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id="toc-Discretization_methods-sublist" class="vector-toc-list"> <li id="toc-Finite_volume_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Finite_volume_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Finite volume method</span> </div> </a> <ul id="toc-Finite_volume_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite_element_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Finite_element_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Finite element method</span> </div> </a> <ul id="toc-Finite_element_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite_difference_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Finite_difference_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.3</span> <span>Finite difference method</span> </div> </a> <ul id="toc-Finite_difference_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spectral_element_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Spectral_element_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.4</span> <span>Spectral element method</span> </div> </a> <ul id="toc-Spectral_element_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lattice_Boltzmann_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lattice_Boltzmann_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.5</span> <span>Lattice Boltzmann method</span> </div> </a> <ul id="toc-Lattice_Boltzmann_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vortex_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Vortex_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.6</span> <span>Vortex method</span> </div> </a> <ul id="toc-Vortex_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Boundary_element_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Boundary_element_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.7</span> <span>Boundary element method</span> </div> </a> <ul id="toc-Boundary_element_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-High-resolution_discretization_schemes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#High-resolution_discretization_schemes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.8</span> <span>High-resolution discretization schemes</span> </div> </a> <ul id="toc-High-resolution_discretization_schemes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Turbulence_models" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Turbulence_models"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Turbulence models</span> </div> </a> <ul id="toc-Turbulence_models-sublist" class="vector-toc-list"> <li id="toc-Reynolds-averaged_Navier–Stokes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Reynolds-averaged_Navier–Stokes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Reynolds-averaged Navier–Stokes</span> </div> </a> <ul id="toc-Reynolds-averaged_Navier–Stokes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Large_eddy_simulation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Large_eddy_simulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Large eddy simulation</span> </div> </a> <ul id="toc-Large_eddy_simulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Detached_eddy_simulation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Detached_eddy_simulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Detached eddy simulation</span> </div> </a> <ul id="toc-Detached_eddy_simulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Direct_numerical_simulation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Direct_numerical_simulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4</span> <span>Direct numerical simulation</span> </div> </a> <ul id="toc-Direct_numerical_simulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coherent_vortex_simulation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Coherent_vortex_simulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.5</span> <span>Coherent vortex simulation</span> </div> </a> <ul id="toc-Coherent_vortex_simulation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-PDF_methods" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#PDF_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.6</span> <span>PDF methods</span> </div> </a> <ul id="toc-PDF_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vorticity_confinement_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Vorticity_confinement_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.7</span> <span>Vorticity confinement method</span> </div> </a> <ul id="toc-Vorticity_confinement_method-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_eddy_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Linear_eddy_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.8</span> <span>Linear eddy model</span> </div> </a> <ul id="toc-Linear_eddy_model-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Two-phase_flow" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Two-phase_flow"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Two-phase flow</span> </div> </a> <ul id="toc-Two-phase_flow-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Solution_algorithms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Solution_algorithms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Solution algorithms</span> </div> </a> <ul id="toc-Solution_algorithms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Unsteady_aerodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unsteady_aerodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Unsteady aerodynamics</span> </div> </a> <ul id="toc-Unsteady_aerodynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biomedical_engineering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biomedical_engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Biomedical engineering</span> </div> </a> <ul id="toc-Biomedical_engineering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-CPU_versus_GPU" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#CPU_versus_GPU"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.7</span> <span>CPU versus GPU</span> </div> </a> <ul id="toc-CPU_versus_GPU-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Computational fluid dynamics</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 31 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-31" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">31 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D9%8A%D9%86%D8%A7%D9%85%D9%8A%D9%83%D8%A7_%D9%85%D9%88%D8%A7%D8%A6%D8%B9_%D8%AD%D8%B3%D8%A7%D8%A8%D9%8A%D8%A9" title="ديناميكا موائع حسابية – Arabic" lang="ar" hreflang="ar" data-title="ديناميكا موائع حسابية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Ra%C4%8Dunarska_dinamika_fluida" title="Računarska dinamika fluida – Bosnian" lang="bs" hreflang="bs" data-title="Računarska dinamika fluida" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mec%C3%A0nica_de_fluids_computacional" title="Mecànica de fluids computacional – Catalan" lang="ca" hreflang="ca" data-title="Mecànica de fluids computacional" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/V%C3%BDpo%C4%8Detn%C3%AD_dynamika_tekutin" title="Výpočetní dynamika tekutin – Czech" lang="cs" hreflang="cs" data-title="Výpočetní dynamika tekutin" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Numerische_Str%C3%B6mungsmechanik" title="Numerische Strömungsmechanik – German" lang="de" hreflang="de" data-title="Numerische Strömungsmechanik" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A5%CF%80%CE%BF%CE%BB%CE%BF%CE%B3%CE%B9%CF%83%CF%84%CE%B9%CE%BA%CE%AE_%CF%81%CE%B5%CF%85%CF%83%CF%84%CE%BF%CE%B4%CF%85%CE%BD%CE%B1%CE%BC%CE%B9%CE%BA%CE%AE" title="Υπολογιστική ρευστοδυναμική – Greek" lang="el" hreflang="el" data-title="Υπολογιστική ρευστοδυναμική" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Fluidodin%C3%A1mica_computacional" title="Fluidodinámica computacional – Spanish" lang="es" hreflang="es" data-title="Fluidodinámica computacional" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AF%DB%8C%D9%86%D8%A7%D9%85%DB%8C%DA%A9_%D8%B3%DB%8C%D8%A7%D9%84%D8%A7%D8%AA_%D9%85%D8%AD%D8%A7%D8%B3%D8%A8%D8%A7%D8%AA%DB%8C" title="دینامیک سیالات محاسباتی – Persian" lang="fa" hreflang="fa" data-title="دینامیک سیالات محاسباتی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/M%C3%A9canique_des_fluides_num%C3%A9rique" title="Mécanique des fluides numérique – French" lang="fr" hreflang="fr" data-title="Mécanique des fluides numérique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EC%82%B0_%EC%9C%A0%EC%B2%B4_%EC%97%AD%ED%95%99" title="전산 유체 역학 – Korean" lang="ko" hreflang="ko" data-title="전산 유체 역학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%AD%E0%A4%BF%E0%A4%95%E0%A4%B2%E0%A4%A8%E0%A4%BE%E0%A4%A4%E0%A5%8D%E0%A4%AE%E0%A4%95_%E0%A4%A4%E0%A4%B0%E0%A4%B2_%E0%A4%AF%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80" title="अभिकलनात्मक तरल यांत्रिकी – Hindi" lang="hi" hreflang="hi" data-title="अभिकलनात्मक तरल यांत्रिकी" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Ra%C4%8Dunalna_dinamika_fluida" title="Računalna dinamika fluida – Croatian" lang="hr" hreflang="hr" data-title="Računalna dinamika fluida" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Dinamika_fluida_komputasi" title="Dinamika fluida komputasi – Indonesian" lang="id" hreflang="id" data-title="Dinamika fluida komputasi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Fluidodinamica_computazionale" title="Fluidodinamica computazionale – Italian" lang="it" hreflang="it" data-title="Fluidodinamica computazionale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B2%A3%E0%B2%95%E0%B2%AF%E0%B2%82%E0%B2%A4%E0%B3%8D%E0%B2%B0%E0%B3%80%E0%B2%AF_%E0%B2%A6%E0%B3%8D%E0%B2%B0%E0%B2%B5_%E0%B2%9A%E0%B2%B2%E0%B2%A8%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0" title="ಗಣಕಯಂತ್ರೀಯ ದ್ರವ ಚಲನಶಾಸ್ತ್ರ – Kannada" lang="kn" hreflang="kn" data-title="ಗಣಕಯಂತ್ರೀಯ ದ್ರವ ಚಲನಶಾಸ್ತ್ರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D1%98%D1%83%D1%82%D0%B5%D1%80%D1%81%D0%BA%D0%B0_%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0_%D0%BD%D0%B0_%D1%84%D0%BB%D1%83%D0%B8%D0%B4%D0%B8" title="Компјутерска динамика на флуиди – Macedonian" lang="mk" hreflang="mk" data-title="Компјутерска динамика на флуиди" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mn badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://mn.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BE%D1%86%D0%BE%D0%BD_%D0%B1%D0%BE%D0%B4%D0%BE%D1%85_%D1%88%D0%B8%D0%BD%D0%B3%D1%8D%D0%BD%D0%B8%D0%B9_%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA" title="Тооцон бодох шингэний динамик – Mongolian" lang="mn" hreflang="mn" data-title="Тооцон бодох шингэний динамик" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Numerieke_stromingsleer" title="Numerieke stromingsleer – Dutch" lang="nl" hreflang="nl" data-title="Numerieke stromingsleer" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E5%80%A4%E6%B5%81%E4%BD%93%E5%8A%9B%E5%AD%A6" title="数値流体力学 – Japanese" lang="ja" hreflang="ja" data-title="数値流体力学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Numerisk_fluiddynamikk" title="Numerisk fluiddynamikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Numerisk fluiddynamikk" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Obliczeniowa_mechanika_p%C5%82yn%C3%B3w" title="Obliczeniowa mechanika płynów – Polish" lang="pl" hreflang="pl" data-title="Obliczeniowa mechanika płynów" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fluidodin%C3%A2mica_computacional" title="Fluidodinâmica computacional – Portuguese" lang="pt" hreflang="pt" data-title="Fluidodinâmica computacional" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro badge-Q17437796 badge-featuredarticle 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.sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Computational_physics" title="Computational physics">Computational physics</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Rayleigh-Taylor_instability.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Rayleigh-Taylor_instability.jpg/220px-Rayleigh-Taylor_instability.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Rayleigh-Taylor_instability.jpg/330px-Rayleigh-Taylor_instability.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Rayleigh-Taylor_instability.jpg/440px-Rayleigh-Taylor_instability.jpg 2x" data-file-width="500" data-file-height="500" /></a></span></td></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Computational_mechanics" title="Computational mechanics">Mechanics</a></li> <li><a href="/wiki/Computational_electromagnetics" title="Computational electromagnetics">Electromagnetics</a></li> <li><a href="/wiki/Multiphysics_simulation" title="Multiphysics simulation">Multiphysics</a></li> <li><a href="/wiki/Computational_particle_physics" title="Computational particle physics">Particle physics</a></li> <li><a href="/wiki/Computational_thermodynamics" title="Computational thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Computer_simulation" title="Computer simulation">Simulation</a><br /></li></ul></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Potentials</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Morse/Long-range_potential" title="Morse/Long-range potential">Morse/Long-range potential</a></li> <li><a href="/wiki/Lennard-Jones_potential" title="Lennard-Jones potential">Lennard-Jones potential</a></li> <li><a href="/wiki/Yukawa_potential" title="Yukawa potential">Yukawa potential</a></li> <li><a href="/wiki/Morse_potential" title="Morse potential">Morse potential</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a class="mw-selflink selflink">Fluid dynamics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference</a></li> <li><a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element</a></li> <li><a href="/wiki/Boundary_element_method" title="Boundary element method">Boundary element</a></li> <li><a href="/wiki/Lattice_Boltzmann_methods" title="Lattice Boltzmann methods">Lattice Boltzmann</a></li> <li><a href="/wiki/Riemann_solver" title="Riemann solver">Riemann solver</a></li> <li><a href="/wiki/Dissipative_particle_dynamics" title="Dissipative particle dynamics">Dissipative particle dynamics</a></li> <li><a href="/wiki/Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">Smoothed particle hydrodynamics</a></li> <li><a href="/wiki/Turbulence_modeling" title="Turbulence modeling">Turbulence models</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Monte_Carlo_method" title="Monte Carlo method">Monte Carlo methods</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Monte_Carlo_integration" title="Monte Carlo integration">Integration</a></li> <li><a href="/wiki/Gibbs_sampling" title="Gibbs sampling">Gibbs sampling</a></li> <li><a href="/wiki/Metropolis%E2%80%93Hastings_algorithm" title="Metropolis–Hastings algorithm">Metropolis algorithm</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Particle</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/N-body_simulation" title="N-body simulation">N-body</a></li> <li><a href="/wiki/Particle-in-cell" title="Particle-in-cell">Particle-in-cell</a></li> <li><a href="/wiki/Molecular_dynamics" title="Molecular dynamics">Molecular dynamics</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Sergei_K._Godunov" class="mw-redirect" title="Sergei K. Godunov">Godunov</a></li> <li><a href="/wiki/Stanislaw_Ulam" class="mw-redirect" title="Stanislaw Ulam">Ulam</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">von Neumann</a></li> <li><a href="/wiki/Boris_Galerkin" title="Boris Galerkin">Galerkin</a></li> <li><a href="/wiki/Edward_Norton_Lorenz" title="Edward Norton Lorenz">Lorenz</a></li> <li><a href="/wiki/Kenneth_G._Wilson" title="Kenneth G. Wilson">Wilson</a></li> <li><a href="/wiki/Berni_Alder" title="Berni Alder">Alder</a></li> <li><a href="/wiki/Robert_D._Richtmyer" title="Robert D. Richtmyer">Richtmyer</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Computational_physics" title="Template:Computational physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Computational_physics" title="Template talk:Computational physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Computational_physics" title="Special:EditPage/Template:Computational physics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Computational fluid dynamics</b> (<b>CFD</b>) is a branch of <a href="/wiki/Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a> that uses <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a> and <a href="/wiki/Data_structure" title="Data structure">data structures</a> to analyze and solve problems that involve <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid flows</a>. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid (<a href="/wiki/Liquid" title="Liquid">liquids</a> and <a href="/wiki/Gas" title="Gas">gases</a>) with surfaces defined by <a href="/wiki/Boundary_value_problem#Boundary_value_conditions" title="Boundary value problem">boundary conditions</a>. With high-speed <a href="/wiki/Supercomputer" title="Supercomputer">supercomputers</a>, better solutions can be achieved, and are often required to solve the largest and most complex problems. