**for**

**11.8 The Joukowski
Airfoil**

The Russian scientist __Nikolai
Egorovich Joukowsky__ studied the function

.

He showed that the image of a circle passing through
and containing the point
is mapped onto a curve shaped like the cross section of an airplane
wing. We call this curve the Joukowski
airfoil. If the streamlines for a flow around the circle
are known, then their images under the mapping
will be streamlines for a flow around the Joukowski airfoil, as shown
in Figure 11.60.

** Figure
11.60** Image of a fluid flow under
.

The mapping
is two-to-one, because , for . The
region is
mapped one-to-one onto the w plane slit along the segment of the real
axis . To
visualize this mapping, we investigate the implicit form, which we
obtain by using the substitutions

, and

.

Forming the quotient of these two quantities results in the
relationship

.

The inverse of is . If
we use the notation and , then
we can express
as the composition of ,
,
and , that
is

(11-36) .

Which is verified by the calculation

We can easily show that maps
the four points onto , respectively.

However, the composition functions in Equation
(11-36) must be considered in order to
visualize the geometry involved. First, the bilinear
transformation maps
the region onto
the right half-plane , and
the points are
mapped onto , respectively. Second,
the function maps
the right half plane onto the W plane
slit along its negative real axis, and the points , are
mapped onto , respectively. Then
the bilinear transformation maps
the latter region onto the W plane
slit along the portion of the real axis ,
and the points are
mapped onto , respectively. These
three compositions are shown in Figure 11.61.

The composition mappings for .Figure 11.61

The circle with center on the imaginary axis passes through the points and has radius . With the restriction that , then this circle intersects the x axis at the point with angle , with . We want to track the image of in the Z, W, and w planes. First, the image of this circle under is the line that passes through the origin and is inclined at the angle . Second, the function maps the line onto the ray inclined at the angle . Finally, the transformation given by maps the ray onto the arc of the circle that passes through the points and intersects the u axis at with angle , where . The restriction on the angle , and hence , is necessary in order for the arc to have a low profile. The arc lies in the center of the Joukowski airfoil and is shown in Figure 11.62.

The images of the circles and under the composition mappings for .Figure 11.62

If we let b be fixed, , then the larger circle with center given by (just a bit to the left of the imaginary axis) will pass through the points and has radius . Set b=a and the circle also intersects the x axis at the point at the angle . The image of circle under is the circle , which is tangent to at the origin in the Z-plane. The function maps the circle onto the cardioid in the W-plane. Finally, maps the cardioid onto the Joukowski airfoil that passes through the point and surrounds the point , as shown in Figure 11.62. An observer traversing counterclockwise will traverse the image curves and clockwise but will traverse counterclockwise. Thus the points 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 around a circle (see Figure 11.51). This flow is adjusted with a linear transformation so that it flows horizontally around the circle , as shown in Figure 11.63. Then the mapping creates a flow around the Joukowski airfoil, as illustrated in Figure 11.64.

The horizontal flow around the circle .Figure 11.63

The horizontal flow around the Joukowski airfoil .Figure 11.64

**11.8.1 Flow with
Circulation**

The function , where
and k is real, is the complex
potential for a uniform horizontal flow past the unit circle
,
with circulation strength k and
velocity at infinity .

For illustrative purposes, we let
and use the substitution . Now
the complex potential has the form

(11-37)

and the corresponding velocity function is

.

We can express the complex potential
in form:

and we have the formulas for the velocity potential
stream function

, and

.

For the flow given
by , where
c is a constant, we have

. (Streamlines.)

Setting
in this equation, we get for
all ,
so the unit circle is a natural boundary curve for the flow.

Points at which the flow has zero velocity
are called stagnation points. To find them we
solve ; for
the function in Equation (11-37) we
have

Multiplying through by
and rearranging terms gives

Now we invoke the quadratic equation to obtain

(stagnation
point(s).)

If , then there are two stagnation points on the unit circle . If , then there is one stagnation point on the unit circle. If , then the stagnation point lies outside the unit circle. We are mostly interested in the case with two stagnation points. When , the two stagnation points are , which is the flow discussed in Example 11.25 (see Section 11.7). The cases are shown in Figure 11.65.

Flows past the unit circle with circulation .Figure 11.65

We are now ready to combine the preceding ideas. For illustrative purposes, we consider a circle with center that passes through the points and has radius . We use the linear transformation to map the flow with circulation (or ) around onto the flow around the circle , as shown in Figure 11.66.

Flow with circulation around .Figure 11.66

Then we use the
mapping to
map this flow around the Joukowski airfoil, as shown in Figure 11.67
and compare it to the flows shown in Figures 11.63 and
11.64.

If the second transformation in the composition given
by is
modified to be , then
the image of the flow shown in Figure 11.66 will be the flow around
the modified airfoil shown in Figure 11.68. The advantage
of this latter airfoil is that the sides of its tailing edge form an
angle of
radians, or ,
which is more realistic than the angle of
of the traditional Joukowski airfoil.

Flow with circulation around a traditional Joukowski airfoil.Figure 11.67

Flow with circulation around a modified Joukowski airfoil.Figure 11.68

The following *Mathematica* subroutine will form the
functions that are needed to graph a Joukowski airfoil.

**Example 1.** For a
fixed value dx, increasing the
parameter dy will bend the
airfoil.

**Example 2.** For a
fixed value dy, increasing the
parameter dx will fatten out the
airfoil.

**Example
3.** Increasing both parameters dx
and dy will bend and fatten out the
airfoil.

**Example 4.** Consider
the modified Joukowski airfoil when is
used to map the Z plane onto the
W plane. Refer to Figure 11.69
and discuss why the angle of the trailing edge of the modified
Joukowski airfoil
forms an angle of
radians.

The images of the circles and under the modified Joukowski airfoil.Figure 11.69

**Exercises
for Section 11.8. The Joukowski
Airfoil**** **

__Joukowski
Transformation and Airfoils__

**The Next Module
is**

**The
Schwarz-Christoffel Transformation**

**Return to the Complex
Analysis Modules **

__Return
to the Complex Analysis Project__

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

(c) 2012 John H. Mathews, Russell W. Howell