The density and momentum at each site are defined in equation (3.2) as the sum of the occupation numbers at a site and the sum of the occupation number multiplied by its velocity. This allows the density and velocity to be calculated at each site. The density and velocity found in this way are very noisy and need to be averaged. In section 3.5.2 the mean density and velocity are found by taking ensemble averages. In practice it is usually more convenient to average over a region of the grid. This region must be small compared to the typical length scale of the flow being simulated. The larger the region, or averaging cell, the less noisy the results will be. The size of a cell is, however, restricted by the limits imposed on the overall grid size by computer memory and time restrictions. Typically a cell will be no smaller than 16 by 16 sites. Figure 3-8
Table: The total number of particles on link ,
the x and y components of
the averaged velocity , its
magnitude and inclination from link , the total number of
particles M and the average density for the four averaging
cells shown in figure 3-8.
shows four averaging cells (here shown as 6 by 6 sites for convenience) on a portion of a square lattice. Table 3-1 shows the microscopic details of the total number of particles on each of the links in each cell and the total number of particles in each cell, M. Also shown in the table are the details of the averaged velocity and the averaged density . The angle is the angle between the averaged velocity direction and the x-axis (the direction of ). Note that the density is defined to be the mean number of particles per site not the mean number of particles per unit volume. Even after averaging over a large cell the results can still by noisy. Ensemble averaging is often used as well as cell averaging in an attempt to further reduce the noise.