Page 297 - Computational Fluid Dynamics for Engineers

9.6 Conformal Mapping Methods 287

t
7] = an" 1 (^\ (9.6.19b)
Where:
1 2
= R / cos ( - J - e x cos y + 1 (9.6.20a)
D 3
x
1/2
= fl sinj-j -e smy (9.6.20b)
D 4
R = e x Dl + Dl\ (9.6.20C)

- 1
t
9 = an ( —- J (9.6.20d)
Di = e x cos 2y - 2 cos y (9.6.20e)

D 2 = e x sin 2y - 2 sin y (9.6.20f)
As a test case, consider the NACA 0012 airfoil, the coordinates of which are
given by:
4
y = ±0.6 k).2969\/^ - 0.126x - 0.3516x 2 + 0.2843x 3 - 0.1015x l (9.6.21)
Here, x is the distance from the leading edge normalized by the chord c and y
is normalized thickness. To determine the nose radius, write:

-7= =0.6 0.2969 - 0.126v^ - 0.3516x 3/2 + ... (9.6.22)
yjx
In the limit, as x —> 0, yjyfx = 0.6 x 0.2969 = 0.17814 and the normalized nose
radius is:
= -(0.6 x 0.2969) 2 = 0.015867 (9.6.23)
f 0
The shift in coordinates is obtained by replacing

x = x + y (9.6.24)

Appendix B gives a listing of a program to generate a C-mesh by the wind
tunnel mapping for the NACA0012 airfoil. In this program XW and YW denote
£ and 77 respectively. NX and NY denote the number of stations in x and y
directions, respectively, and the parameter HTC denotes the height-to-chord
ratio and is used to control the range of x and £. The large the value of HTC,
the smaller the airfoil becomes in the physical domain, as shown in Fig. 9.15a.
In general, HTC varies from 15 to 20 for full potential methods and it varies
from 50 to 100 for Euler methods. It can also be used to control the grid in
the 77-direction. As a rule of thumb, the larger the value of HTC, the larger
the distance from the wall to the outer boundary. Figure 9.15c shows a sample
mesh generated with HTC of 15 and a singular point at the center of the leading
edge circle (RLE) of the NACA0012 airfoil. Figure 9.15d shows a close up of
the mesh near the airfoil.