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Joukowsky transform

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In <a href="Applied_mathematics" title="Applied mathematics">applied mathematics</a>, the Joukowsky transform (sometimes transliterated Joukovsky, Joukowski or Zhukovsky) is a <a href="Conformal_map" title="Conformal map">conformal map</a> historically used to understand some principles of <a href="Airfoil" title="Airfoil">airfoil</a> design. It is named after <a href="Nikolay_Yegorovich_Zhukovsky" class="mw-redirect" title="Nikolay Yegorovich Zhukovsky">Nikolai Zhukovsky</a>, who published it in 1910.<a href="#cite_note-1">[1]</a>

The transform is

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<annotation encoding="application/x-tex">{\displaystyle z=\zeta +{\frac {1}{\zeta }},}</annotation>

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</math><img src="./a73ff7ce2f0ed759d082be0a69d14c7557718a66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.768ex; height:5.676ex;" alt="{\displaystyle z=\zeta +{\frac {1}{\zeta }},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+iy}">

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<annotation encoding="application/x-tex">{\displaystyle z=x+iy}</annotation>

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</math><img src="./08e90bb6b36fef59c6113eed2a08f10d77240741.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\displaystyle z=x+iy}" loading="lazy"> is a <a href="Complex_number" title="Complex number">complex variable</a> in the new space and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta =\chi +i\eta }">

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<annotation encoding="application/x-tex">{\displaystyle \zeta =\chi +i\eta }</annotation>

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</math><img src="./638a942b3b57872612e8e544ae1a2b97ac6c59db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.461ex; height:2.676ex;" alt="{\displaystyle \zeta =\chi +i\eta }" loading="lazy"> is a complex variable in the original space.

In <a href="Aerodynamics" title="Aerodynamics">aerodynamics</a>, the transform is used to solve for the two-dimensional <a href="Potential_flow" title="Potential flow">potential flow</a> around a class of airfoils known as Joukowsky airfoils. A Joukowsky airfoil is generated in the <a href="Complex_plane" title="Complex plane">complex plane</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-plane) by applying the Joukowsky transform to a circle in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

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<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

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</math><img src="./d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy">-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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta =-1}">

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<annotation encoding="application/x-tex">{\displaystyle \zeta =-1}</annotation>

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</math><img src="./daae2fdfe150aa3cda5fb7a57999b95c4a55978f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.164ex; height:2.509ex;" alt="{\displaystyle \zeta =-1}" loading="lazy"> (where the derivative is zero) and intersects the point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta =1.}">

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<annotation encoding="application/x-tex">{\displaystyle \zeta =1.}</annotation>

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</math><img src="./7cc1220234c54437adfad9fc79b1cf7de5182c4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.003ex; height:2.509ex;" alt="{\displaystyle \zeta =1.}" loading="lazy"> This can be achieved for any allowable centre position <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{x}+i\mu _{y}}">

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<annotation encoding="application/x-tex">{\displaystyle \mu _{x}+i\mu _{y}}</annotation>

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</math><img src="./8fbe254e7c96cdd0d3c69119947bdca4ba8e84b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.668ex; height:2.843ex;" alt="{\displaystyle \mu _{x}+i\mu _{y}}" loading="lazy"> by varying the radius of the circle.

Joukowsky airfoils have a <a href="Cusp_(singularity)" title="Cusp (singularity)">cusp</a> at their <a href="Trailing_edge" title="Trailing edge">trailing edge</a>. 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.

<div class="mw-heading mw-heading2"><h2 id="General_Joukowsky_transform">General Joukowsky transform</h2></div>

The Joukowsky transform of any complex number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

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<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

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</math><img src="./d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> is as follows:

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./8e2b2f8c9848f1657d05a4cc14dd96c9a3e87507.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.005ex; width:44.689ex; height:25.176ex;" alt="{\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}}}" loading="lazy"></dd></dl>

So the real (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">) and imaginary (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">) components are:

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<annotation encoding="application/x-tex">{\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}}}</annotation>

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</math><img src="./f0a50c5ba8c65bcd954c518839f845f2acb3e598.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:23.894ex; height:13.176ex;" alt="{\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}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Sample_Joukowsky_airfoil">Sample Joukowsky airfoil</h3></div>

The transformation of all complex numbers on the <a href="Unit_circle" title="Unit circle">unit circle</a> is a special case.

