199 mathematical modeling of mold-filling and solidification - Acta ...

Acta Metallurgica Slovaca, 15, 2009, 3 (199 - 207) 202
For the incompressible solution algorithm the following equation is used:
dρ 1
=
dP β
where β is the bulk modulus.
e) The dissipation rate equation for NKE - (New k-ε Model) is given by:
∂( ρε ) ∂( ρuε
) ∂( ρvε
) ∂( ρwε
) ∂ ⎛ µ ⎞ ⎛ ⎞ ⎛ ⎞
+ + + = ⎜ t ∂ε
∂ µ
⎟ +
+ ⎜ ⎟
⎜ t ∂ε
∂ µ
⎟ t ∂ε
+
∂t
∂x
∂y
∂z
∂x
⎝ σ ε ∂x
⎠ ∂y
⎝ σε ∂y
⎠ ∂z
⎝ σ ε ∂z
⎠
2
ε ε Cµ
(1 C3)
β1ρκ
⎛ ∂T
∂T
∂T
⎞
C1ε
µ t Φ C2ρ
+
⎜ g x + g y + g z
⎟
κ κ σ t ⎝ ∂x
∂y
∂z
⎠
(5)
The new functions; deformation tensor S ij , and the antisymetric tensor W ij , are based
on the velocity components u k in the flow field.
1
Sij
= ( ui,
j + u j,
i )
2
Wij
= 1
( ui,
j − u j,
i ) + Cµ Ω mε
mij
2
where, C µ , is the turbulent constant; Ω m is the angular velocity of the coordinate system and ε mij
is the alternating tensor operator.
The invariants are:
η = κ
2S ij S ij and κ
ξ =
ε
2W ij
ε
Wij
C 1
µ =
, ⎛ η
2 2
4 + 1.5 η + ξ
⎟ ⎞
C 1ε
= max ⎜C1M
⎝ η + 5 ⎠
where, C 1ε , C 2 , σ k , σ ε , σ t , C 3 , C 4 , C 1M and β 1 are constants.
2.2 Free surface modeling
In the last years due to software and hardware technological evolutions, intensive
studies and applications concerning the mold filling by metallic fluids can be found in the
literature. During the mold filling process, liquid metal and air coexist in the mold and the
interface position changes rapidly with time [13]. It is essential to introduce a free-surfacetracking
algorithm to analyze the filling process. The Volume of Fluid Method (VOF) is the
most widely used method for free-surface tracking during mold filling [13]. In a three
dimensional rectangular coordinate frame, the transport equation on the VOF, F(r,t), for an
incompressible fluid can be written as:
∂ F r
+ ∇ F.
v = 0
∂ t
(6)
The F(r,t) function governed by the above equation is unity in the region occupied by
the fluid and zero in the empty region. For the given computational domain, the F(r,t) field