# Joukowsky transform
The **Joukowsky transform** (also written Joukowski or Zhukovsky transform) is the conformal map of the complex plane
\[ z = \zeta + \frac{a^{2}}{\zeta} \]
where \( a > 0 \) is a parameter. This classical method of two-dimensional airfoil theory uses the transform to convert the analytically solvable potential flow past a circular cylinder into the flow past a body shaped like an airfoil, meaning that lift and pressure on a wing are obtained by working out the simpler circular case and then transforming the solution.[1](https://trinh.github.io/BathMAFluids/ch-chapter04-potentialflows-8.html) • [2](https://www.grc.nasa.gov/WWW/K-12/airplane/map.html)
| Key fact | Detail |
|---|---|
| Mapping | \( z = \zeta + a^{2}/\zeta \); conformal everywhere except the pole \( \zeta = 0 \) and the critical points \( \zeta = \pm a \), where \( G'(\zeta) = 0 \)[1](https://trinh.github.io/BathMAFluids/ch-chapter04-potentialflows-8.html) |
| Circle to ellipse | A circle of radius \( r > a \) maps to an ellipse with principal radii \( r + a^{2}/r \) and \( r - a^{2}/r \); as \( r \to a \) the image collapses to a flat plate of length \( 4a \)[1](https://trinh.github.io/BathMAFluids/ch-chapter04-potentialflows-8.html) |
| Inverse map | \( \zeta = \tfrac{1}{2}\left(z \pm \sqrt{z^{2} - 4a^{2}}\right) \)[1](https://trinh.github.io/BathMAFluids/ch-chapter04-potentialflows-8.html) |
| Shape control | Shifting the circle's center along the x-axis sets airfoil thickness; shifting along the y-axis sets camber[3](https://www.diam.unige.it/~irro/conformi_e.html) |
| Lift rule | Kutta–Joukowski theorem: \( L = \rho U_{\infty} \Gamma \), lift per unit span perpendicular to the free stream[4](https://ntrs.nasa.gov/api/citations/19830006993/downloads/19830006993.pdf) |
| Circulation fix | The Kutta condition, smooth flow leaving the trailing edge, uniquely determines the circulation and hence the lift; for a flat plate at angle \( \alpha \), \( \Gamma = -4\pi u_{0} R \sin\alpha \)[5](https://galileo.phys.virginia.edu/classes/311/notes/aero/node4.html) • [6](https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf) |
## Geometry of the mapping
Writing a point of a circle as \( \zeta = r e^{\imath\theta} \), the transform gives
\[ z = \left(r + \frac{a^{2}}{r}\right)\cos\theta + \imath\left(r - \frac{a^{2}}{r}\right)\sin\theta, \]
which is the equation of an ellipse provided \( r > a \). The exterior of the circle maps to the exterior of the ellipse, and orientation is preserved, anticlockwise in \( \zeta \) mapping to anticlockwise in \( z \).[1](https://trinh.github.io/BathMAFluids/ch-chapter04-potentialflows-8.html) In the limit \( r \to a \) the ellipse degenerates to the line segment \( z = 2a\cos\theta \), running from \( z = 2a \) to \( z = -2a \), a flat plate of length \( 4a \).[1](https://trinh.github.io/BathMAFluids/ch-chapter04-potentialflows-8.html) The Caltech aerodynamics notes state the same degenerate case as a cylinder of radius \( \zeta_T \) mapping to the segment \( y = 0 \), \( -2\zeta_T \le x \le 2\zeta_T \).[7](https://shepherd.caltech.edu/T5/resources/AE101_Guide.pdf)
The map fails to be conformal at exactly the points that matter for airfoil shapes. If the generating circle passes through one of the critical points \( \zeta = \pm a \) and encloses the other, the image has exactly one cusp and is smooth everywhere else; the cusp becomes the sharp trailing edge of the airfoil.[8](https://demonstrations.wolfram.com/TheJoukowskiMappingAirfoilsFromCircles/) In the NASA Glenn formulation with the mapping \( A = z + 1/z \), the two poles sit at \( x = 1 \) and \( x = -1 \): a cylinder touching a pole produces a sharp edge, both poles inside the cylinder produce a closed body, and only one pole inside produces a non-physical shape.[2](https://www.grc.nasa.gov/WWW/K-12/airplane/map.html)
