Zhukovsky’s Airfoil

A simple transformation with remarkable properties was used by Nikolai Zhukovsky around 1910 to study the flow around aircraft wings. It is defined by

\displaystyle \omega = \textstyle{\frac{1}{2}}\displaystyle{\left(z +\frac{1}{z}\right)}

and is usually called the Joukowsky Map. We begin with a discussion of the theory of fluid flow in two dimensions. Readers familiar with 2D potential flow may skip to the section Joukowsky Airfoil.


Visualization of airflow around a Joukowsky airfoil. Image generated using code on this website.

Analytic Functions and Harmonic Functions

Complex variable theory is a powerful tool in modelling fluid flow in two dimensions. The independent complex variable is {z = x + i y}. If a function {f(z)} is analytic, its derivative {d f/d z} does not depend on the direction of the increment {d z}. Therefore

\displaystyle \frac{\partial f}{\partial x} = \frac{\partial f}{\partial iy}

Writing {f(z) = u(x,y) + i v(x,y)}, this leads to { u_x + i v_x = v_y - i u_y } , which immediately yields the Cauchy-Riemann equations:

\displaystyle u_x = v_y \qquad u_y = - v_x \,.

It then follows that {u_{xx} = v_{yx} = v_{xy} = - u_{yy}}, so that {u_{xx} + u_{yy} = 0}; and similarly {v_{xx} + v_{yy} = 0}. Thus, both {u} and {v} satisfy the 2D Laplace equation; they are harmonic functions. This is remarkable: the real and imaginary parts of any analytic function are harmonic functions.

The Complex Potential

We denote the velocity by {\mathbf{V} = (u(x,y),v(x,y))}. To keep things simple, we will consider steady-state fluid flow ({\partial/\partial t = 0}) that is both irrotational and incompressible:

\displaystyle \delta \equiv \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \qquad\qquad \zeta \equiv \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} = 0 \,.

The flow can be represented by a velocity potential {\phi(x,y)},

\displaystyle \mathbf{V} = \boldsymbol{\nabla}\phi \qquad\mbox{so that}\qquad u = \frac{\partial\phi}{\partial x} \mbox{\quad and\quad } v = \phantom{-}\frac{\partial\phi}{\partial y}

or by a streamfunction {\psi(x,y)},

\displaystyle \mathbf{V} = -\boldsymbol{k\times\nabla}\psi \qquad\mbox{so that}\qquad u = \frac{\partial\psi}{\partial y} \mbox{\quad and\quad } v = -\frac{\partial\psi}{\partial x} \,.

It follows from {\delta = \zeta = 0} that {\phi} and {\psi} are harmonic functions

\displaystyle \nabla^2\phi = \phi_{xx}+\phi_{yy} = 0 \qquad \nabla^2\psi = \psi_{xx}+\psi_{yy} =0

and comprise the real and imaginary parts of an analytical function, the complex potential:

\displaystyle \omega = \phi + i \psi \qquad\mbox{with}\qquad \frac{\mathrm{d}\omega}{\mathrm{d}z} = u - i v \,.

Examples of Potential Flow

Example I: The complex potential {\omega = z} corresponds to

\displaystyle\phi = x \qquad\mbox{and}\qquad \psi = y

yielding a uniform eastward flow

\displaystyle u = \phi_x = \psi_y = 1 \qquad\mbox{and}\qquad v = \phi_y = -\psi_x = 0 \,,

so that {\mathbf{V} = (1,0)}. This is shown in the Figure below (left panel).


Velocity potential {\phi} (dashed blue) and streamfunction {\psi} (solid red) for the complex potentials {\omega = z} (left) and {\omega =1/z} (right).

Example II:
The potential {\omega = 1/z} has velocity potential and streamfunction

\displaystyle \phi = \frac{x}{x^2+y^2} \qquad\mbox{and}\qquad \psi = \frac{-y}{x^2+y^2}

with corresponding velocity components

\displaystyle u = \frac{x^2-y^2}{(x^2+y^2)^2}\,, \qquad v = -\frac{2xy}{(x^2+y^2)^2} \,.

This is a dipole flow, shown in the FIgure above (right panel).

Example III: The potential {\omega = - i\log z} can be expressed as

\displaystyle \omega = - i\log ( r \exp i\theta ) = \theta - i \log r

and corresponds to velocity potential and streamfunction

\displaystyle \phi = \theta \qquad\mbox{and}\qquad \psi = - \log r

with radial and tangential flow components

\displaystyle v_r = \frac{\partial\phi}{\partial r} = 0 \qquad\mbox{and}\qquad v_\theta = \frac{1}{r}\frac{\partial\phi}{\partial r} = \frac{1}{r}

which is flow counterclockwise about the origin. Note that the magnitude of {v_\theta} decreases with radius by precisely the amount to ensure

\displaystyle \zeta = \frac{\partial v_\theta}{\partial r} - \frac{\partial v_r}{\partial \theta} = 0 \,,

so the flow, although circulating, is irrotational.

