It is important to insure that the model is independent of the
grid orientation, particularly
since, as discussed is section 5.1.6, the orientation of the grid
can affect the change in each distribution function and
whether four or six of the seven distribution functions
are changed. As noted in section 5.1.6 a change in some of the 
at one time-step will produce a change in all the at the next
time-step.
The dependence on the grid orientation
was tested using a 64 by 64 grid which was set up with the
x-axis along the direction of 
.
A new set of perpendicular axis x' and y' were defined so that
the x-axis and the x'-axis intersect at an angle, 
.
A square box 40 lu by 40 lu was then superimposed on the grid, the edges
of the square being parallel to the directions of the x'- and y'-axis.
All points outside this square where then defined as boundary sites.
This is shown in figure 5-3.
Figure 5-3: A box, at angle to the x-axis, superimposed on the regular
grid and the co-ordinate systems. The hashed area is filled with boundary
sites.
The non-boundary sites inside the 40 by 40 lu tilted square were filled with
particles with 
and velocity zero and gravity was applied in
the direction of the -y'-axis. Thus the vertical direction
is in the direction of the
y'-axis.
This was done for 
, 
, 
,
, 
and 
. The density variation with height was then found in
the following ways:
When the
density was measured in a horizontal line (parallel to the y-axis) above
the boundary, as was done in the previous section. Density values were found
at heights with separation 
lu.
When 
the
density was measure in a horizontal line (parallel to the x-axis).
Density
values were found at heights with separation 1 lu.
When 
, , and
the horizontal direction (parallel to the y'-axis) does not lie along
a grid direction. The following method was used to find the density at given
heights above the boundary.
of the line ((0,0),(x+1,y)) and
gradient 
of the line 
. is closer to m or to the
point 
of is closer to m.
and 
.
This gives a set of data 
for a = 1, 2, ...,N
where the value of N depends on the angle .
This is illustrated in figure 5-4
Figure 5-4: Part of the hexagonal grid is shown. The thick solid line represents
the line through O with gradient m, the thick dashed line represents the
`bottom' boundary and the solid dots represent the sites which are considered
as lying nearest to the thick solid line. Point P is the last of these
point which is still within the boundary.
where the thick solid line represents
the line through O with gradient m, the thick dashed line represents the
`bottom' boundary and the solid dots represent the sites which are considered
as lying nearest to the thick solid line. Point P is the last of these
point which is within the boundary so the distance D is the length of OP.
For some values of the line parallel to the y'-axis,
passing through O', see figure 5-3, also passes through the
fluid and the density at various heights above the boundary can be found
along this line. Similar results were obtained for both methods
confirming that the angle of the box and the angle at which gravity is acting
are consistent.