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<annotation encoding="application/x-tex">{\displaystyle |\zeta |={\sqrt {\chi ^{2}+\eta ^{2}}}=1,}</annotation>

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which gives

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<annotation encoding="application/x-tex">{\displaystyle \chi ^{2}+\eta ^{2}=1.}</annotation>

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So the real component becomes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x=\chi (1+1)=2\chi }">

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<annotation encoding="application/x-tex">{\textstyle x=\chi (1+1)=2\chi }</annotation>

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</math><img src="./f0d10e0cdfb497f565a02c4e767b41bbf24bcd9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.574ex; height:2.843ex;" alt="{\textstyle x=\chi (1+1)=2\chi }" loading="lazy"> and the imaginary component becomes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle y=\eta (1-1)=0}">

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<annotation encoding="application/x-tex">{\textstyle y=\eta (1-1)=0}</annotation>

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</math><img src="./0f8b62459ef7fe2d30673632b8d98c0fe2ba5481.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.659ex; height:2.843ex;" alt="{\textstyle y=\eta (1-1)=0}" loading="lazy">.

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.

<div class="mw-heading mw-heading2"><h2 id="Velocity_field_and_circulation_for_the_Joukowsky_airfoil">Velocity field and circulation for the Joukowsky airfoil</h2></div>

The solution to <a href="Potential_flow_around_a_circular_cylinder" title="Potential flow around a circular cylinder">potential flow around a circular cylinder</a> is <a href="Analytic_function" title="Analytic function">analytic</a> and well known. It is the superposition of <a href="Uniform_flow" class="mw-redirect" title="Uniform flow">uniform flow</a>, a <a href="Doublet_(potential_flow)" class="mw-redirect" title="Doublet (potential flow)">doublet</a>, and a <a href="Vortex" title="Vortex">vortex</a>.

The <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widetilde {W}}={\widetilde {u}}_{x}-i{\widetilde {u}}_{y},}">

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<annotation encoding="application/x-tex">{\displaystyle {\widetilde {W}}={\widetilde {u}}_{x}-i{\widetilde {u}}_{y},}</annotation>

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</math><img src="./6b5c1b7a7030cfca911ecac0337c03fdec80879f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.806ex; height:3.343ex;" alt="{\displaystyle {\widetilde {W}}={\widetilde {u}}_{x}-i{\widetilde {u}}_{y},}" loading="lazy"> around the circle in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

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<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

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</math><img src="./d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy">-plane is

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<annotation encoding="application/x-tex">{\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}}},}</annotation>

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where

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<annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{x}+i\mu _{y}}</annotation>

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</math><img src="./100f4b6171728e1311a81b7645769c77203ee9b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.168ex; height:2.843ex;" alt="{\displaystyle \mu =\mu _{x}+i\mu _{y}}" loading="lazy"> is the complex coordinate of the centre of the circle,</li>

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<annotation encoding="application/x-tex">{\displaystyle V_{\infty }}</annotation>

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</math><img src="./0b9ee987969cd445eb595eb4b44367b42a8aeeb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.231ex; height:2.509ex;" alt="{\displaystyle V_{\infty }}" loading="lazy"> is the <a href="Uniform_flow" class="mw-redirect" title="Uniform flow">freestream velocity</a> of the fluid,</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> is the <a href="Angle_of_attack" title="Angle of attack">angle of attack</a> of the airfoil with respect to the freestream flow,

