Graphics Reference
In-Depth Information
Assuming (
i
−
1
,j
)and(
i, j
−
1) are fluid cells, this would reduce the
equation to the following:
p
i,j
+
ρ
Δ
x
u
solid
⎛
⎞
Δ
t
u
i
+1
/
2
,j
−
4
p
i,j
−
−
0
−
p
i−
1
,j
−
p
i,j−
1
Δ
t
ρ
⎝
⎠
Δ
x
2
u
i
+1
/
2
,j
−
,
u
i−
1
/
2
,j
Δ
x
+
v
i,j
+1
/
2
−
v
i,j−
1
/
2
Δ
x
=
−
3
p
i,j
−
=
Δ
t
ρ
p
i−
1
,j
−
p
i,j−
1
Δ
x
2
u
i
+1
/
2
,j
−
+
u
i
+1
/
2
,j
−
.
u
i−
1
/
2
,j
Δ
x
+
v
i,j
+1
/
2
−
v
i,j−
1
/
2
Δ
x
u
solid
−
Δ
x
We can observe a few things about this example that hold in general and
how this will let us implement it in code. First, for the air cell bound-
ary condition, we simply just delete mention of that
p
from the equation.
Second, for the solid cell boundary condition, we delete mention of that
p
but also reduce the coecient in front of
p
i,j
, which is normally four, by
one—in other words, the coecient in front of
p
i,j
is equal to the number of
non-solid grid cell neighbors (this is the same in three dimensions). Third,
scale = 1 / dx;
loop over i,j,k where cell(i,j,k)==FLUID:
if cell(i-1,j,k)==SOLID:
rhs(i,j,k) -= scale * (u(i,j,k) - usolid(i,j,k));
if cell(i+1,j,k)==SOLID:
rhs(i,j,k) += scale * (u(i+1,j,k) - usolid(i+1,j,k));
if cell(i,j-1,k)==SOLID:
rhs(i,j,k) -= scale * (v(i,j,k) - vsolid(i,j,k));
if cell(i,j+1,k)==SOLID:
rhs(i,j,k) += scale * (v(i,j+1,k) - vsolid(i,j+1,k));
if cell(i,j,k-1)==SOLID:
rhs(i,j,k) -= scale * (w(i,j,k) - wsolid(i,j,k));
if cell(i,j,k+1)==SOLID:
rhs(i,j,k) += scale * (w(i,j,k+1) - wsolid(i,j,k+1));
Figure 4.3.
Modifying the right-hand side to account for solid velocities.
