Wikipedia / Joukowsky transform
Wikipedia / Joukowsky transform
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.
The transform and its right-inverse are
is a complex variable is 2-to-1; but a local left-inverse is always one of the right-inverse branches.
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 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
is as follows:
) components are:
Sample Joukowsky airfoil
The transformation of all complex numbers on the unit circle is a special case.
which gives
.
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
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.
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 circulation, found using the Kutta condition
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 == 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 This transform is
between the tangents as
, 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 in the previous equation gives
gives it in the form of equation .
Symmetrical Joukowsky airfoils
In 1943 Hsue-shen Tsien:
.
Notes
References
External links