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.  

Exploration

Figure 11.61  The composition mappings for  .  

    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.

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

    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  .  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.

Figure 11.63  The horizontal flow around the circle .  

Figure 11.64  The horizontal flow around the Joukowski airfoil .  

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.  

Figure 11.65  Flows past the unit circle with circulation .

    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.

Figure 11.66  Flow with circulation around .  

    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.

Figure 11.67  Flow with circulation around a traditional Joukowski airfoil.

Figure 11.68  Flow with circulation around a modified Joukowski airfoil.

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.

Explore Solution 1.

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

Explore Solution 2.

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

Explore Solution 3.

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.  

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

Solution 4.