The Joukowski Airfoil


11.8  The Joukowski Airfoil

    The Russian scientist Nikolai Egorovich Joukowsky studied the function

He showed that the image of a circle passing through [Graphics:Images/JoukowskiTransMod_gr_2.gif] and containing the point [Graphics:Images/JoukowskiTransMod_gr_3.gif] 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 [Graphics:Images/JoukowskiTransMod_gr_4.gif] will be streamlines for a flow around the Joukowski airfoil, as shown in Figure 11.60.  

           [Graphics:Images/JoukowskiFigure11.060_gr_1.gif]          [Graphics:Images/JoukowskiFigure11.060_gr_2.gif]

                                        Figure 11.60  Image of a fluid flow under [Graphics:Images/JoukowskiTransMod_gr_5.gif].  

    The mapping [Graphics:Images/JoukowskiTransMod_gr_6.gif] is two-to-one, because  [Graphics:Images/JoukowskiTransMod_gr_7.gif],  for  [Graphics:Images/JoukowskiTransMod_gr_8.gif].  The region  [Graphics:Images/JoukowskiTransMod_gr_9.gif]  is mapped one-to-one onto the w plane slit along the segment of the real axis [Graphics:Images/JoukowskiTransMod_gr_10.gif].  To visualize this mapping, we investigate the implicit form, which we obtain by using the substitutions  

            [Graphics:Images/JoukowskiTransMod_gr_11.gif],  and    

Forming the quotient of these two quantities results in the relationship


    The inverse of  [Graphics:Images/JoukowskiTransMod_gr_14.gif]  is  [Graphics:Images/JoukowskiTransMod_gr_15.gif].  If we use the notation  [Graphics:Images/JoukowskiTransMod_gr_16.gif]  and  [Graphics:Images/JoukowskiTransMod_gr_17.gif],  then we can express [Graphics:Images/JoukowskiTransMod_gr_18.gif] as the composition of  [Graphics:Images/JoukowskiTransMod_gr_19.gif], [Graphics:Images/JoukowskiTransMod_gr_20.gif], and [Graphics:Images/JoukowskiTransMod_gr_21.gif],  that is

(11-36)        [Graphics:Images/JoukowskiTransMod_gr_22.gif].  

Which is verified by the calculation


    We can easily show that [Graphics:Images/JoukowskiTransMod_gr_24.gif]  maps the four points  [Graphics:Images/JoukowskiTransMod_gr_25.gif]  onto  [Graphics:Images/JoukowskiTransMod_gr_26.gif],  respectively.  
However, the composition functions in Equation (11-36) must be considered in order to visualize the geometry involved.  First, the bilinear transformation  [Graphics:Images/JoukowskiTransMod_gr_27.gif]  maps the region  [Graphics:Images/JoukowskiTransMod_gr_28.gif]  onto the right half-plane  [Graphics:Images/JoukowskiTransMod_gr_29.gif],  and the points  [Graphics:Images/JoukowskiTransMod_gr_30.gif]  are mapped onto  [Graphics:Images/JoukowskiTransMod_gr_31.gif],  respectively.  Second, the function  [Graphics:Images/JoukowskiTransMod_gr_32.gif]  maps the right half plane onto the W plane slit along its negative real axis, and the points [Graphics:Images/JoukowskiTransMod_gr_33.gif],  are mapped onto  [Graphics:Images/JoukowskiTransMod_gr_34.gif],  respectively.  Then the bilinear transformation [Graphics:Images/JoukowskiTransMod_gr_35.gif]  maps the latter region onto the W plane slit along the portion of the real axis [Graphics:Images/JoukowskiTransMod_gr_36.gif], and the points  [Graphics:Images/JoukowskiTransMod_gr_37.gif]  are mapped onto  [Graphics:Images/JoukowskiTransMod_gr_38.gif],  respectively.  These three compositions are shown in Figure 11.61.  


Figure 11.61  The composition mappings for  [Graphics:Images/JoukowskiTransMod_gr_47.gif].  