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as <a href="/wiki/Transonic" title="Transonic">transonic</a> or <a href="/wiki/Turbulence" title="Turbulence">turbulent</a> flows. Initial validation of such software is typically performed using experimental apparatus such as <a href="/wiki/Wind_tunnel" title="Wind tunnel">wind tunnels</a>. In addition, previously performed <a href="/wiki/Closed-form_solution" class="mw-redirect" title="Closed-form solution">analytical</a> or <a href="/wiki/Empirical_research" title="Empirical research">empirical</a> analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as <a href="/wiki/Flight_test" title="Flight test">flight tests</a>. </p><p>CFD is applied to a wide range of research and engineering problems in many fields of study and industries, including <a href="/wiki/Aerodynamics" title="Aerodynamics">aerodynamics</a> and aerospace analysis, <a href="/wiki/Hypersonics" class="mw-redirect" title="Hypersonics">hypersonics</a>, <a href="/wiki/Numerical_weather_prediction" title="Numerical weather prediction">weather simulation</a>, natural science and <a href="/wiki/Environmental_engineering" title="Environmental engineering">environmental engineering</a>, industrial system design and analysis, <a href="/wiki/Biological_engineering" title="Biological engineering">biological engineering</a>, fluid flows and <a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a>, <a href="/wiki/Engine" title="Engine">engine</a> and <a href="/wiki/Combustion" title="Combustion">combustion</a> analysis, and <a href="/wiki/Visual_effects" title="Visual effects">visual effects</a> for film and games. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Background_and_history">Background and history</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=1" title="Edit section: Background and history"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:CFD_Shuttle.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/CFD_Shuttle.jpg/220px-CFD_Shuttle.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/CFD_Shuttle.jpg/330px-CFD_Shuttle.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/CFD_Shuttle.jpg/440px-CFD_Shuttle.jpg 2x" data-file-width="516" data-file-height="387" /></a><figcaption>A computer simulation of high velocity air flow around the <a href="/wiki/Space_Shuttle" title="Space Shuttle">Space Shuttle</a> during re-entry</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:X-43A_(Hyper_-_X)_Mach_7_computational_fluid_dynamic_(CFD).jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/X-43A_%28Hyper_-_X%29_Mach_7_computational_fluid_dynamic_%28CFD%29.jpg/220px-X-43A_%28Hyper_-_X%29_Mach_7_computational_fluid_dynamic_%28CFD%29.jpg" decoding="async" width="220" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/55/X-43A_%28Hyper_-_X%29_Mach_7_computational_fluid_dynamic_%28CFD%29.jpg/330px-X-43A_%28Hyper_-_X%29_Mach_7_computational_fluid_dynamic_%28CFD%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/55/X-43A_%28Hyper_-_X%29_Mach_7_computational_fluid_dynamic_%28CFD%29.jpg/440px-X-43A_%28Hyper_-_X%29_Mach_7_computational_fluid_dynamic_%28CFD%29.jpg 2x" data-file-width="3030" data-file-height="1950" /></a><figcaption>A simulation of the <a href="/wiki/Hyper-X" class="mw-redirect" title="Hyper-X">Hyper-X</a> scramjet vehicle in operation at <a href="/wiki/Mach_number" title="Mach number">Mach</a>-7</figcaption></figure> <p>The fundamental basis of almost all CFD problems is the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a>, which define many single-phase (gas or liquid, but not both) fluid flows. These equations can be simplified by removing terms describing <a href="/wiki/Viscous" class="mw-redirect" title="Viscous">viscous</a> actions to yield the <a href="/wiki/Euler_equations_(fluid_dynamics)" title="Euler equations (fluid dynamics)">Euler equations</a>. Further simplification, by removing terms describing <a href="/wiki/Vorticity" title="Vorticity">vorticity</a> yields the <a href="/wiki/Full_potential_equation" class="mw-redirect" title="Full potential equation">full potential equations</a>. Finally, for small <a href="/wiki/Perturbation_theory" title="Perturbation theory">perturbations</a> in subsonic and <a href="/wiki/Supersonic" class="mw-redirect" title="Supersonic">supersonic</a> flows (not <a href="/wiki/Transonic" title="Transonic">transonic</a> or <a href="/wiki/Hypersonic" class="mw-redirect" title="Hypersonic">hypersonic</a>) these equations can be <a href="/wiki/Linearization" title="Linearization">linearized</a> to yield the linearized potential equations. </p><p>Historically, methods were first developed to solve the linearized potential equations. Two-dimensional (2D) methods, using <a href="/wiki/Conformal_transformation" class="mw-redirect" title="Conformal transformation">conformal transformations</a> of the flow about a <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a> to the flow about an <a href="/wiki/Airfoil" title="Airfoil">airfoil</a> were developed in the 1930s.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>One of the earliest type of calculations resembling modern CFD are those by <a href="/wiki/Lewis_Fry_Richardson" title="Lewis Fry Richardson">Lewis Fry Richardson</a>, in the sense that these calculations used finite differences and divided the physical space in cells. Although they failed dramatically, these calculations, together with Richardson's book <i>Weather Prediction by Numerical Process</i>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> set the basis for modern CFD and numerical meteorology. In fact, early CFD calculations during the 1940s using <a href="/wiki/ENIAC" title="ENIAC">ENIAC</a> used methods close to those in Richardson's 1922 book.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>The computer power available paced development of <a href="/wiki/Three-dimensional_space" title="Three-dimensional space">three-dimensional</a> methods. Probably the first work using computers to model fluid flow, as governed by the Navier–Stokes equations, was performed at <a href="/wiki/Los_Alamos_National_Lab" class="mw-redirect" title="Los Alamos National Lab">Los Alamos National Lab</a>, in the T3 group.<sup id="cite_ref-legacy_T3_5-0" class="reference"><a href="#cite_note-legacy_T3-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-harlow2004fluid_6-0" class="reference"><a href="#cite_note-harlow2004fluid-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> This group was led by <a href="/wiki/Francis_H._Harlow" title="Francis H. Harlow">Francis H. Harlow</a>, who is widely considered one of the pioneers of CFD. From 1957 to late 1960s, this group developed a variety of numerical methods to simulate transient two-dimensional fluid flows, such as <a href="/wiki/Particle-in-cell" title="Particle-in-cell">particle-in-cell</a> method,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <a href="/w/index.php?title=Fluid-in-cell&action=edit&redlink=1" class="new" title="Fluid-in-cell (page does not exist)">fluid-in-cell</a> method,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> <a href="/w/index.php?title=Vorticity_stream_function&action=edit&redlink=1" class="new" title="Vorticity stream function (page does not exist)">vorticity stream function</a> method,<sup id="cite_ref-Fromm1963_9-0" class="reference"><a href="#cite_note-Fromm1963-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Marker-and-cell_method" title="Marker-and-cell method">marker-and-cell method</a>.<sup id="cite_ref-harlow_welch_10-0" class="reference"><a href="#cite_note-harlow_welch-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Fromm's vorticity-stream-function method for 2D, transient, incompressible flow was the first treatment of strongly contorting incompressible flows in the world. </p><p>The first paper with three-dimensional model was published by John Hess and <a href="/wiki/A.M.O._Smith" class="mw-redirect" title="A.M.O. Smith">A.M.O. Smith</a> of <a href="/wiki/Douglas_Aircraft" class="mw-redirect" title="Douglas Aircraft">Douglas Aircraft</a> in 1967.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> This method discretized the surface of the geometry with panels, giving rise to this class of programs being called Panel Methods. Their method itself was simplified, in that it did not include lifting flows and hence was mainly applied to ship hulls and aircraft fuselages. The first lifting Panel Code (A230) was described in a paper written by Paul Rubbert and Gary Saaris of Boeing Aircraft in 1968.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> In time, more advanced three-dimensional Panel Codes were developed at <a href="/wiki/Boeing" title="Boeing">Boeing</a> (PANAIR, A502),<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Lockheed_Corporation" title="Lockheed Corporation">Lockheed</a> (Quadpan),<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Douglas_Aircraft_Company" title="Douglas Aircraft Company">Douglas</a> (HESS),<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <a href="/wiki/McDonnell_Aircraft" class="mw-redirect" title="McDonnell Aircraft">McDonnell Aircraft</a> (MACAERO),<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <a href="/wiki/NASA" title="NASA">NASA</a> (PMARC)<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> and Analytical Methods (WBAERO,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> USAERO<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> and VSAERO<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>). Some (PANAIR, HESS and MACAERO) were higher order codes, using higher order distributions of surface singularities, while others (Quadpan, PMARC, USAERO and VSAERO) used single singularities on each surface panel. The advantage of the lower order codes was that they ran much faster on the computers of the time. Today, VSAERO has grown to be a multi-order code and is the most widely used program of this class. It has been used in the development of many <a href="/wiki/Submarine" title="Submarine">submarines</a>, surface <a href="/wiki/Ship" title="Ship">ships</a>, <a href="/wiki/Automobile" class="mw-redirect" title="Automobile">automobiles</a>, <a href="/wiki/Helicopter" title="Helicopter">helicopters</a>, <a href="/wiki/Aircraft" title="Aircraft">aircraft</a>, and more recently <a href="/wiki/Wind_turbine" title="Wind turbine">wind turbines</a>. Its sister code, USAERO is an unsteady panel method that has also been used for modeling such things as high speed trains and racing <a href="/wiki/Yacht" title="Yacht">yachts</a>. The NASA PMARC code from an early version of VSAERO and a derivative of PMARC, named CMARC,<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> is also commercially available. </p><p>In the two-dimensional realm, a number of Panel Codes have been developed for airfoil analysis and design. The codes typically have a <a href="/wiki/Boundary_layer" title="Boundary layer">boundary layer</a> analysis included, so that viscous effects can be modeled. <a href="/w/index.php?title=Richard_Eppler&action=edit&redlink=1" class="new" title="Richard Eppler (page does not exist)">Richard Eppler</a><span class="noprint" style="font-size:85%; font-style: normal;"> [<a href="https://de.wikipedia.org/wiki/Richard_Eppler" class="extiw" title="de:Richard Eppler">de</a>]</span> developed the PROFILE code, partly with NASA funding, which became available in the early 1980s.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> This was soon followed by <a href="/wiki/Mark_Drela" title="Mark Drela">Mark Drela</a>'s <a href="/wiki/XFOIL" title="XFOIL">XFOIL</a> code.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Both PROFILE and XFOIL incorporate two-dimensional panel codes, with coupled boundary layer codes for airfoil analysis work. PROFILE uses a <a href="/wiki/Conformal_transformation" class="mw-redirect" title="Conformal transformation">conformal transformation</a> method for inverse airfoil design, while XFOIL has both a conformal transformation and an inverse panel method for airfoil design. </p><p>An intermediate step between Panel Codes and Full Potential codes were codes that used the Transonic Small Disturbance equations. In particular, the three-dimensional WIBCO code,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> developed by Charlie Boppe of <a href="/wiki/Grumman_Aircraft" class="mw-redirect" title="Grumman Aircraft">Grumman Aircraft</a> in the early 1980s has seen heavy use. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Starship_simul_3.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Starship_simul_3.png/220px-Starship_simul_3.png" decoding="async" width="220" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Starship_simul_3.png/330px-Starship_simul_3.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Starship_simul_3.png/440px-Starship_simul_3.png 2x" data-file-width="569" data-file-height="449" /></a><figcaption>A simulation of the <a href="/wiki/SpaceX_Starship" title="SpaceX Starship">SpaceX Starship</a> during re-entry</figcaption></figure> <p>Developers turned to Full Potential codes, as panel methods could not calculate the non-linear flow present at <a href="/wiki/Transonic" title="Transonic">transonic</a> speeds. The first description of a means of using the Full Potential equations was published by Earll Murman and <a href="/wiki/Julian_Cole" title="Julian Cole">Julian Cole</a> of Boeing in 1970.<sup id="cite_ref-Murman_Cole_1971_26-0" class="reference"><a href="#cite_note-Murman_Cole_1971-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Frances Bauer, <a href="/wiki/Paul_Garabedian" title="Paul Garabedian">Paul Garabedian</a> and <a href="/wiki/David_Korn_(computer_scientist)" title="David Korn (computer scientist)">David Korn</a> of the Courant Institute at <a href="/wiki/New_York_University" title="New York University">New York University</a> (NYU) wrote a series of two-dimensional Full Potential airfoil codes that were widely used, the most important being named Program H.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> A further growth of Program H was developed by Bob Melnik and his group at <a href="/wiki/Grumman_Aerospace" class="mw-redirect" title="Grumman Aerospace">Grumman Aerospace</a> as Grumfoil.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Antony_Jameson" title="Antony Jameson">Antony Jameson</a>, originally at Grumman Aircraft and the Courant Institute of NYU, worked with David Caughey to develop the important three-dimensional Full Potential code FLO22<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> in 1975. Many Full Potential codes emerged after this, culminating in Boeing's Tranair (A633) code,<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> which still sees heavy use. </p><p>The next step was the Euler equations, which promised to provide more accurate solutions of transonic flows. The methodology used by Jameson in his three-dimensional FLO57 code<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> (1981) was used by others to produce such programs as Lockheed's TEAM program<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> and IAI/Analytical Methods' MGAERO program.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> MGAERO is unique in being a structured <a href="/wiki/Cartesian_coordinate_system" title="Cartesian coordinate system">cartesian</a> mesh code, while most other such codes use structured body-fitted grids (with the exception of NASA's highly successful CART3D code,<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Lockheed's SPLITFLOW code<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Georgia_Institute_of_Technology" class="mw-redirect" title="Georgia Institute of Technology">Georgia Tech</a>'s NASCART-GT).<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Antony_Jameson" title="Antony Jameson">Antony Jameson</a> also developed the three-dimensional AIRPLANE code<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> which made use of unstructured tetrahedral grids. </p><p>In the two-dimensional realm, Mark Drela and Michael Giles, then graduate students at MIT, developed the ISES Euler program<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> (actually a suite of programs) for airfoil design and analysis. This code first became available in 1986 and has been further developed to design, analyze and optimize single or multi-element airfoils, as the MSES program.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> MSES sees wide use throughout the world. A derivative of MSES, for the design and analysis of airfoils in a cascade, is MISES,<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> developed by Harold Youngren while he was a graduate student at MIT. </p><p>The Navier–Stokes equations were the ultimate target of development. Two-dimensional codes, such as NASA Ames' ARC2D code first emerged. A number of three-dimensional codes were developed (ARC3D, <a href="/wiki/Overflow_(software)" title="Overflow (software)">OVERFLOW</a>, CFL3D are three successful NASA contributions), leading to numerous commercial packages. </p><p>Recently CFD methods have gained traction for modeling the flow behavior of granular materials within various chemical processes in engineering. This approach has emerged as a cost-effective alternative, offering a nuanced understanding of complex flow phenomena while minimizing expenses associated with traditional experimental methods.