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is the radius of the circle, calculated using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R={\sqrt {\left(1-\mu _{x}\right)^{2}+\mu _{y}^{2}}}}">

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<annotation encoding="application/x-tex">{\textstyle R={\sqrt {\left(1-\mu _{x}\right)^{2}+\mu _{y}^{2}}}}</annotation>

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</math><img src="./4f133e6b4cf8b407bee9c34f1cdcefef923b7fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:21.923ex; height:4.843ex;" alt="{\textstyle R={\sqrt {\left(1-\mu _{x}\right)^{2}+\mu _{y}^{2}}}}" loading="lazy">,</li>

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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

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</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> is the <a href="Circulation_(fluid_dynamics)" class="mw-redirect" title="Circulation (fluid dynamics)">circulation</a>, found using the <a href="Kutta_condition" title="Kutta condition">Kutta condition</a>, which reduces in this case to <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma =4\pi V_{\infty }R\sin \left(\alpha +\sin ^{-1}{\frac {\mu _{y}}{R}}\right).}">

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<annotation encoding="application/x-tex">{\displaystyle \Gamma =4\pi V_{\infty }R\sin \left(\alpha +\sin ^{-1}{\frac {\mu _{y}}{R}}\right).}</annotation>

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The complex velocity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> around the airfoil in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-plane is, according to the rules of conformal mapping and using the Joukowsky transformation,

<mfrac>

<mover>

</mover>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mover>

</mover>

</mrow>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W={\frac {\widetilde {W}}{\frac {dz}{d\zeta }}}={\frac {\widetilde {W}}{1-{\frac {1}{\zeta ^{2}}}}}.}</annotation>

</semantics>

</math>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W=u_{x}-iu_{y},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W=u_{x}-iu_{y},}</annotation>

</semantics>

</math><img src="./98be72c141ecf747c1e2ee81296aeb4c832a983a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.705ex; height:2.843ex;" alt="{\displaystyle W=u_{x}-iu_{y},}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{x}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{x}}</annotation>

</semantics>

</math><img src="./045896a773b67c720bdff8d660cfcc906f5f6b20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.502ex; height:2.009ex;" alt="{\displaystyle u_{x}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{y}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{y}}</annotation>

</semantics>

</math><img src="./ad3979cb7bef75dc4cfb429817450ae8bc370d00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.379ex; height:2.343ex;" alt="{\displaystyle u_{y}}" loading="lazy"> the velocity components in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> directions respectively (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=x+iy,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=x+iy,}</annotation>

</semantics>

</math><img src="./8ee21a775fda65273b6e1a6786b7f3d51a85de65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.961ex; height:2.509ex;" alt="{\displaystyle z=x+iy,}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> real-valued). From this velocity, other properties of interest of the flow, such as the <a href="Coefficient_of_pressure" class="mw-redirect" title="Coefficient of pressure">coefficient of pressure</a> and <a href="Lift_(force)" title="Lift (force)">lift</a> per unit of span can be calculated.

<div class="mw-heading mw-heading2"><h2 id="Kármán–Trefftz_transform">Kármán–Trefftz transform</h2></div>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

</semantics>

</math><img src="./d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy">-plane to the physical <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha .}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha .}</annotation>

</semantics>

</math><img src="./794850adc0db51d11a6d8cfa857538183424909c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.134ex; height:1.676ex;" alt="{\displaystyle \alpha .}" loading="lazy"> This transform is<a href="#cite_note-MT-2">[2]</a><a href="#cite_note-JB-3">[3]</a>

/* start https://en.wikipedia.org/ */

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</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=nb{\frac {(\zeta +b)^{n}+(\zeta -b)^{n}}{(\zeta +b)^{n}-(\zeta -b)^{n}}},}">

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=nb{\frac {(\zeta +b)^{n}+(\zeta -b)^{n}}{(\zeta +b)^{n}-(\zeta -b)^{n}}},}</annotation>

