Solution 8 (b).

See text and/or instructor's solution manual.

Solution.

        The transformation for the modified Joukowski airfoil can be written as the composition of three functions

[Graphics:../Images/JoukowskiTransModHome_gr_564.gif],   [Graphics:../Images/JoukowskiTransModHome_gr_565.gif]   and   [Graphics:../Images/JoukowskiTransModHome_gr_566.gif]   and their composition is   

                      [Graphics:../Images/JoukowskiTransModHome_gr_567.gif].    

        Let  [Graphics:../Images/JoukowskiTransModHome_gr_568.gif]  be a circle that passes through the points  [Graphics:../Images/JoukowskiTransModHome_gr_569.gif]  and has center   [Graphics:../Images/JoukowskiTransModHome_gr_570.gif]   in the z-plane.  

As shown in Exercise 3, the image of the circle  [Graphics:../Images/JoukowskiTransModHome_gr_571.gif]  under   [Graphics:../Images/JoukowskiTransModHome_gr_572.gif]   is a line  [Graphics:../Images/JoukowskiTransModHome_gr_573.gif]  that passes through the origin,

and the line [Graphics:../Images/JoukowskiTransModHome_gr_574.gif] is inclined at the angle   [Graphics:../Images/JoukowskiTransModHome_gr_575.gif]   in the Z-plane  (as shown in Figure 11.69).

Furthermore, the image of the circle  [Graphics:../Images/JoukowskiTransModHome_gr_576.gif]  is the circle  [Graphics:../Images/JoukowskiTransModHome_gr_577.gif]  under  [Graphics:../Images/JoukowskiTransModHome_gr_578.gif].  

        In Exercise 4 we investigated the situation where the function   [Graphics:../Images/JoukowskiTransModHome_gr_579.gif]   is used.    

Now follow the steps using the mapping   [Graphics:../Images/JoukowskiTransModHome_gr_580.gif].   For the given angle  [Graphics:../Images/JoukowskiTransModHome_gr_581.gif],  

the line  [Graphics:../Images/JoukowskiTransModHome_gr_582.gif]  inclined at the angle  [Graphics:../Images/JoukowskiTransModHome_gr_583.gif]  consists of two rays in the  Z-plane   

                    [Graphics:../Images/JoukowskiTransModHome_gr_584.gif],    and  
                    
                    [Graphics:../Images/JoukowskiTransModHome_gr_585.gif].  

Using the polar coordinates  [Graphics:../Images/JoukowskiTransModHome_gr_586.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_587.gif]

for the mapping   [Graphics:../Images/JoukowskiTransModHome_gr_588.gif]   we have   [Graphics:../Images/JoukowskiTransModHome_gr_589.gif]   and     

                    [Graphics:../Images/JoukowskiTransModHome_gr_590.gif].  

Thus the images of  [Graphics:../Images/JoukowskiTransModHome_gr_591.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_592.gif]  are the two rays  [Graphics:../Images/JoukowskiTransModHome_gr_593.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_594.gif]  respectively, where      

                    [Graphics:../Images/JoukowskiTransModHome_gr_595.gif],    and  
                    
                    [Graphics:../Images/JoukowskiTransModHome_gr_596.gif].  

Observe carefully that in this case   [Graphics:../Images/JoukowskiTransModHome_gr_597.gif].  In this case, the angle between  [Graphics:../Images/JoukowskiTransModHome_gr_598.gif]  is  

                    [Graphics:../Images/JoukowskiTransModHome_gr_599.gif].

Furthermore, the image of the circle  [Graphics:../Images/JoukowskiTransModHome_gr_600.gif]  is the "cardioid like" curve  [Graphics:../Images/JoukowskiTransModHome_gr_601.gif]  under  [Graphics:../Images/JoukowskiTransModHome_gr_602.gif].  

        The transformation  [Graphics:../Images/JoukowskiTransModHome_gr_603.gif]  maps the point   [Graphics:../Images/JoukowskiTransModHome_gr_604.gif]   onto the point   [Graphics:../Images/JoukowskiTransModHome_gr_605.gif].

For the function  [Graphics:../Images/JoukowskiTransModHome_gr_606.gif]   we have   [Graphics:../Images/JoukowskiTransModHome_gr_607.gif].   Since   [Graphics:../Images/JoukowskiTransModHome_gr_608.gif],   

by Theorem 10.1 in Section 10.1 , the mapping is conformal at  [Graphics:../Images/JoukowskiTransModHome_gr_609.gif]  and "angles are preserved."

Hence, the angle of rotation at   [Graphics:../Images/JoukowskiTransModHome_gr_610.gif]  is   [Graphics:../Images/JoukowskiTransModHome_gr_611.gif].

