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Obtaining Macroscopic Quantities

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

  figure1372

   table1379
Table: The total number of particles tex2html_wrap_inline13679 on link tex2html_wrap_inline13399 , the x and y components of the averaged velocity tex2html_wrap_inline12875 , its magnitude and inclination from link tex2html_wrap_inline13689 , the total number of particles M and the average density tex2html_wrap_inline12075 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 tex2html_wrap_inline13679 on each of the links tex2html_wrap_inline13399 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 tex2html_wrap_inline12875 and the averaged density tex2html_wrap_inline12075 . The angle tex2html_wrap_inline12063 is the angle between the averaged velocity direction and the x-axis (the direction of tex2html_wrap_inline13709 ). 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.


next up previous contents
Next: Binary Fluid Models Up: The Lattice Gas Model Previous: Units of Measurement in

James Buick
Tue Mar 17 17:29:36 GMT 1998