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Hierarchy_of_fluid_flow_equations">Hierarchy of fluid flow equations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=2" title="Edit section: Hierarchy of fluid flow equations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Computational_Fluid_Dynamics_for_Phase_Change_Materials" title="Computational Fluid Dynamics for Phase Change Materials">Computational Fluid Dynamics for Phase Change Materials</a></div> <p>CFD can be seen as a group of computational methodologies (discussed below) used to solve equations governing fluid flow. In the application of CFD, a critical step is to decide which set of physical assumptions and related equations need to be used for the problem at hand.<sup id="cite_ref-ferziger_43-0" class="reference"><a href="#cite_note-ferziger-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> To illustrate this step, the following summarizes the physical assumptions/simplifications taken in equations of a flow that is single-phase (see <a href="/wiki/Multiphase_flow" title="Multiphase flow">multiphase flow</a> and <a href="/wiki/Two-phase_flow" title="Two-phase flow">two-phase flow</a>), single-species (i.e., it consists of one chemical species), non-reacting, and (unless said otherwise) compressible. Thermal radiation is neglected, and body forces due to gravity are considered (unless said otherwise). In addition, for this type of flow, the next discussion highlights the hierarchy of flow equations solved with CFD. Note that some of the following equations could be derived in more than one way. </p> <ul><li><a href="/wiki/Conservation_laws" class="mw-redirect" title="Conservation laws">Conservation laws</a> (CL): These are the most fundamental equations considered with CFD in the sense that, for example, all the following equations can be derived from them. For a single-phase, single-species, compressible flow one considers the <a href="/wiki/Conservation_of_mass" title="Conservation of mass">conservation of mass</a>, <a href="/wiki/Conservation_of_linear_momentum" class="mw-redirect" title="Conservation of linear momentum">conservation of linear momentum</a>, and <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a>.</li> <li>Continuum conservation laws (CCL): Start with the CL. Assume that mass, momentum and energy are <i>locally</i> conserved: These quantities are conserved and cannot "teleport" from one place to another but can only move by a continuous flow (see <a href="/wiki/Continuity_equation" title="Continuity equation">continuity equation</a>). Another interpretation is that one starts with the CL and assumes a continuum medium (see <a href="/wiki/Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a>). The resulting system of equations is unclosed since to solve it one needs further relationships/equations: (a) constitutive relationships for the <a href="/wiki/Viscous_stress_tensor" title="Viscous stress tensor">viscous stress tensor</a>; (b) constitutive relationships for the diffusive <a href="/wiki/Heat_flux" title="Heat flux">heat flux</a>; (c) an <a href="/wiki/Equation_of_state" title="Equation of state">equation of state</a> (EOS), such as the <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a> law; and, (d) a caloric equation of state relating temperature with quantities such as <a href="/wiki/Enthalpy" title="Enthalpy">enthalpy</a> or <a href="/wiki/Internal_energy" title="Internal energy">internal energy</a>.</li> <li>Compressible <a href="/wiki/Navier-Stokes_equations" class="mw-redirect" title="Navier-Stokes equations">Navier-Stokes equations</a> (C-NS): Start with the CCL. Assume a Newtonian viscous stress tensor (see <a href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</a>) and a Fourier heat flux (see <a href="/wiki/Heat_flux" title="Heat flux">heat flux</a>).<sup id="cite_ref-cns_44-0" class="reference"><a href="#cite_note-cns-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-panton_45-0" class="reference"><a href="#cite_note-panton-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> The C-NS need to be augmented with an EOS and a caloric EOS to have a closed system of equations.</li> <li>Incompressible Navier-Stokes equations (I-NS): Start with the C-NS. Assume that density is always and everywhere constant.<sup id="cite_ref-landau_46-0" class="reference"><a href="#cite_note-landau-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Another way to obtain the I-NS is to assume that the <a href="/wiki/Mach_number" title="Mach number">Mach number</a> is very small<sup id="cite_ref-landau_46-1" class="reference"><a href="#cite_note-landau-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-panton_45-1" class="reference"><a href="#cite_note-panton-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> and that temperature differences in the fluid are very small as well.<sup id="cite_ref-panton_45-2" class="reference"><a href="#cite_note-panton-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> As a result, the mass-conservation and momentum-conservation equations are decoupled from the energy-conservation equation, so one only needs to solve for the first two equations.<sup id="cite_ref-panton_45-3" class="reference"><a href="#cite_note-panton-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup></li> <li>Compressible <a href="/wiki/Euler_equations_(fluid_dynamics)" title="Euler equations (fluid dynamics)">Euler equations</a> (EE): Start with the C-NS. Assume a frictionless flow with no diffusive heat flux.<sup id="cite_ref-fox_47-0" class="reference"><a href="#cite_note-fox-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup></li> <li>Weakly compressible Navier-Stokes equations (WC-NS): Start with the C-NS. Assume that density variations depend only on temperature and not on pressure.<sup id="cite_ref-poinsot_48-0" class="reference"><a href="#cite_note-poinsot-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> For example, for an <a href="/wiki/Ideal_gas" title="Ideal gas">ideal gas</a>, use <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 \rho =p_{0}/(RT)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =p_{0}/(RT)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ace8d187f2f7877c8f4f514f417cdd2ac3960a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.896ex; height:2.843ex;" alt="{\displaystyle \rho =p_{0}/(RT)}"></span>, 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 p_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b969ada68a88e2aeba9a2d2096abaf1fd53c21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:2.009ex;" alt="{\displaystyle p_{0}}"></span> is a conveniently-defined reference pressure that is always and everywhere constant, <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 \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> is density, <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" 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}"></span> is the specific <a href="/wiki/Gas_constant" title="Gas constant">gas constant</a>, 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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> is temperature. As a result, the WC-NS do not capture acoustic waves. It is also common in the WC-NS to neglect the pressure-work and viscous-heating terms in the energy-conservation equation. The WC-NS are also called the C-NS with the low-Mach-number approximation.</li> <li>Boussinesq equations: Start with the C-NS. Assume that density variations are always and everywhere negligible except in the gravity term of the momentum-conservation equation (where density multiplies the gravitational acceleration).<sup id="cite_ref-kundu_49-0" class="reference"><a href="#cite_note-kundu-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Also assume that various fluid properties such as <a href="/wiki/Viscosity" title="Viscosity">viscosity</a>, <a href="/wiki/Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">thermal conductivity</a>, and <a href="/wiki/Heat_capacity" title="Heat capacity">heat capacity</a> are always and everywhere constant. The Boussinesq equations are widely used in <a href="/wiki/Microscale_meteorology" title="Microscale meteorology">microscale meteorology</a>.</li> <li>Compressible <a href="/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equations" title="Reynolds-averaged Navier–Stokes equations">Reynolds-averaged Navier–Stokes equations</a> and compressible Favre-averaged Navier-Stokes equations (C-RANS and C-FANS): Start with the C-NS. Assume that any flow variable <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, such as density, velocity and pressure, can be represented as <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 f=F+f''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>F</mi> <mo>+</mo> <msup> <mi>f</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=F+f''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e00378aa9cae19d7d0b9a90ad81cf1105e7d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.416ex; height:2.843ex;" alt="{\displaystyle f=F+f''}"></span>, 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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is the ensemble-average<sup id="cite_ref-panton_45-4" class="reference"><a href="#cite_note-panton-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> of any flow variable, 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 f''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbdf186092f4353b7630fa8dda903e493cbbdc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.458ex; height:2.843ex;" alt="{\displaystyle f''}"></span> is a perturbation or fluctuation from this average.<sup id="cite_ref-panton_45-5" class="reference"><a href="#cite_note-panton-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-ras_50-0" class="reference"><a href="#cite_note-ras-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> <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 f''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbdf186092f4353b7630fa8dda903e493cbbdc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.458ex; height:2.843ex;" alt="{\displaystyle f''}"></span> is not necessarily small. If <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is a classic ensemble-average (see <a href="/wiki/Reynolds_decomposition" title="Reynolds decomposition">Reynolds decomposition</a>) one obtains the Reynolds-averaged Navier–Stokes equations. And if <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is a density-weighted ensemble-average one obtains the Favre-averaged Navier-Stokes equations.<sup id="cite_ref-ras_50-1" class="reference"><a href="#cite_note-ras-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> As a result, and depending on the Reynolds number, the range of scales of motion is greatly reduced, something which leads to much faster solutions in comparison to solving the C-NS. However, information is lost, and the resulting system of equations requires the closure of various unclosed terms, notably the <a href="/wiki/Reynolds_stress" title="Reynolds stress">Reynolds stress</a>.</li> <li>Ideal flow or <a href="/wiki/Potential_flow" title="Potential flow">potential flow</a> equations: Start with the EE. Assume zero fluid-particle rotation (zero vorticity) and zero flow expansion (zero divergence).<sup id="cite_ref-panton_45-6" class="reference"><a href="#cite_note-panton-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> The resulting flowfield is entirely determined by the geometrical boundaries.<sup id="cite_ref-panton_45-7" class="reference"><a href="#cite_note-panton-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> Ideal flows can be useful in modern CFD to initialize simulations.</li> <li>Linearized compressible Euler equations (LEE):<sup id="cite_ref-bailly_51-0" class="reference"><a href="#cite_note-bailly-51"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> Start with the EE. Assume that any flow variable <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>, such as density, velocity and pressure, can be represented as <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 f=f_{0}+f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=f_{0}+f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c8185b41196b3363a1ab663fdbfa265e162d93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.416ex; height:2.843ex;" alt="{\displaystyle f=f_{0}+f'}"></span>, 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 f_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6423b30a4c5770c59b5ab92dcb4ce378755440ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{0}}"></span> is the value of the flow variable at some reference or base state, 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 f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span> is a perturbation or fluctuation from this state. Furthermore, assume that this perturbation <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 f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span> is very small in comparison with some reference value. Finally, assume that <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 f_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6423b30a4c5770c59b5ab92dcb4ce378755440ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{0}}"></span> satisfies "its own" equation, such as the EE. The LEE and its many variations are widely used in <a href="/wiki/Computational_aeroacoustics" title="Computational aeroacoustics">computational aeroacoustics</a>.</li> <li>Sound wave or <a href="/wiki/Acoustic_wave_equation" title="Acoustic wave equation">acoustic wave equation</a>: Start with the LEE. Neglect all gradients of <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 f_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6423b30a4c5770c59b5ab92dcb4ce378755440ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{0}}"></span> 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 f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span>, and assume that the Mach number at the reference or base state is very small.<sup id="cite_ref-poinsot_48-1" class="reference"><a href="#cite_note-poinsot-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> The resulting equations for density, momentum and energy can be manipulated into a pressure equation, giving the well-known sound wave equation.</li> <li><a href="/wiki/Shallow_water_equations" title="Shallow water equations">Shallow water equations</a> (SW): Consider a flow near a wall where the wall-parallel length-scale of interest is much larger than the wall-normal length-scale of interest. Start with the EE. Assume that density is always and everywhere constant, neglect the velocity component perpendicular to the wall, and consider the velocity parallel to the wall to be spatially-constant.</li> <li><a href="/wiki/Boundary_layer" title="Boundary layer">Boundary layer</a> equations (BL): Start with the C-NS (I-NS) for compressible (incompressible) boundary layers. Assume that there are thin regions next to walls where spatial gradients perpendicular to the wall are much larger than those parallel to the wall.<sup id="cite_ref-kundu_49-1" class="reference"><a href="#cite_note-kundu-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup></li> <li>Bernoulli equation: Start with the EE. Assume that density variations depend only on pressure variations.<sup id="cite_ref-kundu_49-2" class="reference"><a href="#cite_note-kundu-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> See <a href="/wiki/Bernoulli%27s_Principle" class="mw-redirect" title="Bernoulli's Principle">Bernoulli's Principle</a>.</li> <li>Steady Bernoulli equation: Start with the Bernoulli Equation and assume a steady flow.<sup id="cite_ref-kundu_49-3" class="reference"><a href="#cite_note-kundu-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Or start with the EE and assume that the flow is steady and integrate the resulting equation along a streamline.<sup id="cite_ref-fox_47-1" class="reference"><a href="#cite_note-fox-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-landau_46-2" class="reference"><a href="#cite_note-landau-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Stokes_Flow" class="mw-redirect" title="Stokes Flow">Stokes Flow</a> or creeping flow equations: Start with the C-NS or I-NS. Neglect the inertia of the flow.<sup id="cite_ref-panton_45-8" class="reference"><a href="#cite_note-panton-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-landau_46-3" class="reference"><a href="#cite_note-landau-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> Such an assumption can be justified when the <a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> is very low. As a result, the resulting set of equations is linear, something which simplifies greatly their solution.</li> <li>Two-dimensional channel flow equation: Consider the flow between two infinite parallel plates. Start with the C-NS. Assume that the flow is steady, two-dimensional, and fully developed (i.e., the velocity profile does not change along the streamwise direction).<sup id="cite_ref-panton_45-9" class="reference"><a href="#cite_note-panton-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> Note that this widely-used fully-developed assumption can be inadequate in some instances, such as some compressible, microchannel flows, in which case it can be supplanted by a <i>locally</i> fully-developed assumption.