    The circle  [Graphics:Images/JoukowskiTransMod_gr_48.gif]  with center  [Graphics:Images/JoukowskiTransMod_gr_49.gif]  on the imaginary axis passes through the points  [Graphics:Images/JoukowskiTransMod_gr_50.gif]  and has radius  [Graphics:Images/JoukowskiTransMod_gr_51.gif].  With the restriction that  [Graphics:Images/JoukowskiTransMod_gr_52.gif],  then this circle intersects the x axis at the point [Graphics:Images/JoukowskiTransMod_gr_53.gif] with angle  [Graphics:Images/JoukowskiTransMod_gr_54.gif], with  [Graphics:Images/JoukowskiTransMod_gr_55.gif].  We want to track the image of  [Graphics:Images/JoukowskiTransMod_gr_56.gif]  in the Z,  W, and w planes.  First, the image of this circle  [Graphics:Images/JoukowskiTransMod_gr_57.gif]  under  [Graphics:Images/JoukowskiTransMod_gr_58.gif]  is the line  [Graphics:Images/JoukowskiTransMod_gr_59.gif]  that passes through the origin and is inclined at the angle  [Graphics:Images/JoukowskiTransMod_gr_60.gif].  Second, the function  [Graphics:Images/JoukowskiTransMod_gr_61.gif]  maps the line  [Graphics:Images/JoukowskiTransMod_gr_62.gif]  onto the ray  [Graphics:Images/JoukowskiTransMod_gr_63.gif]  inclined at the angle  [Graphics:Images/JoukowskiTransMod_gr_64.gif].  Finally, the transformation given by  [Graphics:Images/JoukowskiTransMod_gr_65.gif]  maps the ray  [Graphics:Images/JoukowskiTransMod_gr_66.gif]  onto the arc of the circle  [Graphics:Images/JoukowskiTransMod_gr_67.gif]  that passes through the points  [Graphics:Images/JoukowskiTransMod_gr_68.gif]  and intersects the u axis at  [Graphics:Images/JoukowskiTransMod_gr_69.gif] with angle [Graphics:Images/JoukowskiTransMod_gr_70.gif], where    .  The restriction on the angle [Graphics:Images/JoukowskiTransMod_gr_72.gif], and hence [Graphics:Images/JoukowskiTransMod_gr_73.gif], is necessary in order for the arc [Graphics:Images/JoukowskiTransMod_gr_74.gif] to have a low profile.  The arc [Graphics:Images/JoukowskiTransMod_gr_75.gif] lies in the center of the Joukowski airfoil and is shown in Figure 11.62.

Figure 11.62  The images of the circles [Graphics:Images/JoukowskiTransMod_gr_76.gif] and [Graphics:Images/JoukowskiTransMod_gr_77.gif] under the composition mappings for  [Graphics:Images/JoukowskiTransMod_gr_78.gif].  

    If we let b be fixed, [Graphics:Images/JoukowskiTransMod_gr_79.gif], then the larger circle [Graphics:Images/JoukowskiTransMod_gr_80.gif] with center given by [Graphics:Images/JoukowskiTransMod_gr_81.gif]  (just a bit to the left of the imaginary axis) will pass through the points    [Graphics:Images/JoukowskiTransMod_gr_82.gif] and has radius  [Graphics:Images/JoukowskiTransMod_gr_83.gif].  Set  b=a  and the circle [Graphics:Images/JoukowskiTransMod_gr_84.gif] also intersects the x axis at the point  [Graphics:Images/JoukowskiTransMod_gr_85.gif]  at the angle  [Graphics:Images/JoukowskiTransMod_gr_86.gif].  The image of circle [Graphics:Images/JoukowskiTransMod_gr_87.gif] under  [Graphics:Images/JoukowskiTransMod_gr_88.gif]  is the circle  [Graphics:Images/JoukowskiTransMod_gr_89.gif],  which is tangent to  [Graphics:Images/JoukowskiTransMod_gr_90.gif] at the origin in the Z-plane.  The function  [Graphics:Images/JoukowskiTransMod_gr_91.gif]  maps the circle  [Graphics:Images/JoukowskiTransMod_gr_92.gif]  onto the cardioid  [Graphics:Images/JoukowskiTransMod_gr_93.gif]  in the W-plane.  Finally,  [Graphics:Images/JoukowskiTransMod_gr_94.gif]  maps the cardioid  [Graphics:Images/JoukowskiTransMod_gr_95.gif]  onto the Joukowski airfoil  [Graphics:Images/JoukowskiTransMod_gr_96.gif]  that passes through the point  [Graphics:Images/JoukowskiTransMod_gr_97.gif]  and surrounds the point  [Graphics:Images/JoukowskiTransMod_gr_98.gif],  as shown in Figure 11.62.  An observer traversing [Graphics:Images/JoukowskiTransMod_gr_99.gif] counterclockwise will traverse the image curves  [Graphics:Images/JoukowskiTransMod_gr_100.gif]  and  [Graphics:Images/JoukowskiTransMod_gr_101.gif]  clockwise but will traverse  [Graphics:Images/JoukowskiTransMod_gr_102.gif]  counterclockwise.  Thus the points  [Graphics:Images/JoukowskiTransMod_gr_103.gif]  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  [Graphics:Images/JoukowskiTransMod_gr_104.gif]  so that it flows horizontally around the circle [Graphics:Images/JoukowskiTransMod_gr_105.gif], as shown in Figure 11.63.  Then the mapping  [Graphics:Images/JoukowskiTransMod_gr_106.gif]  creates a flow around the Joukowski airfoil, as illustrated in Figure 11.64.

Figure 11.63  The horizontal flow around the circle [Graphics:Images/JoukowskiTransMod_gr_107.gif].  

Figure 11.64  The horizontal flow around the Joukowski airfoil [Graphics:Images/JoukowskiTransMod_gr_108.gif].  



