11 .8 • T HE JOUKOWSKI AIRFOIL 485( c) Show that the line Lo is inclined at the angle ao = ~ -Arctan a.
4. Show that a line t hrough the origin is mapped onto a ray by the mapping w = z^2.- Let Ro be a ray through the origin inclined at an angle f3o.
(a) Show that the image of the ray Ro under w =
2
1
+
2
z is an arc Ao of- z
a circle that passes through 2 and -2.
(b) Show that the arc Ao is inclined at the angle /30.
6. Show that a circle passing through the origin is mapped onto a cardioid by w = z^2 •
Show that the cusp in the cardioid forms an angle of 0°.- Let H 1 be a cardioid whose cusp is at the origin. The image of H 1 under w =
2- 2
z will be a J oukowski airfoil. Show that t railing edge forms an angle of 0°.
- 2
1-z
- Consider the modified Joukowski airfoil when W = 82 (Z) = Z^1 •^925 is used to
map the Z plane onto the W plane. Refer to Figure 11.6 9 and discuss why
w = J(z)
v- I t w=SfW)
v
Figure 11 .69 The images of the ci.rcles Co and C, under the modified Joukowski trans-