<sup id="cite_ref-harley_52-0" class="reference"><a href="#cite_note-harley-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup></li> <li>One-dimensional Euler equations or one-dimensional gas-dynamic equations (1D-EE): Start with the EE. Assume that all flow quantities depend only on one spatial dimension.<sup id="cite_ref-1d-ee_53-0" class="reference"><a href="#cite_note-1d-ee-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Fanno_flow" title="Fanno flow">Fanno flow</a> equation: Consider the flow inside a duct with constant area and adiabatic walls. Start with the 1D-EE. Assume a steady flow, no gravity effects, and introduce in the momentum-conservation equation an empirical term to recover the effect of wall friction (neglected in the EE). To close the Fanno flow equation, a model for this friction term is needed. Such a closure involves problem-dependent assumptions.<sup id="cite_ref-cavazzuti_54-0" class="reference"><a href="#cite_note-cavazzuti-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Rayleigh_flow" title="Rayleigh flow">Rayleigh flow</a> equation. Consider the flow inside a duct with constant area and either non-adiabatic walls without volumetric heat sources or adiabatic walls with volumetric heat sources. Start with the 1D-EE. Assume a steady flow, no gravity effects, and introduce in the energy-conservation equation an empirical term to recover the effect of wall heat transfer or the effect of the heat sources (neglected in the EE).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Methodology">Methodology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=3" title="Edit section: Methodology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In all of these approaches the same basic procedure is followed. </p> <ul><li>During <a href="/wiki/Preprocessor_(CAE)" title="Preprocessor (CAE)">preprocessing</a> <ul><li>The <a href="/wiki/Geometry" title="Geometry">geometry</a> and physical bounds of the problem can be defined using <a href="/wiki/Computer-aided_design" title="Computer-aided design">computer aided design</a> (CAD). From there, data can be suitably processed (cleaned-up) and the fluid volume (or fluid domain) is extracted.</li> <li>The <a href="/wiki/Volume" title="Volume">volume</a> occupied by the fluid is divided into discrete cells (the mesh). The mesh may be uniform or non-uniform, structured or unstructured, consisting of a combination of hexahedral, tetrahedral, prismatic, pyramidal or polyhedral elements.</li> <li>The physical modeling is defined – for example, the equations of fluid motion + <a href="/wiki/Enthalpy" title="Enthalpy">enthalpy</a> + radiation + species conservation</li> <li>Boundary conditions are defined. This involves specifying the fluid behaviour and properties at all bounding surfaces of the fluid domain. For transient problems, the initial conditions are also defined.</li></ul></li> <li>The <a href="/wiki/Computer_simulation" title="Computer simulation">simulation</a> is started and the equations are solved iteratively as a steady-state or transient.</li> <li>Finally a postprocessor is used for the analysis and visualization of the resulting solution.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Discretization_methods">Discretization methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=4" title="Edit section: Discretization methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Discretization_of_Navier%E2%80%93Stokes_equations" title="Discretization of Navier–Stokes equations">Discretization of Navier–Stokes equations</a></div> <p>The stability of the selected discretisation is generally established numerically rather than analytically as with simple linear problems. Special care must also be taken to ensure that the discretisation handles discontinuous solutions gracefully. The <a href="/wiki/Euler_equations_(fluid_dynamics)" title="Euler equations (fluid dynamics)">Euler equations</a> and <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a> both admit shocks and contact surfaces. </p><p>Some of the discretization methods being used are: </p> <div class="mw-heading mw-heading4"><h4 id="Finite_volume_method">Finite volume method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=5" title="Edit section: Finite volume method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite_volume_method" title="Finite volume method">Finite volume method</a></div> <p>The finite volume method (FVM) is a common approach used in CFD codes, as it has an advantage in <a href="/wiki/Random-access_memory" title="Random-access memory">memory</a> usage and solution speed, especially for large problems, high <a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> turbulent flows, and source term dominated flows (like combustion).<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> </p><p>In the finite volume method, the governing partial differential equations (typically the Navier-Stokes equations, the mass and energy conservation equations, and the turbulence equations) are recast in a conservative form, and then solved over discrete control volumes. This <a href="/wiki/Discretization" title="Discretization">discretization</a> guarantees the conservation of fluxes through a particular control volume. The finite volume equation yields governing equations in the form, </p> <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 {\frac {\partial }{\partial t}}\iiint Q\,dV+\iint F\,d\mathbf {A} =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>∭</mo> <mi>Q</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>V</mi> <mo>+</mo> <mo>∬</mo> <mi>F</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t}}\iiint Q\,dV+\iint F\,d\mathbf {A} =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87ae5e6a7210df995e1ea06d0c20ddca0384c740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.515ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial }{\partial t}}\iiint Q\,dV+\iint F\,d\mathbf {A} =0,}"></span></dd></dl> <p>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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is the vector of conserved variables, <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is the vector of fluxes (see <a href="/wiki/Euler_equations_(fluid_dynamics)" title="Euler equations (fluid dynamics)">Euler equations</a> or <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a>), <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> is the volume of the control volume element, 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 \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0795cc96c75d81520a120482662b90f024c9a1a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {A} }"></span> is the surface area of the control volume element. </p> <div class="mw-heading mw-heading4"><h4 id="Finite_element_method">Finite element method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=6" title="Edit section: Finite element method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></div> <p>The finite element method (FEM) is used in structural analysis of solids, but is also applicable to fluids. However, the FEM formulation requires special care to ensure a conservative solution. The FEM formulation has been adapted for use with fluid dynamics governing equations.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_57-0" class="reference"><a href="#cite_note-:0-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> Although FEM must be carefully formulated to be conservative, it is much more stable than the finite volume approach.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> FEM also provides more accurate solutions for smooth problems comparing to FVM. <sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> Another advantage of FEM is that it can handle complex geometries and boundary conditions. However, FEM can require more memory and has slower solution times than the FVM.<sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> </p><p>In this method, a weighted residual equation is formed: </p> <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 R_{i}=\iiint W_{i}Q\,dV^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>∭</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>Q</mi> <mspace width="thinmathspace" /> <mi>d</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i}=\iiint W_{i}Q\,dV^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f00d9e41717eef2a3dd249037afe27ffcd0cec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.999ex; height:5.676ex;" alt="{\displaystyle R_{i}=\iiint W_{i}Q\,dV^{e}}"></span></dd></dl> <p>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 R_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db421291be9d0103404ced7495b363437b67b6b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.509ex;" alt="{\displaystyle R_{i}}"></span> is the equation residual at an element vertex <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 i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>, <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is the conservation equation expressed on an element basis, <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 W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span> is the weight factor, 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 V^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{e}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c412a6ac22071e6689a5c7484277e156387eb93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.915ex; height:2.343ex;" alt="{\displaystyle V^{e}}"></span> is the volume of the element. </p> <div class="mw-heading mw-heading4"><h4 id="Finite_difference_method">Finite difference method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=7" title="Edit section: Finite difference method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite_difference_method" title="Finite difference method">Finite difference method</a></div> <p>The finite difference method (FDM) has historical importance<sup id="cite_ref-:0_57-1" class="reference"><a href="#cite_note-:0-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> and is simple to program. It is currently only used in few specialized codes, which handle complex geometry with high accuracy and efficiency by using embedded boundaries or overlapping grids (with the solution interpolated across each grid).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2010)">citation needed</span></a></i>]</sup> </p> <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 {\frac {\partial Q}{\partial t}}+{\frac {\partial F}{\partial x}}+{\frac {\partial G}{\partial y}}+{\frac {\partial H}{\partial z}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>Q</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>F</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>G</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial Q}{\partial t}}+{\frac {\partial F}{\partial x}}+{\frac {\partial G}{\partial y}}+{\frac {\partial H}{\partial z}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d187e297ce84052a761e865bc8f613c236b0b812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.868ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial Q}{\partial t}}+{\frac {\partial F}{\partial x}}+{\frac {\partial G}{\partial y}}+{\frac {\partial H}{\partial z}}=0}"></span></dd></dl> <p>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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is the vector of conserved variables, 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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>, <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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, 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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> are the fluxes 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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}"></span>, <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" 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}"></span>, 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 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 skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> directions respectively. </p> <div class="mw-heading mw-heading4"><h4 id="Spectral_element_method">Spectral element method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=8" title="Edit section: Spectral element method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Spectral_element_method" title="Spectral element method">Spectral element method</a></div> <p>Spectral element method is a finite element type method. It requires the mathematical problem (the partial differential equation) to be cast in a weak formulation. This is typically done by multiplying the differential equation by an arbitrary test function and integrating over the whole domain. Purely mathematically, the test functions are completely arbitrary - they belong to an infinite-dimensional function space. Clearly an infinite-dimensional function space cannot be represented on a discrete spectral element mesh; this is where the spectral element discretization begins. The most crucial thing is the choice of interpolating and testing functions. In a standard, low order FEM in 2D, for quadrilateral elements the most typical choice is the bilinear test or interpolating function of the form <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 v(x,y)=ax+by+cxy+d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>+</mo> <mi>c</mi> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(x,y)=ax+by+cxy+d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f660c7f445d27145350120e7c48d18450cc4dd98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.496ex; height:2.843ex;" alt="{\displaystyle v(x,y)=ax+by+cxy+d}"></span>. In a spectral element method however, the interpolating and test functions are chosen to be polynomials of a very high order (typically e.g. of the 10th order in CFD applications). This guarantees the rapid convergence of the method. Furthermore, very efficient integration procedures must be used, since the number of integrations to be performed in numerical codes is big. Thus, high order Gauss integration quadratures are employed, since they achieve the highest accuracy with the smallest number of computations to be carried out. At the time there are some academic CFD codes based on the spectral element method and some more are currently under development, since the new time-stepping schemes arise in the scientific world. </p> <div class="mw-heading mw-heading4"><h4 id="Lattice_Boltzmann_method">Lattice Boltzmann method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=9" title="Edit section: Lattice Boltzmann method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lattice_Boltzmann_methods" title="Lattice Boltzmann methods">Lattice Boltzmann methods</a></div> <p>The lattice Boltzmann method (LBM) with its simplified kinetic picture on a lattice provides a computationally efficient description of hydrodynamics. Unlike the traditional CFD methods, which solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. In this method, one works with the discrete in space and time version of the kinetic evolution equation in the Boltzmann <a href="/wiki/Bhatnagar%E2%80%93Gross%E2%80%93Krook_operator" title="Bhatnagar–Gross–Krook operator">Bhatnagar-Gross-Krook (BGK)</a> form. </p> <div class="mw-heading mw-heading4"><h4 id="Vortex_method">Vortex method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=10" title="Edit section: Vortex method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The vortex method, also Lagrangian Vortex Particle Method, is a <a href="/wiki/Meshfree_methods" title="Meshfree methods">meshfree</a> technique for the simulation of incompressible turbulent flows. In it, <a href="/wiki/Vorticity" title="Vorticity">vorticity</a> is discretized onto <a href="/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field" title="Lagrangian and Eulerian specification of the flow field">Lagrangian</a> particles, these computational elements being called vortices, vortons, or vortex particles.<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> Vortex methods were developed as a grid-free methodology that would not be limited by the fundamental smoothing effects associated with grid-based methods. To be practical, however, vortex methods require means for rapidly computing velocities from the vortex elements – in other words they require the solution to a particular form of the <a href="/wiki/N-body_problem" title="N-body problem">N-body problem</a> (in which the motion of N objects is tied to their mutual influences). This breakthrough came in the 1980s with the development of the <a href="/wiki/Barnes%E2%80%93Hut_simulation" title="Barnes–Hut simulation">Barnes-Hut</a> and <a href="/wiki/Fast_multipole_method" title="Fast multipole method">fast multipole method</a> (FMM) algorithms. These paved the way to practical computation of the velocities from the vortex elements. </p><p>Software based on the vortex method offer a new means for solving tough fluid dynamics problems with minimal user intervention.