Page 198 - Computational Fluid Dynamics for Engineers

186 6. Inviscid Flow Equations for Incompressible Flows

4>i+ij - 2(/>jj + </>j-ij 01+1J - 01-1J
Vi + Vi
2Arj
+ (*0 2 = 0 (6.3.12a)

or with (3 = A£/Aq

®l(/>i,j + a2<fo+i,j + ^30i-l j + <l>ij+l - <t>i,j-i = l 0 ° (6.3.12b)
i
+
0
j
-
=
where
2
2
l
a i =-2fa?/? 2 + ), a 2 =f7?|9 2 + ^ ( ^ ) A a 3 = T / / ? - ^ ^ ) / ? (6.3.13)
The boundary conditions are
i = 0, 00, j = cos£j/r/o (6.3.14a)
i = I+l, * - 0 (6.3.14b)
orj
3 = 0, g- (6.3.14c)

j = J + l , (6.3.14d)
% - *
Using Eq. (6.3.14a), Eq. (6.3.12b) at i — 1 can be written as

a i 0 i j + O202J + 01J+1 + 01J-1 = - " 3 00 ,j = - « 3 cos£j/riQ (6.3.15a)
Using Eq. (4.5.14) in Eq. (6.3.14b), Eq. (6.3.12b) at i = I can be written as

a
(ai + -a 2J 4>IJ + f «3 - o 2J 0/-1J + 0/j+i + 0/j-i = ° (6.3.15b)
Using Eq. (4.5.14) in Eq. (6.3.14c), Eq. (6.3.12b) at j = 1 can be written as
4 \ 2
a (6.3.15c)
l + o J 0i,l + «2^i+l,l + «30z-l,l + «0i,2 = 0
Using Eq. (4.5.14) in Eq. (6.3.14d), Eq. (6.3.12b) at j = J can be written as
4 \ 2
a
«1 + o j ^hJ + 20i+l,J + «30i-l,J + g0i,J-l = 0 (6.3.15d)
Using Eq. (4.5.6), with U now equal to </>, Uj =
-J

*1 0iJ
^2 02,i
0 = (6.3.16a)
*i 0z,j

£/ 0/,i