Page 206 - Introduction to Computational Fluid Dynamics

P1: IWV
CB908/Date
0 521 85326 5
0521853265c06
6.3 UNSTRUCTURED MESHES
Further Simplification May 25, 2005 11:10 185
Grouping terms in P and E,k together, we can write Equation 6.94 as

NK NK
V
ρ P + (C ck f ck + d ck ) P = {d ck − (1 − f ck )C ck } E,k
t
k=1 k=1
NK
V
o
+ S V + ρ P o + D k . (6.96)
P
t
k=1
It is possible to simplify this equation further. Thus, let coefficient of Ek be AE k .
Then,
AE k = d ck − (1 − f ck )C ck . (6.97)
Now, for = 1 (i.e., the mass conservation equation), Equation 6.76 gives

NK
V

o
ρ P − ρ + C ck = 0, (6.98)
P
t
k=1
or
NK
V o V
ρ P = ρ P − C ck . (6.99)
t t
k=1
Now, let AP be the multiplier of P in Equation 6.96. Then using Equations
6.97 and 6.99, it follows that 5

NK
V
AP = ρ P + {d ck + f ck C ck } (6.100)
t
k=1
NK
V
= ρ o + {d ck − (1 − f ck )C ck } (6.101)
P
t
k=1
NK
V
o
= ρ + AE k . (6.102)
P
t
k=1
Thus, Equation 6.96 can be compactly written as
NK
NK V

l
o
AP l+1 = AE k l+1 + S V + ρ o + D . (6.103)
P Ek P t P k
k=1 k=1
5 Note the similarity of Equation 6.102 with Equation 6.37 derived for curvilinear grids.