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2010)">citation needed</span></a></i>]</sup> All that is required is specification of problem geometry and setting of boundary and initial conditions. Among the significant advantages of this modern technology; </p> <ul><li>It is practically grid-free, thus eliminating numerous iterations associated with RANS and LES.</li> <li>All problems are treated identically. No modeling or calibration inputs are required.</li> <li>Time-series simulations, which are crucial for correct analysis of acoustics, are possible.</li> <li>The small scale and large scale are accurately simulated at the same time.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Boundary_element_method">Boundary element method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=11" title="Edit section: Boundary element method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Boundary_element_method" title="Boundary element method">Boundary element method</a></div> <p>In the boundary element method, the boundary occupied by the fluid is divided into a surface mesh. </p> <div class="mw-heading mw-heading4"><h4 id="High-resolution_discretization_schemes">High-resolution discretization schemes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=12" title="Edit section: High-resolution discretization schemes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/High-resolution_scheme" title="High-resolution scheme">High-resolution scheme</a></div> <p>High-resolution schemes are used where shocks or discontinuities are present. Capturing sharp changes in the solution requires the use of second or higher-order numerical schemes that do not introduce spurious oscillations. This usually necessitates the application of <a href="/wiki/Flux_limiters" class="mw-redirect" title="Flux limiters">flux limiters</a> to ensure that the solution is <a href="/wiki/Total_variation_diminishing" title="Total variation diminishing">total variation diminishing</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2010)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Turbulence_models">Turbulence models</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=13" title="Edit section: Turbulence models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In computational modeling of turbulent flows, one common objective is to obtain a model that can predict quantities of interest, such as fluid velocity, for use in engineering designs of the system being modeled. For turbulent flows, the range of length scales and complexity of phenomena involved in turbulence make most modeling approaches prohibitively expensive; the resolution required to resolve all scales involved in turbulence is beyond what is computationally possible. The primary approach in such cases is to create numerical models to approximate unresolved phenomena. This section lists some commonly used computational models for turbulent flows. </p><p>Turbulence models can be classified based on computational expense, which corresponds to the range of scales that are modeled versus resolved (the more turbulent scales that are resolved, the finer the resolution of the simulation, and therefore the higher the computational cost). If a majority or all of the turbulent scales are not modeled, the computational cost is very low, but the tradeoff comes in the form of decreased accuracy. </p><p>In addition to the wide range of length and time scales and the associated computational cost, the governing equations of fluid dynamics contain a <a href="/wiki/Nonlinear_system" title="Nonlinear system">non-linear</a> convection term and a non-linear and non-local pressure gradient term. These nonlinear equations must be solved numerically with the appropriate boundary and initial conditions. </p> <div class="mw-heading mw-heading4"><h4 id="Reynolds-averaged_Navier–Stokes"><span id="Reynolds-averaged_Navier.E2.80.93Stokes"></span>Reynolds-averaged Navier–Stokes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=14" title="Edit section: Reynolds-averaged Navier–Stokes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equations" title="Reynolds-averaged Navier–Stokes equations">Reynolds-averaged Navier–Stokes equations</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:DrivAer_SST-URANS-DDES_Comparison.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/DrivAer_SST-URANS-DDES_Comparison.png/258px-DrivAer_SST-URANS-DDES_Comparison.png" decoding="async" width="258" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/DrivAer_SST-URANS-DDES_Comparison.png/387px-DrivAer_SST-URANS-DDES_Comparison.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/DrivAer_SST-URANS-DDES_Comparison.png/516px-DrivAer_SST-URANS-DDES_Comparison.png 2x" data-file-width="1440" data-file-height="1160" /></a><figcaption>External aerodynamics of the <a rel="nofollow" class="external text" href="https://www.mw.tum.de/en/aer/research-groups/automotive/drivaer/">DrivAer</a> model, computed using <a href="/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equations" title="Reynolds-averaged Navier–Stokes equations">URANS</a> (top) and <a href="/wiki/Detached_eddy_simulation" title="Detached eddy simulation">DDES</a> (bottom)</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Verus_Engineering_Porsche_987.2_Ventus_2_Package.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Verus_Engineering_Porsche_987.2_Ventus_2_Package.png/220px-Verus_Engineering_Porsche_987.2_Ventus_2_Package.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Verus_Engineering_Porsche_987.2_Ventus_2_Package.png/330px-Verus_Engineering_Porsche_987.2_Ventus_2_Package.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Verus_Engineering_Porsche_987.2_Ventus_2_Package.png/440px-Verus_Engineering_Porsche_987.2_Ventus_2_Package.png 2x" data-file-width="6000" data-file-height="3376" /></a><figcaption>A simulation of aerodynamic package of a <a href="/wiki/Porsche_Cayman" class="mw-redirect" title="Porsche Cayman">Porsche Cayman (987.2)</a></figcaption></figure> <p><a href="/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equations" title="Reynolds-averaged Navier–Stokes equations">Reynolds-averaged Navier–Stokes</a> (RANS) equations are the oldest approach to turbulence modeling. An ensemble version of the governing equations is solved, which introduces new <i>apparent stresses</i> known as <a href="/wiki/Reynolds_stresses" class="mw-redirect" title="Reynolds stresses">Reynolds stresses</a>. This adds a second-order tensor of unknowns for which various models can provide different levels of closure. It is a common misconception that the RANS equations do not apply to flows with a time-varying mean flow because these equations are 'time-averaged'. In fact, statistically unsteady (or non-stationary) flows can equally be treated. This is sometimes referred to as URANS. There is nothing inherent in Reynolds averaging to preclude this, but the turbulence models used to close the equations are valid only as long as the time over which these changes in the mean occur is large compared to the time scales of the turbulent motion containing most of the energy. </p><p>RANS models can be divided into two broad approaches: </p> <dl><dt><a href="/wiki/Turbulence_modeling" title="Turbulence modeling">Boussinesq hypothesis</a></dt> <dd>This method involves using an algebraic equation for the Reynolds stresses which include determining the turbulent viscosity, and depending on the level of sophistication of the model, solving transport equations for determining the turbulent kinetic energy and dissipation. Models include k-ε (<a href="/wiki/Brian_Launder" title="Brian Launder">Launder</a> and <a href="/wiki/Brian_Spalding" title="Brian Spalding">Spalding</a>),<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> Mixing Length Model (<a href="/wiki/Ludwig_Prandtl" title="Ludwig Prandtl">Prandtl</a>),<sup id="cite_ref-wilcox_63-0" class="reference"><a href="#cite_note-wilcox-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> and Zero Equation Model (Cebeci and <a href="/wiki/Apollo_M._O._Smith" title="Apollo M. O. Smith">Smith</a>).<sup id="cite_ref-wilcox_63-1" class="reference"><a href="#cite_note-wilcox-63"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> The models available in this approach are often referred to by the number of transport equations associated with the method. For example, the Mixing Length model is a "Zero Equation" model because no transport equations are solved; 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 k-\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>−</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k-\epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68e8e5faeda241f70e2e7ff63e811a6f7a434e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.996ex; height:2.343ex;" alt="{\displaystyle k-\epsilon }"></span> is a "Two Equation" model because two transport equations (one for <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> and one for <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 \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" 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 }"></span>) are solved.</dd> <dt><a href="/w/index.php?title=Reynolds_stress_model&action=edit&redlink=1" class="new" title="Reynolds stress model (page does not exist)">Reynolds stress model</a> (RSM)</dt> <dd>This approach attempts to actually solve transport equations for the Reynolds stresses. This means introduction of several transport equations for all the Reynolds stresses and hence this approach is much more costly in CPU effort.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2010)">citation needed</span></a></i>]</sup></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Large_eddy_simulation">Large eddy simulation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=15" title="Edit section: Large eddy simulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Large_eddy_simulation" title="Large eddy simulation">Large eddy simulation</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:LESPremixedFlame.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/LESPremixedFlame.jpg/250px-LESPremixedFlame.jpg" decoding="async" width="250" height="89" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/LESPremixedFlame.jpg/375px-LESPremixedFlame.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/LESPremixedFlame.jpg/500px-LESPremixedFlame.jpg 2x" data-file-width="776" data-file-height="275" /></a><figcaption>Volume rendering of a non-premixed swirl flame as simulated by LES</figcaption></figure> <p><a href="/wiki/Large_eddy_simulation" title="Large eddy simulation">Large eddy simulation</a> (LES) is a technique in which the smallest scales of the flow are removed through a filtering operation, and their effect modeled using subgrid scale models. This allows the largest and most important scales of the turbulence to be resolved, while greatly reducing the computational cost incurred by the smallest scales. This method requires greater computational resources than RANS methods, but is far cheaper than DNS. </p> <div class="mw-heading mw-heading4"><h4 id="Detached_eddy_simulation">Detached eddy simulation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=16" title="Edit section: Detached eddy simulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Detached_eddy_simulation" title="Detached eddy simulation">Detached eddy simulation</a></div> <p><a href="/wiki/Detached_eddy_simulation" title="Detached eddy simulation">Detached eddy simulations</a> (DES) is a modification of a RANS model in which the model switches to a subgrid scale formulation in regions fine enough for LES calculations. Regions near solid boundaries and where the turbulent length scale is less than the maximum grid dimension are assigned the RANS mode of solution. As the turbulent length scale exceeds the grid dimension, the regions are solved using the LES mode. Therefore, the grid resolution for DES is not as demanding as pure LES, thereby considerably cutting down the cost of the computation. Though DES was initially formulated for the Spalart-Allmaras model (Philippe R. Spalart et al., 1997), it can be implemented with other RANS models (Strelets, 2001), by appropriately modifying the length scale which is explicitly or implicitly involved in the RANS model. So while Spalart–Allmaras model based DES acts as LES with a wall model, DES based on other models (like two equation models) behave as a hybrid RANS-LES model. Grid generation is more complicated than for a simple RANS or LES case due to the RANS-LES switch. DES is a non-zonal approach and provides a single smooth velocity field across the RANS and the LES regions of the solutions. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cp_IDDES.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Cp_IDDES.gif/220px-Cp_IDDES.gif" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Cp_IDDES.gif/330px-Cp_IDDES.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Cp_IDDES.gif/440px-Cp_IDDES.gif 2x" data-file-width="768" data-file-height="432" /></a><figcaption>IDDES Simulation of the Karel Motorsports BMW. This is a type of DES simulation completed in OpenFOAM. The plot is coefficient of pressure.</figcaption></figure> <div class="mw-heading mw-heading4"><h4 id="Direct_numerical_simulation">Direct numerical simulation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=17" title="Edit section: Direct numerical simulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Direct_numerical_simulation" title="Direct numerical simulation">Direct numerical simulation</a></div> <p><a href="/wiki/Direct_numerical_simulation" title="Direct numerical simulation">Direct numerical simulation</a> (DNS) resolves the entire range of turbulent length scales. This marginalizes the effect of models, but is extremely expensive. The computational cost is proportional to <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 Re^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Re^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7880b611fbf3453026a7310d6e6615975c5bee20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.902ex; height:2.676ex;" alt="{\displaystyle Re^{3}}"></span>.<sup id="cite_ref-Pope_2000_64-0" class="reference"><a href="#cite_note-Pope_2000-64"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> DNS is intractable for flows with complex geometries or flow configurations. </p> <div class="mw-heading mw-heading4"><h4 id="Coherent_vortex_simulation">Coherent vortex simulation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=18" title="Edit section: Coherent vortex simulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The coherent vortex simulation approach decomposes the turbulent flow field into a coherent part, consisting of organized vortical motion, and the incoherent part, which is the random background flow.<sup id="cite_ref-Farge_2001_65-0" class="reference"><a href="#cite_note-Farge_2001-65"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> This decomposition is done using <a href="/wiki/Wavelet" title="Wavelet">wavelet</a> filtering. The approach has much in common with LES, since it uses decomposition and resolves only the filtered portion, but different in that it does not use a linear, low-pass filter. Instead, the filtering operation is based on wavelets, and the filter can be adapted as the flow field evolves. <a href="/wiki/Marie_Farge" title="Marie Farge">Farge</a> and Schneider tested the CVS method with two flow configurations and showed that the coherent portion of the flow exhibited 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 -{\frac {40}{39}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>40</mn> <mn>39</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {40}{39}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/000610bab6f48b2bcf51214a0c089f55cb473ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.969ex; height:5.176ex;" alt="{\displaystyle -{\frac {40}{39}}}"></span> energy spectrum exhibited by the total flow, and corresponded to coherent structures (<a href="/wiki/Vortex_stretching" title="Vortex stretching">vortex tubes</a>), while the incoherent parts of the flow composed homogeneous background noise, which exhibited no organized structures. Goldstein and Vasilyev<sup id="cite_ref-Goldstein_2004_66-0" class="reference"><a href="#cite_note-Goldstein_2004-66"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> applied the FDV model to large eddy simulation, but did not assume that the wavelet filter eliminated all coherent motions from the subfilter scales. By employing both LES and CVS filtering, they showed that the SFS dissipation was dominated by the SFS flow field's coherent portion. </p> <div class="mw-heading mw-heading4"><h4 id="PDF_methods">PDF methods</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=19" title="Edit section: PDF methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Probability_density_function" title="Probability density function">Probability density function</a> (PDF) methods for turbulence, first introduced by <a href="/wiki/Thomas_S._Lundgren" title="Thomas S. Lundgren">Lundgren</a>,<sup id="cite_ref-Lundgren_1969_67-0" class="reference"><a href="#cite_note-Lundgren_1969-67"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> are based on tracking the one-point PDF of the velocity, <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 f_{V}({\boldsymbol {v}};{\boldsymbol {x}},t)d{\boldsymbol {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{V}({\boldsymbol {v}};{\boldsymbol {x}},t)d{\boldsymbol {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b27efff218e2a4d0e46ac8650357a010609a5b80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.736ex; height:2.843ex;" alt="{\displaystyle f_{V}({\boldsymbol {v}};{\boldsymbol {x}},t)d{\boldsymbol {v}}}"></span>, which gives the probability of the velocity at 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 {\boldsymbol {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606b7680d510560a505937143775ea80fa958051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.532ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {x}}}"></span> being between <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 {\boldsymbol {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b2c2d3aac4213f3996d828c6aa8f4eb464a05cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.318ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {v}}}"></span> 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 {\boldsymbol {v}}+d{\boldsymbol {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mo>+</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {v}}+d{\boldsymbol {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d2e4def9cac2ac456b575c330f34d75a97e7df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.692ex; height:2.343ex;" alt="{\displaystyle {\boldsymbol {v}}+d{\boldsymbol {v}}}"></span>. This approach is analogous to the <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory of gases</a>, in which the macroscopic properties of a gas are described by a large number of particles. PDF methods are unique in that they can be applied in the framework of a number of different turbulence models; the main differences occur in the form of the PDF transport equation. For example, in the context of <a href="/wiki/Large_eddy_simulation" title="Large eddy simulation">large eddy simulation</a>, the PDF becomes the filtered PDF.