484 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS
v
Flow with circulation around
a traditional airfoil.Figure 11.67 Flow with circulation around a traditional Joukowski airfoil.
v
Flow wilh circulation around
a modified airfoil.Figure 11.68 F low with circulation around a modified Joukowski airfoil.shown in Figure 11.68. The advantage of this latter airfoil is that the sides of its
tailing edge form an angle of 0.1571' radians, or 27°, which is more realistic than
the angle of 0° of the traditional Joukowski airfoil.
-------~EXERCISES FOR SECTION 11.8- Find the inverse of the Joukowski transformation.
2. Consider the Joukowski transformation w = z + ~.
z
{a) Show that the circles Cr = {lzl = r: r > 1} are mapped onto t he
elli pses
4 u^2 4 v^2
(r + *)^2 + (r - ~) = 1.(b) Show that the ray r > 0, 9 = a is mapped onto a branch of the hyper-
bola
u2 112-----cos (^2) a sin (^2) a =1 ·
- Let Co be a circle that passes through the points 1 and - 1 and has center CQ = ia.
{a) Find the equation of the circle Co.{b) Show that the image of the circle Co under w = z -^1 is a line Lo that
z+l
passes thr-0ugh the origin.