Page 295 - Computational Fluid Dynamics for Engineers

9.6 Conformal Mapping Methods 285

(b) Compute R from Eq. (9.6.10). that is,

2
R = \Jx\> + y p
(c) Determine £ and 77 at p from Eq. (9.6.12), that is,

= V2i?cos ( ^ 1 , 77P = V2i?sin ( J
£ P
The calculation of the (£, 77) net for off-body points requires the determination
of £ = const and 77 = const lines. One procedure is to start at 77 = ryp, where
£p is known, and vary 77 in small increments and compute r and 6 from Eq.
(9.6.5). This procedure is repeated for other values of £p in uniform increments
in 77 except for the region close to the airfoil surface where the 77-increments
are determined by the body points. Note that in the region behind the airfoil
yp = 0, ?7P = 0 and £p = 2xp.
The £p-constant and 77-constant lines determined in this manner produce a
parabolic grid in the computational plane. The inverse transformation follows
from Eq. (9.6.2); for each value of £ and 77, x and y are determined in the
physical plane.

9.6.2 Wind Tunnel Mapping Function

The C-mesh generated by the parabolic mapping function discussed above is
essentially a set of confocal, orthogonal parabolas wrapping around the airfoil.
With this mapping the outer boundary of the grid given by the last parabola ex-
tends far away from the airfoil surface. This feature may be unnecessary and/or
undesirable in some practical applications. A different mapping function that
addresses this problem is the infinite "wind-tunnel" mapping function defined
by
l
Z = n(l-coshC) (9.6.14)
Where
Z = x + iy (9.6.15a)
C = £ + ZT7 (9.6.15b)
and
coshC = 1 -e z (9.6.16)
In this transformation, the singular point located at half the leading edge radius
is at (In 2,0) and all values of y lie between ±n. As shown in Fig. 9.11a, this
mapping allows the airfoil and its wake, indicated by the contour ABCDE, to
be transformed into a nearly straight 77 = TT line in the computational plane.
The far-field boundaries HI and GF are mapped to the r\ = 0 line. Points on
the upper surface of the airfoil are mapped to the left of the 77-axis and points
on the lower surface are mapped to the right of the 77-axis.