Page 257 - Computational Fluid Dynamics for Engineers

8.2 Solution of the Orr-Sommerfeld Equation 247

8.2.1 Numerical Formulation

To formulate the numerical scheme employing the box method, we consider Eq.
(2.5.13) and its boundary conditions given by (8.1.1) and reduce them to an
equivalent first-order system. We define

4>' = f (8.2 !.la)
2,
f = * + & (8.2 Mb)
J = g (8.2 i.lc)
and write Eq. (2.5.13) as
g' = & - &<f> (8.2 .Id)

With these variables, the boundary conditions become
y = 0, 0 = 0, / = 0 (8.2 !.2a)

6
y = 6; s + (£i + &)/ + (6 + 6)0 = 0;
g + &s = 0 (8.2.2b)
where £2, defined in Eq. (P2.7.6b) attains its value at y — 6.
We now consider a nonuniform mesh with y = 0 represented by yo and y = 6
by yj and approximate the quantities (/, s,g,</)) at points yj by (fj,Sj,gj,<f>j).
As in subsection 4.4.3, we write the finite-difference approximations of Eqs.
(8.2.1) for the midpoint y-\ using centered-difference derivatives to obtain
J 2
</>j - <t>j-i - c 3(fj + fj-i) = {n)j = 0 (8.2.3a)

s
s
r
c
c
fj ~ fj-i - 3( j + j-i) ~ i(0? + <l>j-i) = ( 3)j-i = 0 (8.2.3b)
s s r
j ~ J-i - °s(9j + 9j-i) = ( 2)j = 0 (8.2.3c)
9j ~ 9j-i - C4(sj + Sj_i) - c 2 (0j + 0j-i) = (u)j-i = 0 (8.2.3d)
Here, with hj-i denoting yj — yj-i,
C3 = - ^ , Ci = ^ C 3 , C 2 = - ( 6 ) , _ I C 3 , C 4 = (ff , - ! < * (8.2.4)
)
As in the procedure for solving the boundary-layer equations, Eqs. (8.2.3) are
again written in a sequence which ensures that the A-matrix in Eq. (4.4.30) is
not singular.
2
Equations (8.2.3) are imposed for j = 1, , . . . , J and additional conditions at
. 7 = 0 and j = J are obtained from the appropriate boundary conditions which,
for an external flow, follow from Eqs. (8.2.2) and can be written as

00 = (ri)o = 0, /o = (r 2 )o = 0 (8.2.5a)
fj + c 3sj + ci0j = (r 3)j = 0, gj + c 4sj = (r 4 ) j = 0 (8.2.5b)