Page 249 - Computational Fluid Dynamics for Engineers

238 7. Boundary-Layer Equations

shear stress parameter f!^ with those given by Smith [10] which are accurate to
six decimal places. Use four uniform spacings of (h = 0.8, 0.4, 0.2, 0.1) which
correspond, respectively, to 11, 21, 41 and 81 points for a transformed boundary-
layer thickness of 8.
Wall shear values of Smith [10]

f"
J w
1 1.232588
0 0.332000

7-2. The accuracy of the Box scheme, which is of second-order, can be improved
by using Richardson extrapolation as discussed, for example, in [10]. The pur-
pose of attempting to improve accuracy is to get reasonable answers from a
coarse mesh of net points rather than to acquire more significant digits in the
solution. With few net point, the iterations converge more rapidly and the com-
putations are generally more efficient.
To describe the application of the Richardson extrapolation, consider the
numerical solution of the momentum equation (7.3.6) subject to boundary con-
7
ditions given by Eqs. (7.3.7) and denote the solution [f?(r),u j(r),Vj'(r)] on a
r
net of spacings r ) and k^. Now let two independent computations be made
/
on net with spacings (h,k) = [h(°\k] and (h,k) = [h^\k\. Then we form
h(0)2 fn [h(l). k]_ h(l)2 fn [h(0). k]
1
r
f?^'^""^'j:r :j' h(0)2 " _ h(l)2 •*> <**.*>
4
0
to obtain solutions that are of fourth-order accuracy in /i, (/i ). Similar results
r
r
also hold for u -,v j'. An even more accurate approximation can be obtained, re-
garding the order of errors in /i, if calculations are made by a third admissible
2
net with (h,k) = [h&\k].We now also compute f^[h^\ h^ \ k] in obvious ana-
log with Eq. (P7.2.1a) and form
(
( 1
fi [fc °U U ( 2 ) ;*] - ^(0)2 _ i2)2 ( P 7 2 l b
- - )
h
6
to obtain solutions that are of sixth-order accuracy in /i, 0(h ).
Clearly, the procedure could be reversed and only the £ spacing defined in
an admissible way. More important, the Richardson extrapolation can be used
6
to get, say, 0(h G + fc ) accuracy, because then
fc(D2 fc(l)2
etc. (P7.2.2)
fc(0)2 _ /j(l)2 fc(0)2 _ fc(l)2 '

Even higher-order accuracy can be obtained by additional computations and
extrapolations.