Joukowsky transform

In applied mathematics, the Joukowsky transform, named after Nikolai Zhukovsky (who published it in 1910), is a conformal map historically used to understand some principles of airfoil design.
The transform is
where formula_2 is a complex variable in the new space and formula_3 is a complex variable in the original space.
This transform is also called the Joukowsky transformation, the Joukowski transform, the Zhukovsky transform and other variations.
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 (formula_4-plane) by applying the Joukowsky transform to a circle in the formula_5-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 formula_6 (where the derivative is zero) and intersects the point formula_7 This can be achieved for any allowable centre position formula_8 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 much 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.
The Joukowsky transform of any complex number formula_5 to formula_4 is as follows:
So the real (formula_12) and imaginary (formula_13) components are:
Sample Joukowsky airfoil.
The transformation of all complex numbers on the unit circle is a special case.
So the real component becomes formula_16 and the imaginary component becomes formula_17.
Thus the complex unit circle maps to a flat plate on the real-number line from −2 to +2.
Transformation from other circles make a wide range of airfoil shapes.
Velocity field and circulation for the Joukowsky airfoil.
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 formula_18 around the circle in the formula_5-plane is
where
The complex velocity formula_28 around the airfoil in the formula_4-plane is, according to the rules of conformal mapping and using the Joukowsky transformation,
Here formula_31 with formula_32 and formula_33 the velocity components in the formula_12 and formula_13 directions respectively (formula_36 with formula_12 and formula_13 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.
A Joukowsky airfoil has a cusp at the trailing edge.
The transformation is named after Russian scientist Nikolai Zhukovsky. His name has historically been romanized in a number of ways, thus the variation in spelling of the transform.
Kármán–Trefftz transform.
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 formula_5-plane to the physical formula_4-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 formula_41 This transform is
where formula_42 is a real constant that determines the positions where formula_43, and formula_44 is slightly smaller than 2. The angle formula_23 between the tangents of the upper and lower airfoil surfaces at the trailing edge is related to formula_44 as
The derivative formula_48, required to compute the velocity field, is
Background.
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 formula_52 From conformal mapping theory, this quadratic map is known to change a half plane in the formula_5-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 formula_44 in the previous equation gives
which is the Kármán–Trefftz transform. Solving for formula_4 gives it in the form of equation .
Symmetrical Joukowsky airfoils.
In 1943 Hsue-shen Tsien published a transform of a circle of radius formula_57 into a symmetrical airfoil that depends on parameter formula_58 and angle of inclination formula_23:
The parameter formula_58 yields a flat plate when zero, and a circle when infinite; thus it corresponds to the thickness of the airfoil.