Joukowsky transform

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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.^[1]

The transform and its right-inverse are

${\displaystyle z=\zeta +{\frac {1}{\zeta }},\qquad \zeta ={\tfrac {1}{2}}z\pm {\sqrt {{\bigl (}{\tfrac {1}{2}}z{\bigr )}^{2}-1}}={\frac {1}{{\tfrac {1}{2}}z\mp {\sqrt {{\bigl (}{\tfrac {1}{2}}z{\bigr )}^{2}-1}}}},}${\displaystyle z=\zeta +{\frac {1}{\zeta }},\qquad \zeta ={\tfrac {1}{2}}z\pm {\sqrt {{\bigl (}{\tfrac {1}{2}}z{\bigr )}^{2}-1}}={\frac {1}{{\tfrac {1}{2}}z\mp {\sqrt {{\bigl (}{\tfrac {1}{2}}z{\bigr )}^{2}-1}}}},}

where ${\displaystyle z=x+iy}${\displaystyle z=x+iy} is a complex variable in the new space and ${\displaystyle \zeta =\chi +i\eta }${\displaystyle \zeta =\chi +i\eta } is a complex variable in the original space. The right-inverse is not a global left-inverse because ${\displaystyle \zeta \mapsto z}${\displaystyle \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 (${\displaystyle z}${\displaystyle z}-plane) by applying the Joukowsky transform to a circle in the ${\displaystyle \zeta }${\displaystyle \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 ${\displaystyle \zeta =-1}${\displaystyle \zeta =-1} (where the derivative is zero) and intersects the point ${\displaystyle \zeta =1.}${\displaystyle \zeta =1.} This can be achieved for any allowable centre position ${\displaystyle \mu _{x}+i\mu _{y}}${\displaystyle \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.

The Joukowsky transform of any complex number ${\displaystyle \zeta }${\displaystyle \zeta } to ${\displaystyle z}${\displaystyle z} is as follows:

${\displaystyle {\begin{aligned}z&=x+iy=\zeta +{\frac {1}{\zeta }}\\&=\chi +i\eta +{\frac {1}{\chi +i\eta }}\\[2pt]&=\chi +i\eta +{\frac {\chi -i\eta }{\chi ^{2}+\eta ^{2}}}\\[2pt]&=\chi \left(1+{\frac {1}{\chi ^{2}+\eta ^{2}}}\right)+i\eta \left(1-{\frac {1}{\chi ^{2}+\eta ^{2}}}\right).\end{aligned}}}${\displaystyle {\begin{aligned}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+{\frac {1}{\chi ^{2}+\eta ^{2}}}\right)+i\eta \left(1-{\frac {1}{\chi ^{2}+\eta ^{2}}}\right).\end{aligned}}}

So the real (${\displaystyle x}${\displaystyle x}) and imaginary (${\displaystyle y}${\displaystyle y}) components are:

${\displaystyle {\begin{aligned}x&=\chi \left(1+{\frac {1}{\chi ^{2}+\eta ^{2}}}\right),\\[2pt]y&=\eta \left(1-{\frac {1}{\chi ^{2}+\eta ^{2}}}\right).\end{aligned}}}${\displaystyle {\begin{aligned}x&=\chi \left(1+{\frac {1}{\chi ^{2}+\eta ^{2}}}\right),\\[2pt]y&=\eta \left(1-{\frac {1}{\chi ^{2}+\eta ^{2}}}\right).\end{aligned}}}

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

$${\displaystyle |\zeta |={\sqrt {\chi ^{2}+\eta ^{2}}}=1,}$${\displaystyle |\zeta |={\sqrt {\chi ^{2}+\eta ^{2}}}=1,}

which gives

$${\displaystyle \chi ^{2}+\eta ^{2}=1.}$${\displaystyle \chi ^{2}+\eta ^{2}=1.}

So the real component becomes ${\textstyle x=\chi (1+1)=2\chi }${\textstyle x=\chi (1+1)=2\chi } and the imaginary component becomes ${\textstyle y=\eta (1-1)=0}${\textstyle 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.

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 ${\displaystyle {\widetilde {W}}={\widetilde {u}}_{x}-i{\widetilde {u}}_{y},}${\displaystyle {\widetilde {W}}={\widetilde {u}}_{x}-i{\widetilde {u}}_{y},} around the circle in the ${\displaystyle \zeta }${\displaystyle \zeta }-plane is $${\displaystyle {\widetilde {W}}=V_{\infty }e^{-i\alpha }+{\frac {i\Gamma }{2\pi (\zeta -\mu )}}-{\frac {V_{\infty }R^{2}e^{i\alpha }}{(\zeta -\mu )^{2}}},}$${\displaystyle {\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

• ${\displaystyle \mu =\mu _{x}+i\mu _{y}}${\displaystyle \mu =\mu _{x}+i\mu _{y}} is the complex coordinate of the centre of the circle,
• ${\displaystyle V_{\infty }}${\displaystyle V_{\infty }} is the freestream velocity of the fluid,

${\displaystyle \alpha }${\displaystyle \alpha } is the angle of attack of the airfoil with respect to the freestream flow,