## From circle to airfoil
**Center placement sets the shape.** A circle centered at the origin maps to an ellipse or, at the critical radius, a flat plate. Moving the center of the circle along the x-axis gives the airfoil its thickness, and moving it along the y-axis gives camber, the curvature of the chord line.[3](https://www.diam.unige.it/~irro/conformi_e.html) The generic Joukowski airfoil has a rounded leading edge and a cusp at the trailing edge.[3](https://www.diam.unige.it/~irro/conformi_e.html) The transformation constant must be matched to the circle geometry to guarantee a valid airfoil; a small constant gives a thick, near-cylindrical section, a value approaching the circle radius gives a thin streamlined section, and equality with the radius with no center shift gives the flat plate.[9](https://eaglepubs.erau.edu/compsimforaero/chapter/b-5/)
**Why the flow maps too.** If \( Z = F(z) \) is analytic with an inverse, the complex potential \( W(Z) = w(F^{-1}(Z)) \) is also analytic, and streamlines in one plane map to streamlines in the other; a solid boundary, being a streamline, maps to a solid boundary.[6](https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf) Circulation remains unchanged when the variable is transformed conformally; for this reason, the flow after mapping continues to meet the [Kutta condition](https://www.edgepedia.org/kutta-condition) and produces the proper lift.[10](https://ondara.net/course/course-0/subunit/u6-7-conformal-mapping) At infinity the two flows have exactly the same form, and the angle of attack is the same in either plane.[11](https://ntrs.nasa.gov/api/citations/19880004688/downloads/19880004688.pdf) Velocities transform through the derivative of the mapping, and the pressure coefficient on the airfoil surface follows from Bernoulli's equation for inviscid incompressible flow.[9](https://eaglepubs.erau.edu/compsimforaero/chapter/b-5/)
## The Kutta condition and lift
[Potential flow](https://www.edgepedia.org/potential-flow) around a circle is known exactly, but the circulation around it is not unique; it must be prescribed. For airfoils the choice is fixed by the Kutta condition: in flow over a two-dimensional body with a sharp trailing edge, the circulation develops to just the magnitude that moves the rear [stagnation point](https://www.edgepedia.org/stagnation-point) to the trailing edge, so the flow leaves the edge smoothly.[12](https://stern.lab.uiowa.edu/sites/stern.lab.uiowa.edu/files/2025-01/Chapter%208.5%20Airfoil%20and%20Wing%20Theory%20and%20Forces%20on%20Lifting%20Bodies.pdf) Without this condition, flow would wrap around the sharp trailing edge from one side to the other with infinite velocity there.[5](https://galileo.phys.virginia.edu/classes/311/notes/aero/node4.html) For a finite-angle trailing edge the edge must be a stagnation point; for a cusped (zero-angle) trailing edge the velocity may be finite but must be equal on the two sides of the separating streamline.[7](https://shepherd.caltech.edu/T5/resources/AE101_Guide.pdf)
Once the circulation \( \Gamma \) is fixed, the Kutta–Joukowski theorem gives the lift per unit span,
\[ L = \rho\, U_{\infty}\, \Gamma, \]
with \( \rho \) the fluid density and \( U_{\infty} \) the undisturbed velocity at infinity, directed perpendicular to the free stream.[4](https://ntrs.nasa.gov/api/citations/19830006993/downloads/19830006993.pdf) For the flat-plate case the MIT notes derive \( \Gamma = -4\pi u_{0} R \sin\alpha \) from the requirement of finite velocity at the trailing edge; the singularity at the other end cannot be removed simultaneously.[6](https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf) For the flat plate analyzed through the transform, the center of pressure and the aerodynamic center both fall at the quarter-chord location.[9](https://eaglepubs.erau.edu/compsimforaero/chapter/b-5/)
## History and role
In his 1902 paper, Martin Kutta mapped the [potential flow around a circular cylinder](https://www.edgepedia.org/potential-flow-around-a-circular-cylinder) onto a flat plate at an angle of attack, imposed a finite velocity at the trailing edge based on experiment, and derived the lift formula that became a key result of classical thin-airfoil theory. A few years later Nikolai Zhukovsky (Joukowsky) generalized the technique to convert a suitably placed circle into a finite-thickness, cusped airfoil profile, the classic Joukowsky airfoil.[13](https://eaglepubs.erau.edu/introductiontoaerospaceflightvehicles/chapter/classic-airfoil-theory/)
