## M

Figure 7. Discrete velocities of the D3Q15 model on a three-dimensional square lattice. (Figure is taken from Freundiger [34]).

constraints would not necessarily be isotropic a Galilean invariant in general, as observed in some newly proposed LBE models for non-ideal gases. Two other parameters, a¡ and y4, remain adjustable. In addition, there are six relaxation parameters s¡ in the model as opposed to one in the LBE BGK model. Two of them, s3 and s8, determine the bulk and the shear viscosities, respectively. Also, because c;=-2, s9=sH [this reference]. The remaining three relaxation parameters, s}, s¡ and s7, can be adjusted without having any effect on the transport coefficients in the order of k2. However, they do have effects in higher-order terms. Therefore, one can keep values of these three relaxation parameters only slightly larger than 1 (no severe over-relaxation effects are produced by these modes) so that the corresponding kinetic modes are well separated from those modes more directly affecting hydrodynamic transport.

It is interesting to note that the present model degenerates to the BGK LBE model [8,9] if we use a single relaxation parameter for all the modes, i.e., sa= 1/r, and choose a¡= 4 and y 4= -18.

Therefore, in the BGK LBE model, all the modes relax with exactly the same relaxation parameter so there is no separation in time scales among the kinetic modes. This may severely affect the dynamics and the stability of the system, due to the coupling among these modes.

### 3.6 Lattice Boltzmann model in Three Dimensions

Each point on a unit cubic lattice space has six nearest neighbours, (±1,0,0), (0,±1,0), and (0,0,±1), twelve next nearest neighbours, (±1,±1,0), (±1,0,±1), and (0,±1,±1), and eight third nearest neighbours, (±1,±1,±1). Elementary discrete velocity sets for lattice Boltzmann models in three dimensions are constructed from the set of twenty-six vectors pointing from the origin to the above neighbours and the zero vector (0,0,0). The twenty-seven velocities are usually grouped into four subsets labelled by their squared modulus, 0,1,2, and 3. We also use the notation DdQq for the q-velocity model in d-dimensional space in what follows [9]. The fifteen discrete velocities in the D3Q15 (see Figure 7) model are:

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