• ${\displaystyle R}${\displaystyle R} is the radius of the circle, calculated using ${\textstyle R={\sqrt {\left(1-\mu _{x}\right)^{2}+\mu _{y}^{2}}}}${\textstyle R={\sqrt {\left(1-\mu _{x}\right)^{2}+\mu _{y}^{2}}}},
• ${\displaystyle \Gamma }${\displaystyle \Gamma } is the circulation, found using the Kutta condition, which reduces in this case to $${\displaystyle \Gamma =4\pi V_{\infty }R\sin \left(\alpha +\sin ^{-1}{\frac {\mu _{y}}{R}}\right).}$${\displaystyle \Gamma =4\pi V_{\infty }R\sin \left(\alpha +\sin ^{-1}{\frac {\mu _{y}}{R}}\right).}

The complex velocity ${\displaystyle W}${\displaystyle W} around the airfoil in the ${\displaystyle z}${\displaystyle z}-plane is, according to the rules of conformal mapping and using the Joukowsky transformation, $${\displaystyle W={\frac {\widetilde {W}}{\frac {dz}{d\zeta }}}={\frac {\widetilde {W}}{1-{\frac {1}{\zeta ^{2}}}}}.}$${\displaystyle W={\frac {\widetilde {W}}{\frac {dz}{d\zeta }}}={\frac {\widetilde {W}}{1-{\frac {1}{\zeta ^{2}}}}}.}

Here ${\displaystyle W=u_{x}-iu_{y},}${\displaystyle W=u_{x}-iu_{y},} with ${\displaystyle u_{x}}${\displaystyle u_{x}} and ${\displaystyle u_{y}}${\displaystyle u_{y}} the velocity components in the ${\displaystyle x}${\displaystyle x} and ${\displaystyle y}${\displaystyle y} directions respectively (${\displaystyle z=x+iy,}${\displaystyle z=x+iy,} with ${\displaystyle x}${\displaystyle x} and ${\displaystyle y}${\displaystyle 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.

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 ${\displaystyle \zeta }${\displaystyle \zeta }-plane to the physical ${\displaystyle z}${\displaystyle 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 ${\displaystyle \alpha .}${\displaystyle \alpha .} This transform is^{[2] [3]}

where ${\displaystyle b}${\displaystyle b} is a real constant that determines the positions where ${\displaystyle dz/d\zeta =0}${\displaystyle dz/d\zeta =0}, and ${\displaystyle n}${\displaystyle n} is slightly smaller than 2. The angle ${\displaystyle \alpha }${\displaystyle \alpha } between the tangents of the upper and lower airfoil surfaces at the trailing edge is related to ${\displaystyle n}${\displaystyle n} as^[2]

${\displaystyle \alpha =2\pi -n\pi ,\quad n=2-{\frac {\alpha }{\pi }}.}${\displaystyle \alpha =2\pi -n\pi ,\quad n=2-{\frac {\alpha }{\pi }}.}

The derivative ${\displaystyle dz/d\zeta }${\displaystyle dz/d\zeta }, required to compute the velocity field, is

${\displaystyle {\frac {dz}{d\zeta }}={\frac {4n^{2}}{\zeta ^{2}-1}}{\frac {\left(1+{\frac {1}{\zeta }}\right)^{n}\left(1-{\frac {1}{\zeta }}\right)^{n}}{\left[\left(1+{\frac {1}{\zeta }}\right)^{n}-\left(1-{\frac {1}{\zeta }}\right)^{n}\right]^{2}}}.}${\displaystyle {\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}}}.}

First, add and subtract 2 from the Joukowsky transform, as given above:

${\displaystyle {\begin{aligned}z+2&=\zeta +2+{\frac {1}{\zeta }}={\frac {1}{\zeta }}(\zeta +1)^{2},\\[3pt]z-2&=\zeta -2+{\frac {1}{\zeta }}={\frac {1}{\zeta }}(\zeta -1)^{2}.\end{aligned}}}${\displaystyle {\begin{aligned}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{aligned}}}

Dividing the left and right hand sides gives

${\displaystyle {\frac {z-2}{z+2}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{2}.}${\displaystyle {\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 ${\displaystyle \zeta =+1.}${\displaystyle \zeta =+1.} From conformal mapping theory, this quadratic map is known to change a half plane in the ${\displaystyle \zeta }${\displaystyle \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 ${\displaystyle n}${\displaystyle n} in the previous equation gives^[2]

${\displaystyle {\frac {z-n}{z+n}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{n},}${\displaystyle {\frac {z-n}{z+n}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{n},}

which is the Kármán–Trefftz transform. Solving for ${\displaystyle z}${\displaystyle z} gives it in the form of equation A .

In 1943 Hsue-shen Tsien published a transform of a circle of radius ${\displaystyle a}${\displaystyle a} into a symmetrical airfoil that depends on parameter ${\displaystyle \epsilon }${\displaystyle \epsilon } and angle of inclination ${\displaystyle \alpha }${\displaystyle \alpha }:^[4]

${\displaystyle z=e^{i\alpha }\left(\zeta -\epsilon +{\frac {1}{\zeta -\epsilon }}+{\frac {2\epsilon ^{2}}{a+\epsilon }}\right).}${\displaystyle z=e^{i\alpha }\left(\zeta -\epsilon +{\frac {1}{\zeta -\epsilon }}+{\frac {2\epsilon ^{2}}{a+\epsilon }}\right).}

The parameter ${\displaystyle \epsilon }${\displaystyle \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 ${\displaystyle a=1+\epsilon }${\displaystyle a=1+\epsilon }.

• Joukowski Transform NASA Applet
• Joukowsky Transform Interactive WebApp

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