In this work, the D3Q19 model is used which applies for a three-dimensional grid and provides 19 distinct propagation directions. Discretising time and space with At and Ax = At yields the Lattice-Boltzmann equation ... 

The simplest such collision operator is the lattice BGK (Bhatnagar-Gross-Krook) model [77], Cij = -Sij/x, where the collisional relaxation time t is related to the viscosity. Here we will work within the more general framework of the multirelaxation time (MRT) model [110], for which the lattice BGK model is a special case. 

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