1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11.8 • THE JOUKOWSKI AIRFOIL 481

y

Figure U.63 The horizontal flow around the circle C1.

v

Flow around the airfoil.

Figure 11.64 The horizontal flow around the Joukowski airfoil A1.

the circle K 1 , which is tangent to Lo at the origin. The function W = S2 (Z)
maps the circle K 1 onto the cardioid H1. Finally, w = S3 (W) maps the cardioid
H 1 onto the Joukowski airfoil A1 that passes through the point w2 = 2 and

surrounds the point w 4 = -2, as shown in Figure 11.62. An observer traversing

C 1 counterclockwise will traverse the image curves K 1 and H 1 clockwise but will
traverse A 1 counterclockwise. Thus the points Z4, Z4, W4, and W4 will always
be to the observer's left.
Now we are ready to visualize the flow around the Joukowski airfoil. We
start with the fluid flow a.round a circle (see Figure 11.51). This flow is adjusted
with a linear transformation z* = az + b so that it flows horizontally a.round the

circle C 1 , as shown in Figure 11 .63. Then the mapping w = J (z*) creates a flow

around the Joukowski airfoil, as illustrated in Figure 11.64.

11.8.1 Flow with Circulation

The function F ( z) = sz + ~ +

2

k. Log z, where s > 0 and k is real, is the

z 7rt

complex potential for a uniform horizontal flow past the unit circle lzl = 1, with

circulation strength k and velocity at infinity V"" = s. For illustrative purposes,

we let s = 1 and use the substitution a = ;:. Now the complex potential has

the form

F (z) = z + ~ + aiLogz,

z

(11-37)