11.8.1  Flow with Circulation

    The function  [Graphics:Images/JoukowskiTransMod_gr_109.gif],  where [Graphics:Images/JoukowskiTransMod_gr_110.gif] and k is real, is the complex potential for a uniform horizontal flow past the unit circle [Graphics:Images/JoukowskiTransMod_gr_111.gif], with circulation strength k and velocity at infinity  [Graphics:Images/JoukowskiTransMod_gr_112.gif].  
For illustrative purposes, we let [Graphics:Images/JoukowskiTransMod_gr_113.gif] and use the substitution  [Graphics:Images/JoukowskiTransMod_gr_114.gif].  Now the complex potential has the form  

(11-37)        [Graphics:Images/JoukowskiTransMod_gr_115.gif]

and the corresponding velocity function is  


    We can express the complex potential in  [Graphics:Images/JoukowskiTransMod_gr_117.gif]  form:


and we have the formulas for the velocity potential [Graphics:Images/JoukowskiTransMod_gr_119.gif] stream function  [Graphics:Images/JoukowskiTransMod_gr_120.gif]   

            [Graphics:Images/JoukowskiTransMod_gr_121.gif],  and  


    For the flow given by  [Graphics:Images/JoukowskiTransMod_gr_123.gif],  where c is a constant, we have  

            [Graphics:Images/JoukowskiTransMod_gr_124.gif].     (Streamlines.)

Setting [Graphics:Images/JoukowskiTransMod_gr_125.gif] in this equation, we get  [Graphics:Images/JoukowskiTransMod_gr_126.gif]  for all [Graphics:Images/JoukowskiTransMod_gr_127.gif], 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   [Graphics:Images/JoukowskiTransMod_gr_128.gif];  for the function in Equation (11-37) we have  


Multiplying through by [Graphics:Images/JoukowskiTransMod_gr_130.gif] and rearranging terms gives   


Now we invoke the quadratic equation to obtain  

            [Graphics:Images/JoukowskiTransMod_gr_132.gif]    (stagnation point(s).)

    If  [Graphics:Images/JoukowskiTransMod_gr_133.gif],  then there are two stagnation points on the unit circle [Graphics:Images/JoukowskiTransMod_gr_134.gif].  If [Graphics:Images/JoukowskiTransMod_gr_135.gif], then there is one stagnation point on the unit circle.  If [Graphics:Images/JoukowskiTransMod_gr_136.gif], then the stagnation point lies outside the unit circle.  We are mostly interested in the case with two stagnation points.  When [Graphics:Images/JoukowskiTransMod_gr_137.gif], the two stagnation points are [Graphics:Images/JoukowskiTransMod_gr_138.gif], which is the flow discussed in Example 11.25 (see Section 11.7). The cases [Graphics:Images/JoukowskiTransMod_gr_139.gif] are shown in Figure 11.65.  

Figure 11.65  Flows past the unit circle with circulation [Graphics:Images/JoukowskiTransMod_gr_140.gif].

    We are now ready to combine the preceding ideas.  For illustrative purposes, we consider a [Graphics:Images/JoukowskiTransMod_gr_141.gif] circle with center  [Graphics:Images/JoukowskiTransMod_gr_142.gif]  that passes through the points  [Graphics:Images/JoukowskiTransMod_gr_143.gif]  and has radius  [Graphics:Images/JoukowskiTransMod_gr_144.gif].  We use the linear transformation  [Graphics:Images/JoukowskiTransMod_gr_145.gif]  to map the flow with circulation [Graphics:Images/JoukowskiTransMod_gr_146.gif] (or [Graphics:Images/JoukowskiTransMod_gr_147.gif]) around [Graphics:Images/JoukowskiTransMod_gr_148.gif] onto the flow around the circle [Graphics:Images/JoukowskiTransMod_gr_149.gif], as shown in Figure 11.66.

Figure 11.66  Flow with circulation around [Graphics:Images/JoukowskiTransMod_gr_150.gif].  

    Then we use the mapping  [Graphics:Images/JoukowskiTransMod_gr_151.gif]  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  [Graphics:Images/JoukowskiTransMod_gr_152.gif]  is modified to be  [Graphics:Images/JoukowskiTransMod_gr_153.gif],  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 [Graphics:Images/JoukowskiTransMod_gr_154.gif] radians, or [Graphics:Images/JoukowskiTransMod_gr_155.gif], which is more realistic than the angle of [Graphics:Images/JoukowskiTransMod_gr_156.gif] 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  [Graphics:Images/JoukowskiTransMod_gr_202.gif]  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 [Graphics:Images/JoukowskiTransMod_gr_203.gif] forms an angle of [Graphics:Images/JoukowskiTransMod_gr_204.gif] radians.  

Figure 11.69  The images of the circles [Graphics:Images/JoukowskiTransMod_gr_205.gif] and [Graphics:Images/JoukowskiTransMod_gr_206.gif] under the modified Joukowski airfoil.

Solution 4.


Exercises for Section 11.8. The Joukowski Airfoil


Library Research Experience for Undergraduates

Ideal Fluid Flow

Joukowski Transformation and Airfoils

Complex Potential





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