The Joukowsky Transform

We combine the complex potentials {\omega = z} and {\omega = 1/z} to produce a potential with some remarkable properties, the Joukowsky potential

\displaystyle \omega = \frac{1}{2}\left( z + \frac{1}{z} \right) \,.

This mapping is also called the Zhukovsky transform (Nikolai Zhukovsky studied it around 1910). The velocity potential and streamfunction corresponding to {\omega} are

\displaystyle \phi(x,y) = \Re\{\omega\} = x + \frac{x}{x^2+y^2} \quad\mbox{and}\quad \psi(x,y) = \Im\{\omega\} = y - \frac{y}{x^2+y^2}

These functions are plotted below. We note that the unit circle {|z|=1} corresponds to constant {\psi = 0}: there is no flow across this curve and the fluid flows around the unit circle or, in three dimensions, around an infinite cylinder.


Velocity potential {\phi} (dashed blue) and streamfunction {\psi} (solid red) for the complex potential {\omega = \frac{1}{2}(z+1/z)}.

It is obvious from the definition that {\omega(1/z) = \omega(z)}. If we restrict the mapping to the exterior of the unit circle {|z|=1}, it is one-to-one. Rewriting the map as {z^2 - 2\omega z + 1 = 0}, we have two solutions

\displaystyle z_{+} = \omega + \sqrt{\omega^2-1} \qquad\mbox{and}\qquad z_{-} = \omega - \sqrt{\omega^2-1} \,.

Since {z_{+}z_{-}=1}, one of these is inside {|x|=1} and the other is outside.

Letting {z = r\exp(i\theta)} the mapping may be written

\displaystyle \omega = \frac{1}{2}\biggl[r e^{i\theta} + \frac{1}{r}e^{-i\theta}\biggr] = \frac{1}{2}\left[r+\frac{1}{r}\right]\cos\theta + i\frac{1}{2}\left[r-\frac{1}{r}\right]\sin\theta \,.

Now writing {\omega = \phi + i\psi} we easily show that

\displaystyle \frac{\phi^2}{a^2} + \frac{\psi^2}{b^2} = 1

where {a = \frac{1}{2}(r+1/r)} and {b = \frac{1}{2}(r-1/r)} (so {\omega = a\cos\theta + b\sin\theta}). Thus, circles centered at the origin with {r>1} transform to ellipses
with semi-axes {a} and {b}. Moreover,

\displaystyle \frac{\phi^2}{\cos^2\theta} - \frac{\psi^2}{\sin^2\theta} = 1

so lines through the origin are mapped to hyperbolae.

Joukowsky Airfoil

Under the Joukowsky transform, the unit circle maps to the line segment {\omega \in [-1,+1]} (Figure below, left panel). When the centre of the circle is moved to {z_0} but still passes through {z=1}, the image curve represents an airfoil (right panel).


Left: Unit circle maps to line segment. Right: Shifted circle transforms to a curve representing an airfoil.

The mapping that converts a circle into an airfoil also maps the flow around the circle or, in 3D, the cylinder to flow around the airfoil. In the figure below we show a single streamline (left panel) and the corresponding streamline around the airfoil (right panel).


Left: Streamline around cylinder. Right: Corresponding streamline around Airfoil.

Adding Circulation

We saw in Example III above that a potential {\omega = - i k\log z} implies circulation around the wing. This results in lift. In the figure below, the flow without circulation (left panel, {k=0}) and with circulation (right panel, {k=0.5}) are shown.


Streamlines around cylinder without circulation (left) and with circulation added to provide lift (right).

A more comprehensive solution for a Joukowsky airfoil is shown in the final figure. This solution was generated using code on the Mathematica StackExchange site referenced below.


Streamlines around a Joukowsky airfoil (see text).


{\bullet} Complex-analysis.com: Website with visualizations:  URL

{\bullet} Mathematica StackExchange site: URL

{\bullet} Spiegel, Murray R., 1964: Schaums Outline Series of Complex Variables. Schaum Publishing Co., New York, 313pp.

{\bullet} Strang, Gilbert, 1986: Introduction to Applied Mathematics. Wellesley-Cambridge Press, 758pp. ISBN: 0-961-40880-4.

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