Page 231 - Computational Fluid Dynamics for Engineers

220 7. Boundary-Layer Equations

7.3.2 Newton's Method

Unlike the unsteady heat conduction equation, Eq. (4.2.4), the boundary-layer
equations are nonlinear. So after we write the finite-difference approximations,
we follow the third step (see subsection 4.4.3) and linearize Eqs. (7.3.15), (7.3.19)
and (7.3.21).
l
v
J
If we assume J 1 - 1 , uT j~ -> a n d 7~ 1 to be known for 0 < j < , then Eqs.
/
(7.3.15), (7.3.19) and (7.3.21) form a system of 3J + 3 equations for the so-
1
lution of 3 J + 3 unknowns (/J , vtt, vj), j = 0 , 1 , . . . , . To solve this nonlin-
J
ear system, we use Newton's method; we introduce the iterates [/'} , u^ , ^ ],
v = 0,1, ,..., with initial value (y = 0) equal to those at the previous x-station
2
n l
x ~ (which is usually the best initial guess available). For the higher iterates
we set
y
y
y
y
fj = /jf + ofj \ u j J =uy + Su)- ', v j J = v j + Sv j (7.3.22)
We then insert the right-hand sides of these expressions in place of f? uf
and v^ in Eqs. (7.3.15) and (7.3.19) and drop the terms that are quadratic in
<$/• , Syr" and 6v^. This procedure yields the following linear system (the
superscript n is dropped from / j , UJ, Vj and v from (5 quantities for simplicity).
fi
Sfj ~ Sfj-i ~ y ( < ^ + ^-i) = ^ (7.3.23a)

v
6i/j — Suj^i —^-(^Vj + 8 j-i) = (^3)j-i (7.3.23b)
(SI)J6VJ + (S2)JSVJ-I + (ss)jSfj + (s 4)j6fj-i + (s 5)j6uj + (SG)J6UJ-I = (r 2)j
(7.3.23c)
where
7 3 24a
(ri); = / j ^ - / j " ) + M ^ i / 2 ( " - )
(ra^-i = ^ - uf ] + h jVf\ /2 (7.3.24b)

~h?{bMvf - bf} lVf\) + «!(/«)W 1 / 2
R
to); = l-l/2
2 n )
- M u ) ^ 1 / 2 + « ( ^ _ 1 / 2 / j - 1 / 2 " ^ - i / a ^ i / a ) J
(7.3.24c)
In writing the system given by Eqs. (7.3.23) we have used a certain order
for them. The reason for this choice, as we shall see later, is to ensure that the
AQ matrix in Eq. (7.3.28a) is not singular.
The coefficients of the linearized momentum equation are

f V)
+
1
- -
(«i)i = V ^ T i ~ T T I / * (7 3 25a)