Page 234 - Computational Fluid Dynamics for Engineers

7.4 Computer Program BLP 223

by specifying locations with intervals which can be uniform or nonuniform. Its
distribution depends on the variation of u e with £ so that the pressure gradient
parameter ra(£) can be calculated accurately. To ensure this requirement, it is
necessary to take small Z\£-steps (k n) where there are rapid variations in u e(£)
and where flow approaches separation.
For laminar flows, it is often sufficient to use a uniform grid in the 77-direction.
A choice of transformed boundary-layer thickness rj e equal to 8 often ensures
/f
that the dimensionless slope of the velocity profile at the edge, f (rj e), is suf-
- 3
ficiently small (< 10 ) and that approximately 41 j-points satisfies numerical
accuracy requirements. For turbulent flows, however, a uniform grid is not sat-
isfactory because the boundary-layer thickness rj e and dimensionless wall shear
parameter f!^ are much larger in turbulent flows than laminar flows. Since short
steps in 77 must be taken to maintain computational accuracy when f!^ is large,
the steps near the wall in a turbulent boundary-layer must be shorter than the
corresponding steps in a laminar boundary-layer under similar conditions.
A convenient and useful 77-grid is a geometric progression having the property
that the ratios of lengths of any two adjacent intervals is a constant; that is,
hj — Khj-i [2]. The distance to the j-th line is given by the formula;
K* - 1
Vj = h i j = l , 2 , . . . , J K>\ (7.4.2)
K - \
There are two parameters: hi, the length of the first Z\ry-step, and K, the ratio
J
of two successive steps. The total number of points, , can be calculated by the
formula:
ln[l + (jr-l)(rfe/hi)]
J = + 1 (7.4.3)

1.24

1.20 h

1.16

1.12

1.08 h

1.04

1.00
0.2 1 1000 10000
Tie/hi x 10- 2
Fig. 7.4. Variation of K with h\ for different r/ e-values.