168
6. Numerical Solutions of Advection-Dominated Problems
where
(I = i,j, k)
(6.2.44)
The second line integral on the left-hand side ofEq. (6.2.29) can be calculated
in the same way:
r mC(Yydx - Vxdy) = L (ßi~Ci + ßtjCj + ßt"Ck ),
J(L i )
ei
(6.2.45)
where
1= i,j, k,
(6.2.46)
In the derivation of Eq. (6.2.46), we assumed that div(mV) = O. If div(mV) #- 0,
the terms in Eq. (5.2.14) should also be added to ßi! (Sun and Yeh, 1983).
In order to calculate the right side of Eq. (6.2.29), we can divide it into three
parts which are commensurate with Eq. (5.2.17), that is,
fi [ o(mC) ] fi oC fi om .
-;:) - + M dxdy =
m;:;;-dxdy +
C;:;;-dxdy
~
ut
~
ut
~
ut
+fr Mdxdy.
J(Di)
(6.2.47)
Using the expressions of C and oC/ot in different parts of subdomain (DJ, the
three integrals on the right-hand side of the above equation can be obtained
easily. The result is
(6.2.48)
where
(6.2.49)
and
(6.2.50)