Civil Engineering Reference
In-Depth Information
FIGURE 6.21
Indexing scheme for set of quantities defined in Eq. (6.40).
For the purpose of our analysis we adopt a specific representation of the discretization employed by
the SIMPLE algorithm which is defined according to
Fig. 6.21
.
Accordingly, it follows that given an
explicit function
(
T
), a discrete set of upstream boundary values
T
B
, specified top surface and midplane
boundary conditions, and a three-dimensional flow field U defined over the region of the meltpool, a
unique discrete temperature field is determined according to the weighted sum
6
1
6
a
p
--------
T
p
a
i
T
i
,
(6.39)
i
1
where
0.1
()
l
(
U
i
)
,0
----------------------------
a
i
i
max
1.0
()
l
max
U
i
0
[
,
]
ip
1
V
B
l
(6.40)
,
i
and
6
1
--
a
p
a
i
.
(6.41)
i
1
Next, by adopting arguments similar to those in the previous section, it follows that the temperature
field
T
E
associated with an arbitrary downstream surface (e.g.,
Fig. 6.20
)
can be interpreted as being
defined by a function
T
E
T
E
(
T
B
,
,
V
B
,
U
)
T
E
(
T
B
,
,
,
,
/
T
,
).
(6.42)
are the density, coefficient of viscosity, surface tension coefficient,
and coefficient of expansion, respectively, of the liquid metal. The dependence of the flow field
U
on
these quantities is through the Navier-Stokes and Continuity equations. As in the previous section, we
consider a partitioning of the solution domain into a set of distributed subdomains and the process of
determining a discrete temperature field
T
p
over a closed solution domain by adopting Eq. (6.39) as an
The quantities
,
,
/
T
, and