<sup id="cite_ref-Colucci_1998_68-0" class="reference"><a href="#cite_note-Colucci_1998-68"><span class="cite-bracket">[</span>68<span class="cite-bracket">]</span></a></sup> PDF methods can also be used to describe chemical reactions,<sup id="cite_ref-Fox_2003_69-0" class="reference"><a href="#cite_note-Fox_2003-69"><span class="cite-bracket">[</span>69<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Pope_1985_70-0" class="reference"><a href="#cite_note-Pope_1985-70"><span class="cite-bracket">[</span>70<span class="cite-bracket">]</span></a></sup> and are particularly useful for simulating chemically reacting flows because the chemical source term is closed and does not require a model. The PDF is commonly tracked by using Lagrangian particle methods; when combined with large eddy simulation, this leads to a <a href="/wiki/Langevin_equation" title="Langevin equation">Langevin equation</a> for subfilter particle evolution. </p> <div class="mw-heading mw-heading4"><h4 id="Vorticity_confinement_method">Vorticity confinement method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=20" title="Edit section: Vorticity confinement method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Vorticity_confinement" title="Vorticity confinement">Vorticity confinement</a></div> <p>The <a href="/wiki/Vorticity_confinement" title="Vorticity confinement">vorticity confinement</a> (VC) method is an Eulerian technique used in the simulation of turbulent wakes. It uses a solitary-wave like approach to produce a stable solution with no numerical spreading. VC can capture the small-scale features to within as few as 2 grid cells. Within these features, a nonlinear difference equation is solved as opposed to the <a href="/wiki/Finite_difference_equation" class="mw-redirect" title="Finite difference equation">finite difference equation</a>. VC is similar to <a href="/wiki/Shock_capturing_methods" class="mw-redirect" title="Shock capturing methods">shock capturing methods</a>, where conservation laws are satisfied, so that the essential integral quantities are accurately computed. </p> <div class="mw-heading mw-heading4"><h4 id="Linear_eddy_model">Linear eddy model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=21" title="Edit section: Linear eddy model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Linear eddy model is a technique used to simulate the convective mixing that takes place in turbulent flow.<sup id="cite_ref-71" class="reference"><a href="#cite_note-71"><span class="cite-bracket">[</span>71<span class="cite-bracket">]</span></a></sup> Specifically, it provides a mathematical way to describe the interactions of a scalar variable within the vector flow field. It is primarily used in one-dimensional representations of turbulent flow, since it can be applied across a wide range of length scales and Reynolds numbers. This model is generally used as a building block for more complicated flow representations, as it provides high resolution predictions that hold across a large range of flow conditions. </p> <div class="mw-heading mw-heading3"><h3 id="Two-phase_flow">Two-phase flow</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=22" title="Edit section: Two-phase flow"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Bubble-rising.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Bubble-rising.jpg/220px-Bubble-rising.jpg" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Bubble-rising.jpg/330px-Bubble-rising.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Bubble-rising.jpg/440px-Bubble-rising.jpg 2x" data-file-width="1920" data-file-height="1080" /></a><figcaption>Simulation of bubble horde using <a href="/wiki/Volume_of_fluid_method" title="Volume of fluid method">volume of fluid method</a></figcaption></figure> <p>The modeling of <a href="/wiki/Two-phase_flow" title="Two-phase flow">two-phase flow</a> is still under development. Different methods have been proposed, including the <a href="/wiki/Volume_of_fluid_method" title="Volume of fluid method">Volume of fluid method</a>, the <a href="/wiki/Level-set_method" title="Level-set method">level-set method</a> and <a href="/w/index.php?title=Front_tracking&action=edit&redlink=1" class="new" title="Front tracking (page does not exist)">front tracking</a>.<sup id="cite_ref-72" class="reference"><a href="#cite_note-72"><span class="cite-bracket">[</span>72<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-73" class="reference"><a href="#cite_note-73"><span class="cite-bracket">[</span>73<span class="cite-bracket">]</span></a></sup> These methods often involve a tradeoff between maintaining a sharp interface or conserving mass <sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag may use weasel words or too-vague attribution. (November 2010)">according to whom?</span></a></i>]</sup>. This is crucial since the evaluation of the density, viscosity and surface tension is based on the values averaged over the interface.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2010)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Solution_algorithms">Solution algorithms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=23" title="Edit section: Solution algorithms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Discretization in the space produces a system of <a href="/wiki/Ordinary_differential_equations" class="mw-redirect" title="Ordinary differential equations">ordinary differential equations</a> for unsteady problems and algebraic equations for steady problems. Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of (usually) nonlinear algebraic equations. Applying a <a href="/wiki/Newton%27s_method#Nonlinear_systems_of_equations" title="Newton's method">Newton</a> or <a href="/wiki/Fixed_point_iteration" class="mw-redirect" title="Fixed point iteration">Picard</a> iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefinite in the presence of incompressibility. Such systems, particularly in 3D, are frequently too large for direct solvers, so iterative methods are used, either stationary methods such as <a href="/wiki/Successive_over-relaxation" title="Successive over-relaxation">successive overrelaxation</a> or <a href="/wiki/Krylov_subspace" title="Krylov subspace">Krylov subspace</a> methods. Krylov methods such as <a href="/wiki/Generalized_minimal_residual_method" title="Generalized minimal residual method">GMRES</a>, typically used with <a href="/wiki/Preconditioner" title="Preconditioner">preconditioning</a>, operate by minimizing the residual over successive subspaces generated by the preconditioned operator. </p><p><a href="/wiki/Multigrid_method" title="Multigrid method">Multigrid</a> has the advantage of asymptotically optimal performance on many problems. Traditional<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Manual_of_Style/Words_to_watch#Unsupported_attributions" title="Wikipedia:Manual of Style/Words to watch"><span title="The material near this tag may use weasel words or too-vague attribution. (November 2010)">according to whom?</span></a></i>]</sup> solvers and preconditioners are effective at reducing high-frequency components of the residual, but low-frequency components typically require many iterations to reduce. By operating on multiple scales, multigrid reduces all components of the residual by similar factors, leading to a mesh-independent number of iterations.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2010)">citation needed</span></a></i>]</sup> </p><p>For indefinite systems, preconditioners such as <a href="/wiki/Incomplete_LU_factorization" title="Incomplete LU factorization">incomplete LU factorization</a>, <a href="/wiki/Additive_Schwarz_method" title="Additive Schwarz method">additive Schwarz</a>, and <a href="/wiki/Multigrid_method" title="Multigrid method">multigrid</a> perform poorly or fail entirely, so the problem structure must be used for effective preconditioning.<sup id="cite_ref-74" class="reference"><a href="#cite_note-74"><span class="cite-bracket">[</span>74<span class="cite-bracket">]</span></a></sup> Methods commonly used in CFD are the <a href="/wiki/SIMPLE_algorithm" title="SIMPLE algorithm">SIMPLE</a> and <a href="/wiki/Uzawa_iteration" title="Uzawa iteration">Uzawa algorithms</a> which exhibit mesh-dependent convergence rates, but recent advances based on block LU factorization combined with multigrid for the resulting definite systems have led to preconditioners that deliver mesh-independent convergence rates.<sup id="cite_ref-75" class="reference"><a href="#cite_note-75"><span class="cite-bracket">[</span>75<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Unsteady_aerodynamics">Unsteady aerodynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=24" title="Edit section: Unsteady aerodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>CFD made a major break through in late 70s with the introduction of LTRAN2, a 2-D code to model oscillating airfoils based on <a href="/wiki/Transonic" title="Transonic">transonic</a> small perturbation theory by Ballhaus and associates.<sup id="cite_ref-76" class="reference"><a href="#cite_note-76"><span class="cite-bracket">[</span>76<span class="cite-bracket">]</span></a></sup> It uses a Murman-Cole switch algorithm for modeling the moving shock-waves.<sup id="cite_ref-Murman_Cole_1971_26-1" class="reference"><a href="#cite_note-Murman_Cole_1971-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> Later it was extended to 3-D with use of a rotated difference scheme by AFWAL/Boeing that resulted in LTRAN3.<sup id="cite_ref-77" class="reference"><a href="#cite_note-77"><span class="cite-bracket">[</span>77<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-78" class="reference"><a href="#cite_note-78"><span class="cite-bracket">[</span>78<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Biomedical_engineering">Biomedical engineering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=25" title="Edit section: Biomedical engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vel-Streamline-FC.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Vel-Streamline-FC.jpg/220px-Vel-Streamline-FC.jpg" decoding="async" width="220" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Vel-Streamline-FC.jpg/330px-Vel-Streamline-FC.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Vel-Streamline-FC.jpg/440px-Vel-Streamline-FC.jpg 2x" data-file-width="972" data-file-height="584" /></a><figcaption>Simulation of blood flow in a human <a href="/wiki/Aorta" title="Aorta">aorta</a></figcaption></figure> <p>CFD investigations are used to clarify the characteristics of aortic flow in details that are beyond the capabilities of experimental measurements. To analyze these conditions, CAD models of the human vascular system are extracted employing modern imaging techniques such as <a href="/wiki/MRI" class="mw-redirect" title="MRI">MRI</a> or <a href="/wiki/Computed_Tomography" class="mw-redirect" title="Computed Tomography">Computed Tomography</a>. A 3D model is reconstructed from this data and the fluid flow can be computed. Blood properties such as density and viscosity, and realistic boundary conditions (e.g. systemic pressure) have to be taken into consideration. Therefore, making it possible to analyze and optimize the flow in the cardiovascular system for different applications.<sup id="cite_ref-79" class="reference"><a href="#cite_note-79"><span class="cite-bracket">[</span>79<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="CPU_versus_GPU">CPU versus GPU</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=26" title="Edit section: CPU versus GPU"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Traditionally, CFD simulations are performed on CPUs.<sup id="cite_ref-80" class="reference"><a href="#cite_note-80"><span class="cite-bracket">[</span>80<span class="cite-bracket">]</span></a></sup> </p><p>In a more recent trend, simulations are also performed on GPUs. These typically contain slower but more processors. For CFD algorithms that feature good parallelism performance (i.e. good speed-up by adding more cores) this can greatly reduce simulation times. Fluid-implicit particle<sup id="cite_ref-81" class="reference"><a href="#cite_note-81"><span class="cite-bracket">[</span>81<span class="cite-bracket">]</span></a></sup> and lattice-Boltzmann methods<sup id="cite_ref-82" class="reference"><a href="#cite_note-82"><span class="cite-bracket">[</span>82<span class="cite-bracket">]</span></a></sup> are typical examples of codes that scale well on GPUs. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=27" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 25em;"> <ul><li><a href="/wiki/Application_of_CFD_in_thermal_power_plants" title="Application of CFD in thermal power plants">Application of CFD in thermal power plants</a></li> <li><a href="/wiki/Blade_element_theory" title="Blade element theory">Blade element theory</a></li> <li><a href="/wiki/Boundary_conditions_in_fluid_dynamics" title="Boundary conditions in fluid dynamics">Boundary conditions in fluid dynamics</a></li> <li><a href="/wiki/Cavitation_modelling" title="Cavitation modelling">Cavitation modelling</a></li> <li><a href="/wiki/Central_differencing_scheme" title="Central differencing scheme">Central differencing scheme</a></li> <li><a href="/wiki/Computational_magnetohydrodynamics" title="Computational magnetohydrodynamics">Computational magnetohydrodynamics</a></li> <li><a href="/wiki/Discrete_element_method" title="Discrete element method">Discrete element method</a></li> <li><a href="/wiki/Finite_element_method" title="Finite element method">Finite element method</a></li> <li><a href="/wiki/Finite_volume_method_for_unsteady_flow" title="Finite volume method for unsteady flow">Finite volume method for unsteady flow</a></li> <li><a href="/wiki/Fluid_animation" title="Fluid animation">Fluid animation</a></li> <li><a href="/wiki/Immersed_boundary_method" title="Immersed boundary method">Immersed boundary method</a></li> <li><a href="/wiki/Lattice_Boltzmann_methods" title="Lattice Boltzmann methods">Lattice Boltzmann methods</a></li> <li><a href="/wiki/List_of_finite_element_software_packages" title="List of finite element software packages">List of finite element software packages</a></li> <li><a href="/wiki/Meshfree_methods" title="Meshfree methods">Meshfree methods</a></li> <li><a href="/wiki/Moving_particle_semi-implicit_method" title="Moving particle semi-implicit method">Moving particle semi-implicit method</a></li> <li><a href="/wiki/Multi-particle_collision_dynamics" title="Multi-particle collision dynamics">Multi-particle collision dynamics</a></li> <li><a href="/wiki/Multidisciplinary_design_optimization" title="Multidisciplinary design optimization">Multidisciplinary design optimization</a></li> <li><a href="/wiki/Numerical_methods_in_fluid_mechanics" title="Numerical methods in fluid mechanics">Numerical methods in fluid mechanics</a></li> <li><a href="/wiki/Shape_optimization" title="Shape optimization">Shape optimization</a></li> <li><a href="/wiki/Smoothed-particle_hydrodynamics" title="Smoothed-particle hydrodynamics">Smoothed-particle hydrodynamics</a></li> <li><a href="/wiki/Stochastic_Eulerian_Lagrangian_method" title="Stochastic Eulerian Lagrangian method">Stochastic Eulerian Lagrangian method</a></li> <li><a href="/wiki/Turbulence_modeling" title="Turbulence modeling">Turbulence modeling</a></li> <li><a href="/wiki/Unified_methods_for_computing_incompressible_and_compressible_flow" title="Unified methods for computing incompressible and compressible flow">Unified methods for computing incompressible and compressible flow</a></li> <li><a href="/wiki/Visualization_(graphics)" title="Visualization (graphics)">Visualization (graphics)</a></li> <li><a href="/wiki/Wind_tunnel" title="Wind tunnel">Wind tunnel</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=28" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMilne-Thomson1973" class="citation book cs1">Milne-Thomson, Louis Melville (1973). <i>Theoretical Aerodynamics</i>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-61980-4" title="Special:BookSources/978-0-486-61980-4"><bdi>978-0-486-61980-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoretical+Aerodynamics&rft.pub=Courier+Corporation&rft.date=1973&rft.isbn=978-0-486-61980-4&rft.aulast=Milne-Thomson&rft.aufirst=Louis+Melville&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (December 2023)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcMurtryGansaugeKersteinKrueger1993" class="citation journal cs1">McMurtry, Patrick A.; Gansauge, Todd C.; Kerstein, Alan R.; Krueger, Steven K. (April 1993). "Linear eddy simulations of mixing in a homogeneous turbulent flow". <i>Physics of Fluids A: Fluid Dynamics</i>. <b>5</b> (4): 1023–1034. