{"id":4060,"date":"2024-05-29T00:44:51","date_gmt":"2024-05-28T22:44:51","guid":{"rendered":"https:\/\/eva.louis-le-grand.net\/maths\/?p=4060"},"modified":"2024-05-29T17:00:43","modified_gmt":"2024-05-29T15:00:43","slug":"moules-aux-tourbillons-de-vapeur","status":"publish","type":"post","link":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/2024\/05\/29\/moules-aux-tourbillons-de-vapeur\/","title":{"rendered":"Moules aux tourbillons de vapeur"},"content":{"rendered":"\n<figure class=\"wp-block-video\"><video controls src=\"blob:https:\/\/eva.louis-le-grand.net\/65c9dde6-67f3-48f5-8c94-f0453aef278e\"><\/video><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">La dimension de Kolmogorov est l&rsquo;\u00e9chelle spatiale \u00e0 partir de laquelle l&rsquo;\u00e9coulement devient visqueux (\ud835\udc45\ud835\udc52\u22481<img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/bffef0158b8a99c515b2d69aa33b43bd2ef80429\" alt=\"{\\displaystyle Re\\approx 1}\">) et permet de dissiper l&rsquo;<a href=\"https:\/\/fr.wikipedia.org\/wiki\/%C3%89nergie_cin%C3%A9tique\">\u00e9nergie cin\u00e9tique<\/a>&nbsp;de l&rsquo;\u00e9coulement.\ud835\udc3f\u2243\ud835\udf07\ud835\udf0c\ud835\udc49<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fff1678829b9922db3c30a275d664b88434831aa\" alt=\"{\\displaystyle L\\simeq {\\frac {\\mu }{\\rho V}}}\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Elle correspond \u00e0 la taille des petits tourbillons qui incarnent ce ph\u00e9nom\u00e8ne. Il est \u00e9galement possible de d\u00e9finir plusieurs param\u00e8tres<sup><a href=\"https:\/\/fr.wikipedia.org\/wiki\/Dimension_de_Kolmogorov#cite_note-2\">2<\/a><\/sup>&nbsp;comme l&rsquo;\u00e9chelle spatiale de Kolmogorov&nbsp;\ud835\udf02<img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/e4d701857cf5fbec133eebaf94deadf722537f64\" alt=\"{\\displaystyle \\eta }\">&nbsp;et l&rsquo;\u00e9chelle temporelle de Kolmogorov&nbsp;\ud835\udf0f\ud835\udf02<img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8c729ecd90c196e786c2b8bdec8acde1972f42e2\" alt=\"{\\displaystyle \\tau _{\\eta }}\">, respectivement la taille et la dur\u00e9e de vie des plus petits tourbillons d&rsquo;un \u00e9coulement turbulent tels que<sup><a href=\"https:\/\/fr.wikipedia.org\/wiki\/Dimension_de_Kolmogorov#cite_note-3\">3<\/a><\/sup>:\ud835\udf02=\ud835\udf083\ud835\udf164<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/cf6d4ccf381dc6b46424c57289d6e2e256675b75\" alt=\"{\\displaystyle \\eta ={\\sqrt[{4}]{\\nu ^{3} \\over \\epsilon }}}\"><\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b5aa75c0e58c186cb188697371e39ca8f8cb8e87\" alt=\"{\\displaystyle \\tau _{\\eta }={\\sqrt {\\nu  \\over \\epsilon }}}\"\/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">Avec&nbsp;\ud835\udf08<img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c15bbbb971240cf328aba572178f091684585468\" alt=\"{\\displaystyle \\nu }\">&nbsp;la viscosit\u00e9 cin\u00e9matique et&nbsp;\ud835\udf16<img decoding=\"async\" src=\"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c3837cad72483d97bcdde49c85d3b7b859fb3fd2\" alt=\"{\\displaystyle \\epsilon }\">&nbsp;le taux de dissipation turbulent. Il n&rsquo;y a pas de&nbsp;<a href=\"https:\/\/fr.wikipedia.org\/wiki\/Tourbillon_de_turbulence\">tourbillons<\/a>&nbsp;plus petits, car cette \u00e9chelle est la limite de l&rsquo;\u00e9coulement \u00e9tudi\u00e9, l\u00e0 o\u00f9 l&rsquo;\u00e9nergie initialement donn\u00e9e au fluide se dissipe.