Page 230 - Computational Fluid Dynamics for Engineers

7.3 Numerical Method for the Standard Problem 219

Similarly, the partial differential equation (7.3.12c) is approximated by center-
2
ing about the midpoint (^ n _ 1 ^ ,^_ 1 / 2 ) of the rectangle P\P*iP?>P±' This can
n 1//2
be done in two steps. In the first step we center it about (^ ~ ,ry) without
specifying 77. If we denote its left-hand side by L, then the finite-difference ap-
proximation to Eq. (7.3.12c) is

1 1 ^ - V 2 „»-V> (""-J-' / fn fn—1
- ( I ^ + Z/ - ) =
(7.3.16)
tn~l/2 ra n + l
n
a n = - ai = + a , a 2 = m n + a n (7.3.17a)
k '
Rn-l = _ Ln-l + an^f vy-l _ (^)n-l] _ mn (7.3.17b)

n-l
/
2
-
n l
L ~ = (H + ^ i / ^ + m(l-7i ) (7.3.17c)
Eq. (7.3.16) can be written as
n 71 2 n n n l n n l n m— 1
[(bv)'] + axifv) - a 2(u ) + a (v - f - ~ v ) = R (7.3.18)
f
The identity sign introduces a useful shorthand: [ " _ 1 means that the quantity
]
in square brackets is evaluated at £ — £ n _ 1 .
n 1 2
We next center Eq. (7.3.18) about the point (£, ~ ' ,Vj~i/2)> that is, we
choose r\ — T7 J_ 1/ 2 a n d obtain

hjHbW - b^v^) + ai(fv)]_ 1/2 - a 2(u^_ 1/2
(7.3.19)
^nf n,n—l rn rn—1 .,n \ _ on—1
1
n
R
+« (^"l / 2/^l/2 - ^-1/2^1/2) = : 1-1/2
where
R L n U2 (7.3.20a)
T-l/2 = - T-V2 + « M«)£},2 - ( )"-V2] - ™"

2
^ Z i / 2 = [hjHbjVj - fcj-ivj-i) + ^ ( / * V i / 2 + m[l - ( u ) , - ! / ^ } " '
(7.3.20b)
1
Eqs. (7.3.15) and (7.3.19) are imposed for j = ,2,..., J — 1 at given 77 and
^ —7 —7 7 ~ — ~~~ o /
K o
o f f l ; n
+
io
"
" °
the transformed boundary-layer thickness 77 e , is to be sufficiently large so that
«
"
transiormea oounaary-iayer tnicKness, r] ei is to oe sumciei
»(7/ e )
u —> 1 asymptotically. The latter is usually satisfied when i; is less than
3
approximately 10~ .
The boundary conditions [Eq. (7.3.13)] yield, at f = <f\
/
/o n = - 5 < = 0, t # = l (7.3.21)