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993PhFlA...5.1023M">1993PhFlA...5.1023M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.858667">10.1063/1.858667</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+Fluids+A%3A+Fluid+Dynamics&rft.atitle=Linear+eddy+simulations+of+mixing+in+a+homogeneous+turbulent+flow&rft.volume=5&rft.issue=4&rft.pages=1023-1034&rft.date=1993-04&rft_id=info%3Adoi%2F10.1063%2F1.858667&rft_id=info%3Abibcode%2F1993PhFlA...5.1023M&rft.aulast=McMurtry&rft.aufirst=Patrick+A.&rft.au=Gansauge%2C+Todd+C.&rft.au=Kerstein%2C+Alan+R.&rft.au=Krueger%2C+Steven+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichardson,_L._F.Chapman,_S.1965" class="citation book cs1">Richardson, L. F.; Chapman, S. (1965). <i>Weather prediction by numerical process</i>. Dover Publications.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Weather+prediction+by+numerical+process&rft.pub=Dover+Publications&rft.date=1965&rft.au=Richardson%2C+L.+F.&rft.au=Chapman%2C+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHunt1998" class="citation journal cs1">Hunt, J.C.R. (January 1998). "Lewis Fry Richardson and his contributions to mathematics, meteorology, and models of conflict". <i>Annual Review of Fluid Mechanics</i>. <b>30</b> (1): xiii–xxxvi. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998AnRFM..30D..13H">1998AnRFM..30D..13H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.fluid.30.1.0">10.1146/annurev.fluid.30.1.0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Fluid+Mechanics&rft.atitle=Lewis+Fry+Richardson+and+his+contributions+to+mathematics%2C+meteorology%2C+and+models+of+conflict&rft.volume=30&rft.issue=1&rft.pages=xiii-xxxvi&rft.date=1998-01&rft_id=info%3Adoi%2F10.1146%2Fannurev.fluid.30.1.0&rft_id=info%3Abibcode%2F1998AnRFM..30D..13H&rft.aulast=Hunt&rft.aufirst=J.C.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-legacy_T3-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-legacy_T3_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.lanl.gov/orgs/t/t3/history.shtml#early">"The Legacy of Group T-3"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">March 13,</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Legacy+of+Group+T-3&rft_id=https%3A%2F%2Fwww.lanl.gov%2Forgs%2Ft%2Ft3%2Fhistory.shtml%23early&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-harlow2004fluid-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-harlow2004fluid_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarlow2004" class="citation journal cs1">Harlow, Francis H. (April 2004). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1259097">"Fluid dynamics in Group T-3 Los Alamos National Laboratory"</a>. <i>Journal of Computational Physics</i>. <b>195</b> (2): 414–433. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004JCoPh.195..414H">2004JCoPh.195..414H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jcp.2003.09.031">10.1016/j.jcp.2003.09.031</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Physics&rft.atitle=Fluid+dynamics+in+Group+T-3+Los+Alamos+National+Laboratory&rft.volume=195&rft.issue=2&rft.pages=414-433&rft.date=2004-04&rft_id=info%3Adoi%2F10.1016%2Fj.jcp.2003.09.031&rft_id=info%3Abibcode%2F2004JCoPh.195..414H&rft.aulast=Harlow&rft.aufirst=Francis+H.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1259097&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarlowEvansRichtmyer1955" class="citation book cs1">Harlow, Francis Harvey; Evans, Martha; Richtmyer, Robert D. (1955). <i>A Machine Calculation Method for Hydrodynamic Problems</i>. Los Alamos Scientific Laboratory of the University of California. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fmdp.39015095283399">2027/mdp.39015095283399</a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1288309947">1288309947</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Machine+Calculation+Method+for+Hydrodynamic+Problems&rft.pub=Los+Alamos+Scientific+Laboratory+of+the+University+of+California&rft.date=1955&rft_id=info%3Ahdl%2F2027%2Fmdp.39015095283399&rft_id=info%3Aoclcnum%2F1288309947&rft.aulast=Harlow&rft.aufirst=Francis+Harvey&rft.au=Evans%2C+Martha&rft.au=Richtmyer%2C+Robert+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (December 2023)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGentryMartinDaly1966" class="citation journal cs1">Gentry, Richard A; Martin, Robert E; Daly, Bart J (August 1966). "An Eulerian differencing method for unsteady compressible flow problems". <i>Journal of Computational Physics</i>. <b>1</b> (1): 87–118. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1966JCoPh...1...87G">1966JCoPh...1...87G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2866%2990014-3">10.1016/0021-9991(66)90014-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Physics&rft.atitle=An+Eulerian+differencing+method+for+unsteady+compressible+flow+problems&rft.volume=1&rft.issue=1&rft.pages=87-118&rft.date=1966-08&rft_id=info%3Adoi%2F10.1016%2F0021-9991%2866%2990014-3&rft_id=info%3Abibcode%2F1966JCoPh...1...87G&rft.aulast=Gentry&rft.aufirst=Richard+A&rft.au=Martin%2C+Robert+E&rft.au=Daly%2C+Bart+J&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-Fromm1963-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Fromm1963_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrommHarlow1963" class="citation journal cs1">Fromm, Jacob E.; Harlow, Francis H. (July 1963). "Numerical Solution of the Problem of Vortex Street Development". <i>The Physics of Fluids</i>. <b>6</b> (7): 975–982. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1963PhFl....6..975F">1963PhFl....6..975F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1706854">10.1063/1.1706854</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Physics+of+Fluids&rft.atitle=Numerical+Solution+of+the+Problem+of+Vortex+Street+Development&rft.volume=6&rft.issue=7&rft.pages=975-982&rft.date=1963-07&rft_id=info%3Adoi%2F10.1063%2F1.1706854&rft_id=info%3Abibcode%2F1963PhFl....6..975F&rft.aulast=Fromm&rft.aufirst=Jacob+E.&rft.au=Harlow%2C+Francis+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-harlow_welch-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-harlow_welch_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarlowWelch1965" class="citation journal cs1">Harlow, Francis H.; Welch, J. Eddie (December 1965). "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface". <i>The Physics of Fluids</i>. <b>8</b> (12): 2182–2189. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1965PhFl....8.2182H">1965PhFl....8.2182H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1761178">10.1063/1.1761178</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Physics+of+Fluids&rft.atitle=Numerical+Calculation+of+Time-Dependent+Viscous+Incompressible+Flow+of+Fluid+with+Free+Surface&rft.volume=8&rft.issue=12&rft.pages=2182-2189&rft.date=1965-12&rft_id=info%3Adoi%2F10.1063%2F1.1761178&rft_id=info%3Abibcode%2F1965PhFl....8.2182H&rft.aulast=Harlow&rft.aufirst=Francis+H.&rft.au=Welch%2C+J.+Eddie&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHessSmith1967" class="citation journal cs1">Hess, J.L.; Smith, A.M.O. (1967). "Calculation of potential flow about arbitrary bodies". <i>Progress in Aerospace Sciences</i>. <b>8</b>: 1–138. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967PrAeS...8....1H">1967PrAeS...8....1H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0376-0421%2867%2990003-6">10.1016/0376-0421(67)90003-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Progress+in+Aerospace+Sciences&rft.atitle=Calculation+of+potential+flow+about+arbitrary+bodies&rft.volume=8&rft.pages=1-138&rft.date=1967&rft_id=info%3Adoi%2F10.1016%2F0376-0421%2867%2990003-6&rft_id=info%3Abibcode%2F1967PrAeS...8....1H&rft.aulast=Hess&rft.aufirst=J.L.&rft.au=Smith%2C+A.M.O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRubbertSaaris1972" class="citation book cs1">Rubbert, P.; Saaris, G. (1972). "Review and evaluation of a three-dimensional lifting potential flow computational method for arbitrary configurations". <i>10th Aerospace Sciences Meeting</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1972-188">10.2514/6.1972-188</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Review+and+evaluation+of+a+three-dimensional+lifting+potential+flow+computational+method+for+arbitrary+configurations&rft.btitle=10th+Aerospace+Sciences+Meeting&rft.date=1972&rft_id=info%3Adoi%2F10.2514%2F6.1972-188&rft.aulast=Rubbert&rft.aufirst=P.&rft.au=Saaris%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarmichaelErickson1981" class="citation book cs1">Carmichael, R.; Erickson, L. (1981). "PAN AIR - A higher order panel method for predicting subsonic or supersonic linear potential flows about arbitrary configurations". <i>14th Fluid and Plasma Dynamics Conference</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1981-1255">10.2514/6.1981-1255</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=PAN+AIR+-+A+higher+order+panel+method+for+predicting+subsonic+or+supersonic+linear+potential+flows+about+arbitrary+configurations&rft.btitle=14th+Fluid+and+Plasma+Dynamics+Conference&rft.date=1981&rft_id=info%3Adoi%2F10.2514%2F6.1981-1255&rft.aulast=Carmichael&rft.aufirst=R.&rft.au=Erickson%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoungrenBouchardCoopersmithMiranda1983" class="citation book cs1">Youngren, H.; Bouchard, E.; Coopersmith, R.; Miranda, L. (1983). "Comparison of panel method formulations and its influence on the development of QUADPAN, an advanced low-order method". <i>Applied Aerodynamics Conference</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1983-1827">10.2514/6.1983-1827</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Comparison+of+panel+method+formulations+and+its+influence+on+the+development+of+QUADPAN%2C+an+advanced+low-order+method&rft.btitle=Applied+Aerodynamics+Conference&rft.date=1983&rft_id=info%3Adoi%2F10.2514%2F6.1983-1827&rft.aulast=Youngren&rft.aufirst=H.&rft.au=Bouchard%2C+E.&rft.au=Coopersmith%2C+R.&rft.au=Miranda%2C+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHessFriedman1983" class="citation book cs1">Hess, J.; Friedman, D. (1983). "Analysis of complex inlet configurations using a higher-order panel method". <i>Applied Aerodynamics Conference</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1983-1828">10.2514/6.1983-1828</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Analysis+of+complex+inlet+configurations+using+a+higher-order+panel+method&rft.btitle=Applied+Aerodynamics+Conference&rft.date=1983&rft_id=info%3Adoi%2F10.2514%2F6.1983-1828&rft.aulast=Hess&rft.aufirst=J.&rft.au=Friedman%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Bristow, D.R., "<a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19800007773.pdf">Development of Panel Methods for Subsonic Analysis and Design</a>," NASA CR-3234, 1980.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Ashby, Dale L.; Dudley, Michael R.; Iguchi, Steve K.; Browne, Lindsey and Katz, Joseph, "<a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19920023178.pdf">Potential Flow Theory and Operation Guide for the Panel Code PMARC</a>", NASA NASA-TM-102851 1991.</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Woodward, F.A., Dvorak, F.A. and Geller, E.W., "<a rel="nofollow" class="external text" href="https://web.archive.org/web/20191210001500/http://www.dtic.mil/dtic/tr/fulltext/u2/782202.pdf">A Computer Program for Three-Dimensional Lifting Bodies in Subsonic Inviscid Flow</a>," USAAMRDL Technical Report, TR 74-18, Ft. Eustis, Virginia, April 1974.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKatzMaskew1988" class="citation journal cs1">Katz, Joseph; Maskew, Brian (April 1988). "Unsteady low-speed aerodynamic model for complete aircraft configurations". <i>Journal of Aircraft</i>. <b>25</b> (4): 302–310. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F3.45564">10.2514/3.45564</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Aircraft&rft.atitle=Unsteady+low-speed+aerodynamic+model+for+complete+aircraft+configurations&rft.volume=25&rft.issue=4&rft.pages=302-310&rft.date=1988-04&rft_id=info%3Adoi%2F10.2514%2F3.45564&rft.aulast=Katz&rft.aufirst=Joseph&rft.au=Maskew%2C+Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaskew1982" class="citation journal cs1">Maskew, Brian (February 1982). "Prediction of Subsonic Aerodynamic Characteristics: A Case for Low-Order Panel Methods". <i>Journal of Aircraft</i>. <b>19</b> (2): 157–163. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F3.57369">10.2514/3.57369</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Aircraft&rft.atitle=Prediction+of+Subsonic+Aerodynamic+Characteristics%3A+A+Case+for+Low-Order+Panel+Methods&rft.volume=19&rft.issue=2&rft.pages=157-163&rft.date=1982-02&rft_id=info%3Adoi%2F10.2514%2F3.57369&rft.aulast=Maskew&rft.aufirst=Brian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Maskew, Brian, "<a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19900004884.pdf">Program VSAERO Theory Document: A Computer Program for Calculating Nonlinear Aerodynamic Characteristics of Arbitrary Configurations</a>", NASA CR-4023, 1987.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text">Pinella, David and Garrison, Peter, "Digital Wind Tunnel CMARC; Three-Dimensional Low-Order Panel Codes," Aerologic, 2009.</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Eppler, R.; Somers, D. M., "<a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19800020753.pdf">A Computer Program for the Design and Analysis of Low-Speed Airfoils</a>," NASA TM-80210, 1980.</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text">Drela, Mark, "<a rel="nofollow" class="external text" href="http://www.web.mit.edu/drela/Public/papers/xfoil_sv.pdf">XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils</a>," in Springer-Verlag Lecture Notes in Engineering, No. 54, 1989.</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoppe1977" class="citation book cs1">Boppe, C. (1977). "Calculation of transonic wing flows by grid embedding". <i>15th Aerospace Sciences Meeting</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1977-207">10.2514/6.1977-207</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Calculation+of+transonic+wing+flows+by+grid+embedding&rft.btitle=15th+Aerospace+Sciences+Meeting&rft.date=1977&rft_id=info%3Adoi%2F10.2514%2F6.1977-207&rft.aulast=Boppe&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-Murman_Cole_1971-26"><span class="mw-cite-backlink">^ <a href="#cite_ref-Murman_Cole_1971_26-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Murman_Cole_1971_26-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMurmanCole1971" class="citation journal cs1">Murman, Earll M.; Cole, Julian D. (January 1971). "Calculation of plane steady transonic flows". <i>AIAA Journal</i>. <b>9</b> (1): 114–121. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1971AIAAJ...9..114C">1971AIAAJ...9..114C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F3.6131">10.2514/3.6131</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=AIAA+Journal&rft.atitle=Calculation+of+plane+steady+transonic+flows&rft.volume=9&rft.issue=1&rft.pages=114-121&rft.date=1971-01&rft_id=info%3Adoi%2F10.2514%2F3.6131&rft_id=info%3Abibcode%2F1971AIAAJ...9..114C&rft.aulast=Murman&rft.aufirst=Earll+M.&rft.au=Cole%2C+Julian+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><i>A Theory of Supercritical Wing Sections, with Computer Programs and Examples</i>. Lecture Notes in Economics and Mathematical Systems. Vol. 66. 1972. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-80678-0">10.1007/978-3-642-80678-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-05807-6" title="Special:BookSources/978-3-540-05807-6"><bdi>978-3-540-05807-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Theory+of+Supercritical+Wing+Sections%2C+with+Computer+Programs+and+Examples&rft.series=Lecture+Notes+in+Economics+and+Mathematical+Systems&rft.date=1972&rft_id=info%3Adoi%2F10.1007%2F978-3-642-80678-0&rft.isbn=978-3-540-05807-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (December 2023)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Mead, H. R.; Melnik, R. E., "<a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19860002735.pdf">GRUMFOIL: A computer code for the viscous transonic flow over airfoils</a>," NASA CR-3806, 1985.