</semantics>

</math><img src="./c7e473b09a3a4bed2dcdbee069b9ab9789c35bcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.824ex; height:6.509ex;" alt="{\displaystyle z=nb{\frac {(\zeta +b)^{n}+(\zeta -b)^{n}}{(\zeta +b)^{n}-(\zeta -b)^{n}}},}" loading="lazy">

</td> <td></td> <td class="nowrap">A</td></tr></tbody></table>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

</semantics>

</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> is a real constant that determines the positions where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dz/d\zeta =0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dz/d\zeta =0}</annotation>

</semantics>

</math><img src="./88cab6f5aecdff0ae70629fa6da2b30fe76a46cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.038ex; height:2.843ex;" alt="{\displaystyle dz/d\zeta =0}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is slightly smaller than 2. The angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> between the <a href="Tangent" title="Tangent">tangents</a> of the upper and lower airfoil surfaces at the trailing edge is related to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> as<a href="#cite_note-MT-2">[2]</a>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha =2\pi -n\pi ,\quad n=2-{\frac {\alpha }{\pi }}.}</annotation>

</semantics>

</math><img src="./e84da8bbd8221e6a0cb394da77bcd1747e577f56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.471ex; height:4.676ex;" alt="{\displaystyle \alpha =2\pi -n\pi ,\quad n=2-{\frac {\alpha }{\pi }}.}" loading="lazy"></dd></dl>

The derivative <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dz/d\zeta }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle dz/d\zeta }</annotation>

</semantics>

</math><img src="./15a99c009eaaf933dbf9f82ea91d5aff627bcecc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.777ex; height:2.843ex;" alt="{\displaystyle dz/d\zeta }" loading="lazy">, required to compute the velocity field, is

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

<mrow>

<mrow>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}}}.}</annotation>

</semantics>

</math><img src="./9e8c37052334854a67b3388bf8047e96fbc966a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:40.139ex; height:10.676ex;" alt="{\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}}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Background">Background</h3></div>

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

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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}}}</annotation>

</semantics>

</math><img src="./ef1652730dc32a9ba2669f06a26a1aa45cb43455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:32.583ex; height:12.509ex;" alt="{\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}}}" loading="lazy"></dd></dl>

Dividing the left and right hand sides gives

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {z-2}{z+2}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{2}.}</annotation>

</semantics>

</math><img src="./3c28d839aaf00e4aca5bf579e59a2588938b7c6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.082ex; height:6.509ex;" alt="{\displaystyle {\frac {z-2}{z+2}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{2}.}" loading="lazy"></dd></dl>

The <a href="Right_hand_side" class="mw-redirect" title="Right hand side">right hand side</a> contains (as a factor) the simple second-power law from <a href="Potential_flow" title="Potential flow">potential flow</a> theory, applied at the trailing edge near <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta =+1.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \zeta =+1.}</annotation>

</semantics>

</math><img src="./717ec4914f880dab7f1caa6aa4879cdc2721260c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.811ex; height:2.509ex;" alt="{\displaystyle \zeta =+1.}" loading="lazy"> From conformal mapping theory, this quadratic map is known to change a half plane in the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

</semantics>

</math><img src="./d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy">-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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> in the previous equation gives<a href="#cite_note-MT-2">[2]</a>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {z-n}{z+n}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{n},}</annotation>

</semantics>

</math><img src="./8531f73e98023b152a72f9d042ab4a1073ffb26d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.478ex; height:6.176ex;" alt="{\displaystyle {\frac {z-n}{z+n}}=\left({\frac {\zeta -1}{\zeta +1}}\right)^{n},}" loading="lazy"></dd></dl>

which is the Kármán–Trefftz transform. Solving for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> gives it in the form of equation <a href="#math_A">A</a>.