        The rays   [Graphics:../Images/JoukowskiTransModHome_gr_612.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_613.gif]  tangent to the "cardioid like" curve at  [Graphics:../Images/JoukowskiTransModHome_gr_614.gif]  make angles  [Graphics:../Images/JoukowskiTransModHome_gr_615.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_616.gif],  respectively.

Then the images of  [Graphics:../Images/JoukowskiTransModHome_gr_617.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_618.gif]  are the two arcs  [Graphics:../Images/JoukowskiTransModHome_gr_619.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_620.gif]  respectively,  that pass through the points  [Graphics:../Images/JoukowskiTransModHome_gr_621.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_622.gif].  

Their image arcs  [Graphics:../Images/JoukowskiTransModHome_gr_623.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_624.gif]  will also make angles  [Graphics:../Images/JoukowskiTransModHome_gr_625.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_626.gif],  respectively, at  [Graphics:../Images/JoukowskiTransModHome_gr_627.gif].  

Furthermore, the image of the "cardioid like" curve  [Graphics:../Images/JoukowskiTransModHome_gr_628.gif]  is the modified Joukowski airfoil  [Graphics:../Images/JoukowskiTransModHome_gr_629.gif]  under  [Graphics:../Images/JoukowskiTransModHome_gr_630.gif].  

        Therefore, the angle at the trailing edge of this modified Joukowski airfoil is   [Graphics:../Images/JoukowskiTransModHome_gr_631.gif]  radians.

[Graphics:../Images/JoukowskiTransModHome_gr_632.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_633.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_634.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_635.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_636.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_637.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_638.gif]


[Graphics:../Images/JoukowskiTransModHome_gr_639.gif]

[Graphics:../Images/JoukowskiTransModHome_gr_640.gif]


We are really done.   

 

Aside.  We can explore some graphs.

                    [Graphics:../Images/JoukowskiTransModHome_gr_641.gif]

                      The circles  [Graphics:../Images/JoukowskiTransModHome_gr_642.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_643.gif]  in the z-plane.  

 

                    [Graphics:../Images/JoukowskiTransModHome_gr_644.gif]

                      The line  [Graphics:../Images/JoukowskiTransModHome_gr_645.gif]  and the circle  [Graphics:../Images/JoukowskiTransModHome_gr_646.gif]  in the Z-plane.  

 

                    [Graphics:../Images/JoukowskiTransModHome_gr_647.gif]

                      The rays  [Graphics:../Images/JoukowskiTransModHome_gr_648.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_649.gif]  and the "cardioid like" curve  [Graphics:../Images/JoukowskiTransModHome_gr_650.gif]  in the Z-plane.  

 

                    [Graphics:../Images/JoukowskiTransModHome_gr_651.gif]

                      The arcs  [Graphics:../Images/JoukowskiTransModHome_gr_652.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_653.gif]  and modified Joukowski airfoil  [Graphics:../Images/JoukowskiTransModHome_gr_654.gif]  in the w-plane, where  [Graphics:../Images/JoukowskiTransModHome_gr_655.gif].

 

Aside.  In the above construction we used the function   [Graphics:../Images/JoukowskiTransModHome_gr_656.gif] which makes the modified Joukowski airfoil  [Graphics:../Images/JoukowskiTransModHome_gr_657.gif]  form an angle of  [Graphics:../Images/JoukowskiTransModHome_gr_658.gif]  radians.

We can compare this by using the function    [Graphics:../Images/JoukowskiTransModHome_gr_659.gif]  which makes the modified Joukowski airfoil  [Graphics:../Images/JoukowskiTransModHome_gr_660.gif]  form an angle of  [Graphics:../Images/JoukowskiTransModHome_gr_661.gif]  radians,

and by using the function    [Graphics:../Images/JoukowskiTransModHome_gr_662.gif]  which makes the standard Joukowski airfoil  [Graphics:../Images/JoukowskiTransModHome_gr_663.gif]  which form an angle of  [Graphics:../Images/JoukowskiTransModHome_gr_664.gif]  radians.

                    [Graphics:../Images/JoukowskiTransModHome_gr_665.gif]

                      The arcs  [Graphics:../Images/JoukowskiTransModHome_gr_666.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_667.gif]  and modified Joukowski airfoil  [Graphics:../Images/JoukowskiTransModHome_gr_668.gif]  in the w-plane, where  [Graphics:../Images/JoukowskiTransModHome_gr_669.gif].  

 

                    [Graphics:../Images/JoukowskiTransModHome_gr_670.gif]

                      The arcs  [Graphics:../Images/JoukowskiTransModHome_gr_671.gif]  and  [Graphics:../Images/JoukowskiTransModHome_gr_672.gif]  and Joukowski airfoil  [Graphics:../Images/JoukowskiTransModHome_gr_673.gif]  in the w-plane, where  [Graphics:../Images/JoukowskiTransModHome_gr_674.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2008 John H. Mathews, Russell W. Howell