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Time integration (5)
10. Account for mass transport across element boundaries (face by face!)
to compute new element mass:
n 1 n L n 1/2
M + = M +∆t⋅ ρ ( w−v)
+ indS
11. For stabilization reasons, the density ρ J (depending on face J) is
computed as a weighted average of the neighbor elements’ densities:
1 [(1 )
e (1 )
e
ρJ = − α
J
ρ + + α
J
ρ ′
]
2
∫
n
S
αJ
= α0 ⋅ sign ( F)
F =∆t⋅ ( w−v)
indS
0< α < 1
α
0
= 0 : centred approximation (oscillations!)
α
0
= 1 : full donor (too diffusive!)
0
∫
S J
19
Time integration (6)
12. The new element density is obtained as:
ρ
n+
1
e
M
=
V
13. Similar treatment then applied also to energy transport:
n+
1
e
n+
1
e
n+
1 n L n
( Mi) −( Mi) L L
n+
12 ( Mi) −( Mi)
= [ ∑( ρ i )
J
( w−v) ⋅ ndS]
+
∆t
∫
J
∆t
SJ
M ∆t
i = i + [ ( i ) ( w−v ) ⋅ndS
]
1 1
n
n+ 1 L L L n+
1 2
ρ
n+ n+
∑ J
M M
∫
J
SJ
14. The final pressure is:
n + 1 1 1
( n +
p f ρ , i
n +
= )
15. The nodal forces due to momentum transport (for the next step) are:
∫
n+
12
e L n+
12
tiI
=
I
ρ ( − ) ⋅∇
i
e
V
F N w v v dV
20
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