Joukowsky transform
In mathematics, a type of conformal map
title: "Joukowsky transform" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["conformal-mappings", "aircraft-aerodynamics", "aircraft-wing-design"] description: "In mathematics, a type of conformal map" topic_path: "general/conformal-mappings" source: "https://en.wikipedia.org/wiki/Joukowsky_transform" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary In mathematics, a type of conformal map ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/64/joukowsky_transform.svg" caption="Example of a Joukowsky transform. The circle above is transformed into the Joukowsky airfoil below."] ::
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
: z = \zeta + \frac{1}{\zeta},\qquad \zeta=\tfrac12z \pm\sqrt{\bigl(\tfrac12z\bigr)^2-1} = \frac1{\tfrac12z\mp\sqrt{\bigl(\tfrac12z\bigr)^2-1}},
where z = x + iy is a complex variable in the new space and \zeta = \chi + i \eta is a complex variable in the original space. The right-inverse is not a global left-inverse because \zeta\mapsto z 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 (z-plane) by applying the Joukowsky transform to a circle in the \zeta-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 \zeta = -1 (where the derivative is zero) and intersects the point \zeta = 1. This can be achieved for any allowable centre position \mu_x + i\mu_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 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 \zeta to z is as follows:
:\begin{align} z &= x + iy = \zeta + \frac{1}{\zeta} \ &= \chi + i \eta + \frac{1}{\chi + i\eta} \[2pt] &= \chi + i \eta + \frac{\chi - i\eta}{\chi^2 + \eta^2} \[2pt] &= \chi\left(1 + \frac1{\chi^2 + \eta^2}\right) + i\eta\left(1 - \frac1{\chi^2 + \eta^2}\right). \end{align}
So the real (x) and imaginary (y) components are:
:\begin{align} x &= \chi\left(1 + \frac1{\chi^2 + \eta^2}\right), \[2pt] y &= \eta\left(1 - \frac1{\chi^2 + \eta^2}\right). \end{align}
Sample Joukowsky airfoil
The transformation of all complex numbers on the unit circle is a special case.
|\zeta| = \sqrt{\chi^2 + \eta^2} = 1,
which gives
\chi^2 + \eta^2 = 1.
So the real component becomes x = \chi (1 + 1) = 2\chi and the imaginary component becomes y = \eta (1 - 1) = 0.
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.
The complex conjugate velocity \widetilde{W} = \widetilde{u}x - i\widetilde{u}y, around the circle in the \zeta-plane is \widetilde{W} = V\infty e^{-i\alpha} + \frac{i\Gamma}{2\pi(\zeta - \mu)} - \frac{V\infty R^2 e^{i\alpha}}{(\zeta - \mu)^2},
where
- \mu = \mu_x + i \mu_y is the complex coordinate of the centre of the circle,
- V_\infty is the freestream velocity of the fluid, \alpha is the angle of attack of the airfoil with respect to the freestream flow,
- R is the radius of the circle, calculated using R = \sqrt{\left(1 - \mu_x\right)^2 + \mu_y^2},
- \Gamma is the circulation, found using the Kutta condition, which reduces in this case to \Gamma = 4\pi V_\infty R\sin\left(\alpha + \sin^{-1}\frac{\mu_y}{R}\right).
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 = \frac{\widetilde{W}}{\frac{dz}{d\zeta}} = \frac{\widetilde{W}}{1 - \frac{1}{\zeta^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 + iy, with x and y 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 ==
::figure[src="https://upload.wikimedia.org/wikipedia/commons/5/5b/Karman_Trefftz_transform.svg" caption="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 \zeta-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 \alpha. This transform is
|:|z = nb \frac{(\zeta + b)^n + (\zeta - b)^n}{(\zeta + b)^n - (\zeta - b)^n}, |
where b is a real constant that determines the positions where dz/d\zeta = 0, and n is slightly smaller than 2. The angle \alpha between the tangents of the upper and lower airfoil surfaces at the trailing edge is related to n as
: \alpha = 2\pi - n\pi, \quad n = 2 - \frac{\alpha}{\pi}.
The derivative dz/d\zeta, required to compute the velocity field, is
: \frac{dz}{d\zeta} = \frac{4n^2}{\zeta^2 - 1} \frac{\left(1 + \frac{1}{\zeta}\right)^n \left(1 - \frac{1}{\zeta}\right)^n} {\left[\left(1 + \frac{1}{\zeta}\right)^n - \left(1 - \frac{1}{\zeta}\right)^n \right]^2}.
Background
First, add and subtract 2 from the Joukowsky transform, as given above:
: \begin{align} z + 2 &= \zeta + 2 + \frac{1}{\zeta} = \frac{1}{\zeta} (\zeta + 1)^2, \[3pt] z - 2 &= \zeta - 2 + \frac{1}{\zeta} = \frac{1}{\zeta} (\zeta - 1)^2. \end{align}
Dividing the left and right hand sides gives
: \frac{z - 2}{z + 2} = \left( \frac{\zeta - 1}{\zeta + 1} \right)^2.
The right hand side contains (as a factor) the simple second-power law from potential flow theory, applied at the trailing edge near \zeta = +1. From conformal mapping theory, this quadratic map is known to change a half plane in the \zeta-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 n in the previous equation gives
: \frac{z - n}{z + n} = \left( \frac{\zeta - 1}{\zeta + 1} \right)^n,
which is the Kármán–Trefftz transform. Solving for z gives it in the form of equation .
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 \epsilon and angle of inclination \alpha:
: z = e^{i\alpha} \left(\zeta - \epsilon + \frac{1}{\zeta - \epsilon} + \frac{2\epsilon^2}{a + \epsilon}\right).
The parameter \epsilon 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 a=1+\epsilon.
Notes
References
- {{cite book | last = Anderson | first = John | year = 1991 | title = Fundamentals of Aerodynamics | edition = Second | publisher = McGraw–Hill | location = Toronto | isbn = 0-07-001679-8 | pages = 195–208
- {{cite web | first=D. W. | last=Zingg | title=Low Mach number Euler computations | year=1989 | publisher=NASA TM-102205 | url=https://ntrs.nasa.gov/search.jsp?R=19940000533&hterms=Zingg+Low+Mach+Low+Mach&qs=Ntx%3Dmode%2520matchall%7Cmode%2520matchall%26Ntk%3DTitle%7CAuthor-Name%26N%3D0%26Ntt%3DLow%2520Mach%7CZingg
References
- Joukowsky, N. E.. (1910). "Über die Konturen der Tragflächen der Drachenflieger". Zeitschrift für Flugtechnik und Motorluftschiffahrt.
- Milne-Thomson, Louis M.. (1973). "Theoretical aerodynamics". Dover Publ..
- Blom, J. J. H.. (1981). "Some Characteristic Quantities of Karman-Trefftz Profiles". NASA Technical Memorandum TM-77013.
- Tsien, Hsue-shen. (1943). "Symmetrical Joukowsky airfoils in shear flow". Quarterly of Applied Mathematics.
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