<div class="mw-heading mw-heading2"><h2 id="Symmetrical_Joukowsky_airfoils">Symmetrical Joukowsky airfoils</h2></div>

In 1943 <a href="Hsue-shen_Tsien" class="mw-redirect" title="Hsue-shen Tsien">Hsue-shen Tsien</a> published a transform of a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> into a symmetrical airfoil that depends on parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation>

</semantics>

</math><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"> and angle of inclination <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy">:<a href="#cite_note-4">[4]</a>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=e^{i\alpha }\left(\zeta -\epsilon +{\frac {1}{\zeta -\epsilon }}+{\frac {2\epsilon ^{2}}{a+\epsilon }}\right).}</annotation>

</semantics>

</math><img src="./0a6d5f2e7577c3f2ba7452a8abcbc311a153daf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.091ex; height:6.343ex;" alt="{\displaystyle z=e^{i\alpha }\left(\zeta -\epsilon +{\frac {1}{\zeta -\epsilon }}+{\frac {2\epsilon ^{2}}{a+\epsilon }}\right).}" loading="lazy"></dd></dl>

The parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation>

</semantics>

</math><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=1+\epsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=1+\epsilon }</annotation>

</semantics>

</math><img src="./4b2dfdb6122cb21a43d1924453d23900a105374f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.275ex; height:2.343ex;" alt="{\displaystyle a=1+\epsilon }" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite id="CITEREFJoukowsky1910" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="Nikolay_Yegorovich_Zhukovsky" class="mw-redirect" title="Nikolay Yegorovich Zhukovsky">Joukowsky, N. E.</a> (1910). "Über die Konturen der Tragflächen der Drachenflieger". Zeitschrift für Flugtechnik und Motorluftschiffahrt (in German). 1: 281–284 and (1912) 3: 81–86.</cite>

</li>

<li id="cite_note-MT-2">^ <a href="#cite_ref-MT_2-0">a</a> <a href="#cite_ref-MT_2-1">b</a> <a href="#cite_ref-MT_2-2">c</a> <cite id="CITEREFMilne-Thomson1973" class="citation book cs1"><a href="L._M._Milne-Thomson" title="L. M. Milne-Thomson">Milne-Thomson, Louis M.</a> (1973). <a rel="nofollow" class="external text" href="https://archive.org/details/theoreticalaerod00miln_923">Theoretical aerodynamics</a> (4th ed.). Dover Publ. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/theoreticalaerod00miln_923/page/n74">128</a>–131. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-486-61980-X</bdi>.</cite>

</li>

<li id="cite_note-JB-3"><a href="#cite_ref-JB_3-0">^</a> <cite id="CITEREFBlom1981" class="citation document cs1">Blom, J. J. H. (1981). "Some Characteristic Quantities of Karman-Trefftz Profiles" (Document). NASA Technical Memorandum TM-77013.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFTsien1943" class="citation journal cs1"><a href="Hsue-shen_Tsien" class="mw-redirect" title="Hsue-shen Tsien">Tsien, Hsue-shen</a> (1943). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fqam%2F8537">"Symmetrical Joukowsky airfoils in shear flow"</a>. Quarterly of Applied Mathematics. 1 (2): 130–248. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fqam%2F8537">10.1090/qam/8537</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<ul><li><cite id="CITEREFAnderson1991" class="citation book cs1">Anderson, John (1991). Fundamentals of Aerodynamics (Second ed.). Toronto: McGraw–Hill. pp. 195–208. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-001679-8</bdi>.</cite></li>

<li><cite id="CITEREFZingg1989" class="citation web cs1">Zingg, D. W. (1989). <a rel="nofollow" class="external text" href="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">"Low Mach number Euler computations"</a>. <a href="NASA" title="NASA">NASA</a> TM-102205.</cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110320205004/http://www.grc.nasa.gov/WWW/K-12/airplane/map.html">Joukowski Transform NASA Applet</a></li>

<li><a rel="nofollow" class="external text" href="http://dimanov.com/airfoil/af_prj_jint.html">Joukowsky Transform Interactive WebApp</a></li></ul></div><div><div class="zim-footer">

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