256
9 Dynamical Behaviors of Keyhole and Weld Pool …
where
c p specific heat;
k
thermal conductivity;
T
temperature.
The Level Set method is used to track the time-varying keyhole free surface. The
governing equation of the Level Set method is written as:
∂φ
∂t
+
− →
U · ∇φ = 0
(9.5)
9.2.2 Boundary Conditions in Vacuum Laser Welding
Given the influence of surface pressure, surface tension, and viscous pressure on
keyhole free surface, the free surface is subject to a pressure boundary condition
consistent with that in atmospheric laser welding. The pressure boundary condition
can be expressed as follows:
p f = p r + σ κ + 2μ n · ∇
− →
U
n
(9.6)
where
Subscript f free surface;
p r
recoil pressure;
σ
surface tension coefficient;
κ
free surface curvature;
μ
dynamic viscosity;
− →
U
three-dimensional velocity vector;
n
three-dimensional unit normal vector in free surface.
The free surface, is also subject to a viscous stress boundary condition due to
the thermal capillary force similar to that in atmospheric laser welding. The viscous
stress boundary condition can be expressed as:
(μ∇
− →
U ) f = μ( n
t 1
t 2 )( n 0 0)
T
(∇
− →
U )( n 0 0)( n
t 1
t 2 )
T
+ μ( n
t 1
t 2 )(0
t 1
t 2 )
T
(∇
− →
U )
− μ( n
t 1
t 2 )( n 0 0)
T
(∇
− →
U )
T
( n
t 1
t 2 )( n
t 1
t 2 )
T
+ ( n
t 1
t 2 )
⎛
⎝
0 ∇ s σ ·
t 1 ∇ s σ ·
t 2
0
0
0
0
0
0
⎞
⎠
(9.7)