The method is limited to special airfoil profiles for which a conformal transformation exists.[11](https://ntrs.nasa.gov/api/citations/19880004688/downloads/19880004688.pdf) Real airfoils are not Joukowsky airfoils, but studying Joukowsky shapes gives general trends for ideal flow over real airfoils of similar thickness and camber.[11](https://ntrs.nasa.gov/api/citations/19880004688/downloads/19880004688.pdf) The same machinery extends further: sequences of conformal transformations can map a partially truncated Joukowsky airfoil onto a circle to model trailing-edge stall,[14](https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/wake-singularity-potential-flow-model-for-airfoils-experiencing-trailingedge-stall/6AFCA4349C62C3D4CFE9CB4D42424508) and the transform remains a standard component of unsteady aerodynamic methods, where the airfoil is scaled to the interval \( x \in [-1,1] \) before the Joukowski mapping is applied.[15](https://doi.org/10.2514%2F1.J061894)
## References
1. [Flow past an aerofoil, Fluid Dynamics Lecture Notes (P. Trinh, University of Bath)](https://trinh.github.io/BathMAFluids/ch-chapter04-potentialflows-8.html)
2. [Conformal Mapping, NASA Glenn Research Center](https://www.grc.nasa.gov/WWW/K-12/airplane/map.html)
3. [The Joukowski Transformation, Conformal Mapping notes, University of Genoa](https://www.diam.unige.it/~irro/conformi_e.html)
4. [II. Kutta-Joukowski Condition, NASA Technical Reports Server](https://ntrs.nasa.gov/api/citations/19830006993/downloads/19830006993.pdf)
5. [The Kutta-Joukowsky condition, University of Virginia physics notes](https://galileo.phys.virginia.edu/classes/311/notes/aero/node4.html)
6. [Nonlinear Dynamics II: Continuum Systems, Lecture Notes, MIT OpenCourseWare (18.354J, Spring 2015)](https://ocw.mit.edu/courses/18-354j-nonlinear-dynamics-ii-continuum-systems-spring-2015/6163837c5efa5d6ed969e535ec8f890c_MIT18_354JS15_lectureNotes.pdf)
7. [Incompressible, Inviscid Flow, AE101 Guide, Caltech](https://shepherd.caltech.edu/T5/resources/AE101_Guide.pdf)
8. [The Joukowski Mapping: Airfoils from Circles, Wolfram Demonstrations Project](https://demonstrations.wolfram.com/TheJoukowskiMappingAirfoilsFromCircles/)
9. [B-5: Conformal Mapping, Computer Simulations for Aerodynamics, Embry-Riddle Aeronautical University](https://eaglepubs.erau.edu/compsimforaero/chapter/b-5/)
10. [Joukowski map for airfoil flow, Math for Fluid Mechanics](https://ondara.net/course/course-0/subunit/u6-7-conformal-mapping)
11. [Two-Dimensional Air, NASA Technical Reports Server](https://ntrs.nasa.gov/api/citations/19880004688/downloads/19880004688.pdf)
12. [Chapter 8.5: Airfoil and Wing Theory and Forces on Lifting Bodies, F. Stern, University of Iowa](https://stern.lab.uiowa.edu/sites/stern.lab.uiowa.edu/files/2025-01/Chapter%208.5%20Airfoil%20and%20Wing%20Theory%20and%20Forces%20on%20Lifting%20Bodies.pdf)
13. [Classic Airfoil Theory, Introduction to Aerospace Flight Vehicles, Embry-Riddle Aeronautical University](https://eaglepubs.erau.edu/introductiontoaerospaceflightvehicles/chapter/classic-airfoil-theory/)
14. [A wake singularity potential flow model for airfoils experiencing trailing-edge stall, Journal of Fluid Mechanics](https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/wake-singularity-potential-flow-model-for-airfoils-experiencing-trailingedge-stall/6AFCA4349C62C3D4CFE9CB4D42424508)
15. [Kutta–Joukowski Theorem for Unsteady Linear Aerodynamics, AIAA Journal](https://doi.org/10.2514%2F1.J061894)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Mechanics › Continuum, solid, and fluid mechanics › Fluid mechanics › Inviscid and potential flow › Conformal and complex-variable methods*
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