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Ce processus appel\u00e9 cascade d&rsquo;\u00e9nergie&nbsp;: la division des grands tourbillons en tourbillons plus petits permet un transfert d&rsquo;\u00e9nergie des grandes \u00e9chelles vers les petites \u00e9chelles. Ce processus est limit\u00e9 par l&rsquo;effet de la dissipation mol\u00e9culaire, qui emp\u00eache les variations de vitesse trop importantes. En pratique, ce transfert d&rsquo;\u00e9nergie n&rsquo;est pas \u00e0 sens unique, le ph\u00e9nom\u00e8ne d&rsquo;appariement tourbillonnaire (en anglais backscatter) permettant le transfert ponctuel de petites structures tourbillonnaires (qui fusionnent) vers une ou des plus grosses.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Kolmogorov, en 1941, a \u00e9mis l&rsquo;hypoth\u00e8se que cette&nbsp;<a href=\"https:\/\/fr.wikipedia.org\/wiki\/Cascade_turbulente\">cascade turbulente<\/a>&nbsp;\u00e9tait auto-similaire&nbsp;: les tourbillons se divisent tous de la m\u00eame mani\u00e8re quelle que soit leur \u00e9chelle, tant qu&rsquo;elle n&rsquo;est ni trop petite (sinon il faut tenir compte de la viscosit\u00e9) ni trop grande (les grands tourbillons d\u00e9pendent de la g\u00e9om\u00e9trie de l&rsquo;\u00e9coulement). C&rsquo;est ce qu&rsquo;on appelle la zone inertielle, et par des arguments d&rsquo;analyse dimensionnelle, il a exprim\u00e9 une loi (loi en -5\/3) qui caract\u00e9rise l&rsquo;auto-similarit\u00e9 de la turbulence (un peu comme une courbe fractale, quand on \u00ab\u00a0zoome\u00a0\u00bb sur une turbulence, on ne peut pas savoir \u00e0 quelle \u00e9chelle on se trouve).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>La dimension de Kolmogorov est l&rsquo;\u00e9chelle spatiale \u00e0 partir de laquelle l&rsquo;\u00e9coulement devient visqueux (\ud835\udc45\ud835\udc52\u22481) et permet de dissiper l&rsquo;\u00e9nergie cin\u00e9tique&nbsp;de l&rsquo;\u00e9coulement.\ud835\udc3f\u2243\ud835\udf07\ud835\udf0c\ud835\udc49 Elle correspond \u00e0 la taille des petits tourbillons qui incarnent ce ph\u00e9nom\u00e8ne. Il est \u00e9galement possible de d\u00e9finir plusieurs param\u00e8tres2&nbsp;comme l&rsquo;\u00e9chelle spatiale de Kolmogorov&nbsp;\ud835\udf02&nbsp;et l&rsquo;\u00e9chelle temporelle de Kolmogorov&nbsp;\ud835\udf0f\ud835\udf02, respectivement la taille et la [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":4062,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-4060","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-non-classe"],"_links":{"self":[{"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/posts\/4060","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/comments?post=4060"}],"version-history":[{"count":3,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/posts\/4060\/revisions"}],"predecessor-version":[{"id":4066,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/posts\/4060\/revisions\/4066"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/media\/4062"}],"wp:attachment":[{"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/media?parent=4060"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/categories?post=4060"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/eva.louis-le-grand.net\/maths\/index.php\/wp-json\/wp\/v2\/tags?post=4060"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}