Joukowsky transform

Summary

In applied mathematics, the Joukowsky transform (sometimes transliterated Joukovsky, Joukowski or Zhukovsky) is a conformal map historically used to understand some principles of airfoil design. It is named after Nikolai Zhukovsky, who published it in 1910.[1]

Example of a Joukowsky transform. The circle above is transformed into the Joukowsky airfoil below.

The transform is

where is a complex variable in the new space and is a complex variable in the original space.

In aerodynamics, the transform is used to solve for the two-dimensional potential flow around a class of airfoils known as Joukowsky airfoils. A Joukowsky airfoil is generated in the complex plane (-plane) by applying the Joukowsky transform to a circle in the -plane. The coordinates of the centre of the circle are variables, and varying them modifies the shape of the resulting airfoil. The circle encloses the point (where the derivative is zero) and intersects the point This can be achieved for any allowable centre position by varying the radius of the circle.

Joukowsky airfoils have a cusp at their trailing edge. A closely related conformal mapping, the Kármán–Trefftz transform, generates the broader class of Kármán–Trefftz airfoils by controlling the trailing edge angle. When a trailing edge angle of zero is specified, the Kármán–Trefftz transform reduces to the Joukowsky transform.

General Joukowsky transform edit

The Joukowsky transform of any complex number   to   is as follows:

 

So the real ( ) and imaginary ( ) components are:

 

Sample Joukowsky airfoil edit

The transformation of all complex numbers on the unit circle is a special case.

 

which gives

 

So the real component becomes   and the imaginary component becomes  .

Thus the complex unit circle maps to a flat plate on the real-number line from −2 to +2.

Transformations from other circles make a wide range of airfoil shapes.

Velocity field and circulation for the Joukowsky airfoil edit

The solution to potential flow around a circular cylinder is analytic and well known. It is the superposition of uniform flow, a doublet, and a vortex.

The complex conjugate velocity   around the circle in the  -plane is

 

where

  •   is the complex coordinate of the centre of the circle,
  •   is the freestream velocity of the fluid,

  is the angle of attack of the airfoil with respect to the freestream flow,

  •   is the radius of the circle, calculated using  ,
  •   is the circulation, found using the Kutta condition, which reduces in this case to
     

The complex velocity   around the airfoil in the  -plane is, according to the rules of conformal mapping and using the Joukowsky transformation,

 

Here   with   and   the velocity components in the   and   directions respectively (  with   and   real-valued). From this velocity, other properties of interest of the flow, such as the coefficient of pressure and lift per unit of span can be calculated.

Kármán–Trefftz transform edit

 
Example of a Kármán–Trefftz transform. The circle above in the  -plane is transformed into the Kármán–Trefftz airfoil below, in the  -plane. The parameters used are:     and   Note that the airfoil in the  -plane has been normalised using the chord length.

The Kármán–Trefftz transform is a conformal map closely related to the Joukowsky transform. While a Joukowsky airfoil has a cusped trailing edge, a Kármán–Trefftz airfoil—which is the result of the transform of a circle in the  -plane to the physical  -plane, analogue to the definition of the Joukowsky airfoil—has a non-zero angle at the trailing edge, between the upper and lower airfoil surface. The Kármán–Trefftz transform therefore requires an additional parameter: the trailing-edge angle   This transform is[2][3]

 

 

 

 

 

(A)

where   is a real constant that determines the positions where  , and   is slightly smaller than 2. The angle   between the tangents of the upper and lower airfoil surfaces at the trailing edge is related to   as[2]

 

The derivative  , required to compute the velocity field, is

 

Background edit

First, add and subtract 2 from the Joukowsky transform, as given above:

 

Dividing the left and right hand sides gives

 

The right hand side contains (as a factor) the simple second-power law from potential flow theory, applied at the trailing edge near   From conformal mapping theory, this quadratic map is known to change a half plane in the  -space into potential flow around a semi-infinite straight line. Further, values of the power less than 2 will result in flow around a finite angle. So, by changing the power in the Joukowsky transform to a value slightly less than 2, the result is a finite angle instead of a cusp. Replacing 2 by   in the previous equation gives[2]

 

which is the Kármán–Trefftz transform. Solving for   gives it in the form of equation A.

Symmetrical Joukowsky airfoils edit

In 1943 Hsue-shen Tsien published a transform of a circle of radius   into a symmetrical airfoil that depends on parameter   and angle of inclination  :[4]

 

The parameter   yields a flat plate when zero, and a circle when infinite; thus it corresponds to the thickness of the airfoil. Furthermore the radius of the cylinder  .

Notes edit

  1. ^ Joukowsky, N. E. (1910). "Über die Konturen der Tragflächen der Drachenflieger". Zeitschrift für Flugtechnik und Motorluftschiffahrt (in German). 1: 281–284 and (1912) 3: 81–86.
  2. ^ a b c Milne-Thomson, Louis M. (1973). Theoretical aerodynamics (4th ed.). Dover Publ. pp. 128–131. ISBN 0-486-61980-X.
  3. ^ Blom, J. J. H. (1981). "Some Characteristic Quantities of Karman-Trefftz Profiles" (Document). NASA Technical Memorandum TM-77013.
  4. ^ Tsien, Hsue-shen (1943). "Symmetrical Joukowsky airfoils in shear flow". Quarterly of Applied Mathematics. 1 (2): 130–248. doi:10.1090/qam/8537.

References edit

  • Anderson, John (1991). Fundamentals of Aerodynamics (Second ed.). Toronto: McGraw–Hill. pp. 195–208. ISBN 0-07-001679-8.
  • Zingg, D. W. (1989). "Low Mach number Euler computations". NASA TM-102205.

External links edit

  • Joukowski Transform NASA Applet
  • Joukowsky Transform Interactive WebApp