34
2 Model of Quasi-Steady Weld Pool Dynamics and Numerical Simulation
a P p
P = a E p
E + a W p
W + a N p
N + a S p
S + a T p
T + a B p
B + b
(2.45)
where
a E = ρd e yz
(2.46a)
a W = ρd w yz
(2.46b)
a N = ρd n zx
(2.46c)
a S = ρd s zx
(2.46d)
a T = ρd t xy
(2.46e)
a B = ρd b xy
(2.46f)
a P = a E + a W + a N + a S + a T + a B
(2.46g)
b =
ρu
∗
w
−
ρu
∗
e
yz +
ρv
∗
s
−
ρv
∗
n
zx
+
ρw
∗
b
−
ρw
∗
t
yx
(2.46h)
2.3.4 SIMPLE Algorithm
Patankar proposed a SIMPLE algorithm to solve the fluid flow and heat transfer
problem, namely the semi-implicit method for pressure-linked equations.
The calculation steps for the SIMPLE algorithm are as follows:
(1) The staggered grid is shown in Fig. 2.6. The real points of the main grid are
the nodes of the predicted pressure value (p)
n , and the virtual points represent
the grid points of the velocity values (ρu*)
nn and (ρv*)
n .
(2) The (ρu*)
n+1 and (ρv*)
n+1 are respectively solved from the following equations
ρu
∗
n+1
i+1/2, j
=
ρu
∗
n
i+1/2, j
+ A
∗
t −
t
x
p
∗
i+1, j
(2.47)