@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-RJD29DDL-S
  dc:modified "2024-10-18"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Joukowsky_transform>, <https://fr.wikipedia.org/wiki/Transformation_de_Joukovsky> ;
  skos:altLabel "Joukovsky transform"@en ;
  a skos:Concept ;
  skos:broader psr:-XWWVH3PC-T ;
  skos:definition """In applied mathematics, the <b>Joukowsky transform</b> (sometimes transliterated <i>Joukovsky</i>, <i>Joukowski</i> or <i>Zhukovsky</i>) 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 is  <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle z=\\\\zeta +{\\rac {1}{\\\\zeta }},}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>z</mi>         <mo>=</mo>         <mi>ζ<!-- ζ --></mi>         <mo>+</mo>         <mrow class="MJX-TeXAtom-ORD">           <mfrac>             <mn>1</mn>             <mi>ζ<!-- ζ --></mi>           </mfrac>         </mrow>         <mo>,</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle z=\\\\zeta +{\\rac {1}{\\\\zeta }},}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a73ff7ce2f0ed759d082be0a69d14c7557718a66" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.768ex; height:5.676ex;" alt="{\\\\displaystyle z=\\\\zeta +{\\rac {1}{\\\\zeta }},}"></span></dd></dl> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle z=x+iy}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>z</mi>         <mo>=</mo>         <mi>x</mi>         <mo>+</mo>         <mi>i</mi>         <mi>y</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle z=x+iy}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08e90bb6b36fef59c6113eed2a08f10d77240741" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.315ex; height:2.509ex;" alt="{\\\\displaystyle z=x+iy}"></span> is a complex variable in the new space and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\zeta =\\\\chi +i\\\\eta }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ζ<!-- ζ --></mi>         <mo>=</mo>         <mi>χ<!-- χ --></mi>         <mo>+</mo>         <mi>i</mi>         <mi>η<!-- η --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta =\\\\chi +i\\\\eta }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/638a942b3b57872612e8e544ae1a2b97ac6c59db" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.461ex; height:2.676ex;" alt="{\\\\displaystyle \\\\zeta =\\\\chi +i\\\\eta }"></span> is a complex variable in the original space. In aerodynamics, the transform is used to solve for the two-dimensional potential flow around a class of airfoils known as Joukowsky airfoils.  A <b>Joukowsky airfoil</b> is generated in the complex plane (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle z}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>z</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle z}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\\\\displaystyle z}"></span>-plane) by applying the Joukowsky transform to a circle in the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\zeta }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ζ<!-- ζ --></mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\\\\displaystyle \\\\zeta }"></span>-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 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\zeta =-1}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ζ<!-- ζ --></mi>         <mo>=</mo>         <mo>−<!-- − --></mo>         <mn>1</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta =-1}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daae2fdfe150aa3cda5fb7a57999b95c4a55978f" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.164ex; height:2.509ex;" alt="{\\\\displaystyle \\\\zeta =-1}"></span> (where the derivative is zero) and intersects the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\zeta =1.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>ζ<!-- ζ --></mi>         <mo>=</mo>         <mn>1.</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\zeta =1.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc1220234c54437adfad9fc79b1cf7de5182c4e" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.003ex; height:2.509ex;" alt="{\\\\displaystyle \\\\zeta =1.}"></span> This can be achieved for any allowable centre position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\mu _{x}+i\\\\mu _{y}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>μ<!-- μ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>x</mi>           </mrow>         </msub>         <mo>+</mo>         <mi>i</mi>         <msub>           <mi>μ<!-- μ --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>y</mi>           </mrow>         </msub>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\mu _{x}+i\\\\mu _{y}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbe254e7c96cdd0d3c69119947bdca4ba8e84b7" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.668ex; height:2.843ex;" alt="{\\\\displaystyle \\\\mu _{x}+i\\\\mu _{y}}"></span> by varying the radius of the circle.  
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Joukowsky_transform">https://en.wikipedia.org/wiki/Joukowsky_transform</a>)"""@en, """La transformation de Joukovsky, nommée d'après le savant aérodynamicien russe Nikolaï Joukovski bien qu'elle soit due à Otto Blumenthal, est une transformation conforme utilisée historiquement dans le calcul des profils d'aile d'avion. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Transformation_de_Joukovsky">https://fr.wikipedia.org/wiki/Transformation_de_Joukovsky</a>)"""@fr ;
  skos:prefLabel "Joukowsky transform"@en, "transformation de Joukovsky"@fr ;
  dc:created "2023-08-08"^^xsd:date ;
  skos:inScheme psr: .

psr:-XWWVH3PC-T
  skos:prefLabel "conformal map"@en, "transformation conforme"@fr ;
  a skos:Concept ;
  skos:narrower psr:-RJD29DDL-S .