Joukowsky transform - Alchetron, The Free Social Encyclopedia

In applied mathematics, the Joukowsky transform, named after Nikolai Zhukovsky, is a conformal map historically used to understand some principles of airfoil design.

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 ( z -plane by applying the Joukowsky transform to a circle in the ζ -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 ζ = − 1 (where the derivative is zero) and intersects the point ζ = 1. This can be achieved for any allowable centre position μ x + i μ y 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 ζ to z is as follows

z = x + i y = ζ + 1 ζ = χ + i η + 1 χ + i η = χ + i η + ( χ − i η ) χ 2 + η 2 = χ ( χ 2 + η 2 + 1 ) χ 2 + η 2 + i η ( χ 2 + η 2 − 1 ) χ 2 + η 2 .

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

x = χ ( χ 2 + η 2 + 1 ) χ 2 + η 2 and y = η ( χ 2 + η 2 − 1 ) χ 2 + η 2 .

Sample Joukowsky airfoil

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

| ζ | = χ 2 + η 2 = 1 which gives χ 2 + η 2 = 1.

So the real component becomes x = χ ( 1 + 1 ) 1 = 2 χ and the imaginary component becomes y = η ( 1 − 1 ) 1 = 0.

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 W ~ = u ~ x − i u ~ y , around the circle in the ζ -plane is

W ~ = V ∞ e − i α + i Γ 2 π ( ζ − μ ) − V ∞ R 2 e i α ( ζ − μ ) 2 ,

where

μ = μ x + i μ y is the complex coordinate of the centre of the circle

V ∞ is the freestream velocity of the fluid

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

R is the radius of the circle, calculated using R = ( 1 − μ x ) 2 + μ y 2

Γ is the circulation, found using the Kutta condition, which reduces in this case to

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

W = W ~ d z d ζ = W ~ 1 − 1 ζ 2 .

Here W = u x − i u y , with u x and u y the velocity components in the x and y directions, respectively ( z = x + i y , with x and y real-valued). From this velocity, other properties of interest of the flow, such as the coefficient of pressure or lift 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 ζ -plane to the physical z -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 α . This transform is equal to:

z = n ( 1 + 1 ζ ) n + ( 1 − 1 ζ ) n ( 1 + 1 ζ ) n − ( 1 − 1 ζ ) n , (A)

with n slightly smaller than 2. The angle α , between the tangents of the upper and lower airfoil surface, at the trailing edge is related to n by:

α = 2 π − n π and n = 2 − α π .

The derivative d z / d ζ , required to compute the velocity field, is equal to:

d z d ζ = 4 n 2 ζ 2 − 1 ( 1 + 1 ζ ) n ( 1 − 1 ζ ) n [ ( 1 + 1 ζ ) n − ( 1 − 1 ζ ) n ] 2 .

Background

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

z + 2 = ζ + 2 + 1 ζ = 1 ζ ( ζ + 1 ) 2 , z − 2 = ζ − 2 + 1 ζ = 1 ζ ( ζ − 1 ) 2 .

Dividing the left and right hand sides gives:

z − 2 z + 2 = ( ζ − 1 ζ + 1 ) 2 .

The right hand side contains (as a factor) the simple second-power law from potential flow theory, applied at the trailing edge near ζ = + 1. From conformal mapping theory this quadratic map is known to change a half plane in the ζ -space into potential flow around a semi-infinite straight line. Further, values of the power less than two will result in flow around a finite angle. So, by changing the power in the Joukowsky transform—to a value slightly less than two—the result is a finite angle instead of a cusp. Replacing 2 by n in the previous equation gives:

z − n z + n = ( ζ − 1 ζ + 1 ) n ,

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

Symmetrical Joukowsky airfoils

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

z = e i α ( ζ − ϵ + 1 ζ − ϵ + 2 ϵ 2 a + ϵ ) .

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

References

Joukowsky transform Wikipedia

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