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJamesonCaughey1977" class="citation book cs1">Jameson, A.; Caughey, D. (1977). "A finite volume method for transonic potential flow calculations". <i>3rd Computational Fluid Dynamics Conference</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1977-635">10.2514/6.1977-635</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=A+finite+volume+method+for+transonic+potential+flow+calculations&rft.btitle=3rd+Computational+Fluid+Dynamics+Conference&rft.date=1977&rft_id=info%3Adoi%2F10.2514%2F6.1977-635&rft.aulast=Jameson&rft.aufirst=A.&rft.au=Caughey%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSamantBussolettiJohnsonBurkhart1987" class="citation book cs1">Samant, S.; Bussoletti, J.; Johnson, F.; Burkhart, R.; Everson, B.; Melvin, R.; Young, D.; Erickson, L.; Madson, M. (1987). "TRANAIR - A computer code for transonic analyses of arbitrary configurations". <i>25th AIAA Aerospace Sciences Meeting</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1987-34">10.2514/6.1987-34</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=TRANAIR+-+A+computer+code+for+transonic+analyses+of+arbitrary+configurations&rft.btitle=25th+AIAA+Aerospace+Sciences+Meeting&rft.date=1987&rft_id=info%3Adoi%2F10.2514%2F6.1987-34&rft.aulast=Samant&rft.aufirst=S.&rft.au=Bussoletti%2C+J.&rft.au=Johnson%2C+F.&rft.au=Burkhart%2C+R.&rft.au=Everson%2C+B.&rft.au=Melvin%2C+R.&rft.au=Young%2C+D.&rft.au=Erickson%2C+L.&rft.au=Madson%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJamesonSchmidtTurkel1981" class="citation book cs1">Jameson, A.; Schmidt, Wolfgang; Turkel, ELI (1981). "Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes". <i>14th Fluid and Plasma Dynamics Conference</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1981-1259">10.2514/6.1981-1259</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Numerical+solution+of+the+Euler+equations+by+finite+volume+methods+using+Runge+Kutta+time+stepping+schemes&rft.btitle=14th+Fluid+and+Plasma+Dynamics+Conference&rft.date=1981&rft_id=info%3Adoi%2F10.2514%2F6.1981-1259&rft.aulast=Jameson&rft.aufirst=A.&rft.au=Schmidt%2C+Wolfgang&rft.au=Turkel%2C+ELI&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRajBrennan1989" class="citation journal cs1">Raj, Pradeep; Brennan, James E. (1989). "Improvements to an Euler aerodynamic method for transonic flow analysis". <i>Journal of Aircraft</i>. <b>26</b>: 13–20. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F3.45717">10.2514/3.45717</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Aircraft&rft.atitle=Improvements+to+an+Euler+aerodynamic+method+for+transonic+flow+analysis&rft.volume=26&rft.pages=13-20&rft.date=1989&rft_id=info%3Adoi%2F10.2514%2F3.45717&rft.aulast=Raj&rft.aufirst=Pradeep&rft.au=Brennan%2C+James+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTiddStrashEpsteinLuntz1991" class="citation book cs1">Tidd, D.; Strash, D.; Epstein, B.; Luntz, A.; Nachshon, A.; Rubin, T. (1991). "Application of an efficient 3-D multigrid Euler method (MGAERO) to complete aircraft configurations". <i>9th Applied Aerodynamics Conference</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1991-3236">10.2514/6.1991-3236</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Application+of+an+efficient+3-D+multigrid+Euler+method+%28MGAERO%29+to+complete+aircraft+configurations&rft.btitle=9th+Applied+Aerodynamics+Conference&rft.date=1991&rft_id=info%3Adoi%2F10.2514%2F6.1991-3236&rft.aulast=Tidd&rft.aufirst=D.&rft.au=Strash%2C+D.&rft.au=Epstein%2C+B.&rft.au=Luntz%2C+A.&rft.au=Nachshon%2C+A.&rft.au=Rubin%2C+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMeltonBergerAftosmisWong1995" class="citation book cs1">Melton, John; Berger, Marsha; Aftosmis, Michael; Wong, Michael (1995). "3D applications of a Cartesian grid Euler method". <i>33rd Aerospace Sciences Meeting and Exhibit</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1995-853">10.2514/6.1995-853</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3D+applications+of+a+Cartesian+grid+Euler+method&rft.btitle=33rd+Aerospace+Sciences+Meeting+and+Exhibit&rft.date=1995&rft_id=info%3Adoi%2F10.2514%2F6.1995-853&rft.aulast=Melton&rft.aufirst=John&rft.au=Berger%2C+Marsha&rft.au=Aftosmis%2C+Michael&rft.au=Wong%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarman,_l,_Jr1995" class="citation book cs1">Karman, l, Jr, Steve (1995). "SPLITFLOW - A 3D unstructured Cartesian/Prismatic grid CFD code for complex geometries". <i>33rd Aerospace Sciences Meeting and Exhibit</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1995-343">10.2514/6.1995-343</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=SPLITFLOW+-+A+3D+unstructured+Cartesian%2FPrismatic+grid+CFD+code+for+complex+geometries&rft.btitle=33rd+Aerospace+Sciences+Meeting+and+Exhibit&rft.date=1995&rft_id=info%3Adoi%2F10.2514%2F6.1995-343&rft.aulast=Karman%2C+l%2C+Jr&rft.aufirst=Steve&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarshallRuffin2004" class="citation book cs1">Marshall, David; Ruffin, Stephen (2004). "An Embedded Boundary Cartesian Grid Scheme for Viscous Flows Using a New Viscous Wall Boundary Condition Treatment". <a rel="nofollow" class="external text" href="https://digitalcommons.calpoly.edu/aero_fac/85"><i>42nd AIAA Aerospace Sciences Meeting and Exhibit</i></a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.2004-581">10.2514/6.2004-581</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-62410-078-9" title="Special:BookSources/978-1-62410-078-9"><bdi>978-1-62410-078-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=An+Embedded+Boundary+Cartesian+Grid+Scheme+for+Viscous+Flows+Using+a+New+Viscous+Wall+Boundary+Condition+Treatment&rft.btitle=42nd+AIAA+Aerospace+Sciences+Meeting+and+Exhibit&rft.date=2004&rft_id=info%3Adoi%2F10.2514%2F6.2004-581&rft.isbn=978-1-62410-078-9&rft.aulast=Marshall&rft.aufirst=David&rft.au=Ruffin%2C+Stephen&rft_id=https%3A%2F%2Fdigitalcommons.calpoly.edu%2Faero_fac%2F85&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJamesonBakerWeatherill1986" class="citation book cs1">Jameson, A.; Baker, T.; Weatherill, N. (1986). "Calculation of Inviscid Transonic Flow over a Complete Aircraft". <i>24th Aerospace Sciences Meeting</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1986-103">10.2514/6.1986-103</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Calculation+of+Inviscid+Transonic+Flow+over+a+Complete+Aircraft&rft.btitle=24th+Aerospace+Sciences+Meeting&rft.date=1986&rft_id=info%3Adoi%2F10.2514%2F6.1986-103&rft.aulast=Jameson&rft.aufirst=A.&rft.au=Baker%2C+T.&rft.au=Weatherill%2C+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilesDrelaThompkins,_Jr.1985" class="citation book cs1">Giles, M.; Drela, M.; Thompkins, Jr., W. (1985). "Newton solution of direct and inverse transonic Euler equations". <i>7th Computational Physics Conference</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1985-1530">10.2514/6.1985-1530</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Newton+solution+of+direct+and+inverse+transonic+Euler+equations&rft.btitle=7th+Computational+Physics+Conference&rft.date=1985&rft_id=info%3Adoi%2F10.2514%2F6.1985-1530&rft.aulast=Giles&rft.aufirst=M.&rft.au=Drela%2C+M.&rft.au=Thompkins%2C+Jr.%2C+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDrela1990" class="citation book cs1">Drela, Mark (1990). "Newton solution of coupled viscous/Inviscid multielement airfoil flows". <i>21st Fluid Dynamics, Plasma Dynamics and Lasers Conference</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F6.1990-1470">10.2514/6.1990-1470</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Newton+solution+of+coupled+viscous%2FInviscid+multielement+airfoil+flows&rft.btitle=21st+Fluid+Dynamics%2C+Plasma+Dynamics+and+Lasers+Conference&rft.date=1990&rft_id=info%3Adoi%2F10.2514%2F6.1990-1470&rft.aulast=Drela&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text">Drela, M. and Youngren H., "A User's Guide to MISES 2.53", MIT Computational Sciences Laboratory, December 1998.</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJopForterrePouliquen2006" class="citation journal cs1">Jop, Pierre; Forterre, Yoël; Pouliquen, Olivier (June 2006). <a rel="nofollow" class="external text" href="https://www.nature.com/articles/nature04801">"A constitutive law for dense granular flows"</a>. <i>Nature</i>. <b>441</b> (7094): 727–730. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/0612110">cond-mat/0612110</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006Natur.441..727J">2006Natur.441..727J</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature04801">10.1038/nature04801</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1476-4687">1476-4687</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16760972">16760972</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=A+constitutive+law+for+dense+granular+flows&rft.volume=441&rft.issue=7094&rft.pages=727-730&rft.date=2006-06&rft_id=info%3Abibcode%2F2006Natur.441..727J&rft_id=info%3Aarxiv%2Fcond-mat%2F0612110&rft_id=info%3Apmid%2F16760972&rft_id=info%3Adoi%2F10.1038%2Fnature04801&rft.issn=1476-4687&rft.aulast=Jop&rft.aufirst=Pierre&rft.au=Forterre%2C+Yo%C3%ABl&rft.au=Pouliquen%2C+Olivier&rft_id=https%3A%2F%2Fwww.nature.com%2Farticles%2Fnature04801&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirounMazzei2024" class="citation journal cs1">Biroun, Mehdi H.; Mazzei, Luca (June 2024). <a rel="nofollow" class="external text" href="https://doi.org/10.1016/j.ces.2024.119997">"Unchannelized granular flows: Effect of initial granular column geometry on fluid dynamics"</a>. <i>Chemical Engineering Science</i>. <b>292</b>: 119997. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ces.2024.119997">10.1016/j.ces.2024.119997</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0009-2509">0009-2509</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Chemical+Engineering+Science&rft.atitle=Unchannelized+granular+flows%3A+Effect+of+initial+granular+column+geometry+on+fluid+dynamics&rft.volume=292&rft.pages=119997&rft.date=2024-06&rft_id=info%3Adoi%2F10.1016%2Fj.ces.2024.119997&rft.issn=0009-2509&rft.aulast=Biroun&rft.aufirst=Mehdi+H.&rft.au=Mazzei%2C+Luca&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%2Fj.ces.2024.119997&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-ferziger-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-ferziger_43-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFerziger,_J._H._and_Peric,_M.2002" class="citation book cs1">Ferziger, J. H. and Peric, M. (2002). <i>Computational methods for fluid dynamics</i>. Springer-Verlag.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+methods+for+fluid+dynamics&rft.pub=Springer-Verlag&rft.date=2002&rft.au=Ferziger%2C+J.+H.+and+Peric%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-cns-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-cns_44-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.cfd-online.com/Wiki/Navier-Stokes_equations">"Navier-Stokes equations"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-01-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Navier-Stokes+equations&rft_id=https%3A%2F%2Fwww.cfd-online.com%2FWiki%2FNavier-Stokes_equations&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-panton-45"><span class="mw-cite-backlink">^ <a href="#cite_ref-panton_45-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-panton_45-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-panton_45-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-panton_45-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-panton_45-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-panton_45-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-panton_45-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-panton_45-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-panton_45-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-panton_45-9"><sup><i><b>j</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPanton,_R._L.1996" class="citation book cs1">Panton, R. L. (1996). <i>Incompressible Flow</i>. John Wiley and Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Incompressible+Flow&rft.pub=John+Wiley+and+Sons&rft.date=1996&rft.au=Panton%2C+R.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-landau-46"><span class="mw-cite-backlink">^ <a href="#cite_ref-landau_46-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-landau_46-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-landau_46-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-landau_46-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandau,_L._D._and_Lifshitz,_E._M.2007" class="citation book cs1">Landau, L. D. and Lifshitz, E. M. (2007). <i>Fluid Mechanics</i>. Elsevier.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics&rft.pub=Elsevier&rft.date=2007&rft.au=Landau%2C+L.+D.+and+Lifshitz%2C+E.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-fox-47"><span class="mw-cite-backlink">^ <a href="#cite_ref-fox_47-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-fox_47-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFox,_R._W._and_McDonald,_A._T.1992" class="citation book cs1">Fox, R. W. and McDonald, A. T. (1992). <i>Introduction to Fluid Mechanics</i>. John Wiley and Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Fluid+Mechanics&rft.pub=John+Wiley+and+Sons&rft.date=1992&rft.au=Fox%2C+R.+W.+and+McDonald%2C+A.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-poinsot-48"><span class="mw-cite-backlink">^ <a href="#cite_ref-poinsot_48-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-poinsot_48-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoinsot,_T._and_Veynante,_D.2005" class="citation book cs1">Poinsot, T. and Veynante, D. (2005). <i>Theoretical and numerical combustion</i>. RT Edwards.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theoretical+and+numerical+combustion&rft.pub=RT+Edwards&rft.date=2005&rft.au=Poinsot%2C+T.+and+Veynante%2C+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-kundu-49"><span class="mw-cite-backlink">^ <a href="#cite_ref-kundu_49-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-kundu_49-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-kundu_49-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-kundu_49-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKundu,_P.1990" class="citation book cs1">Kundu, P. (1990). <i>Fluid Mechanics</i>. Academic Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics&rft.pub=Academic+Press&rft.date=1990&rft.au=Kundu%2C+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-ras-50"><span class="mw-cite-backlink">^ <a href="#cite_ref-ras_50-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ras_50-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.cfd-online.com/Wiki/Favre_averaged_Navier-Stokes_equations">"Favre averaged Navier-Stokes equations"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-01-07</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Favre+averaged+Navier-Stokes+equations&rft_id=https%3A%2F%2Fwww.cfd-online.com%2FWiki%2FFavre_averaged_Navier-Stokes_equations&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-bailly-51"><span class="mw-cite-backlink"><b><a href="#cite_ref-bailly_51-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBailly,_C.,_and_Daniel_J.2000" class="citation journal cs1">Bailly, C., and Daniel J. (2000). "Numerical solution of acoustic propagation problems using Linearized Euler Equations". <i>AIAA Journal</i>. <b>38</b> (1): 22–29. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000AIAAJ..38...22B">2000AIAAJ..38...22B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2514%2F2.949">10.2514/2.949</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=AIAA+Journal&rft.atitle=Numerical+solution+of+acoustic+propagation+problems+using+Linearized+Euler+Equations&rft.volume=38&rft.issue=1&rft.pages=22-29&rft.date=2000&rft_id=info%3Adoi%2F10.2514%2F2.949&rft_id=info%3Abibcode%2F2000AIAAJ..38...22B&rft.au=Bailly%2C+C.%2C+and+Daniel+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-harley-52"><span class="mw-cite-backlink"><b><a href="#cite_ref-harley_52-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarley,_J._C._and_Huang,_Y._and_Bau,_H._H._and_Zemel,_J._N.1995" class="citation journal cs1">Harley, J. C. and Huang, Y. and Bau, H. H. and Zemel, J. N. (1995). "Gas flow in micro-channels". <i>Journal of Fluid Mechanics</i>. <b>284</b>: 257–274. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1995JFM...284..257H">1995JFM...284..257H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112095000358">10.1017/S0022112095000358</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122833857">122833857</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=Gas+flow+in+micro-channels&rft.volume=284&rft.pages=257-274&rft.date=1995&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122833857%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112095000358&rft_id=info%3Abibcode%2F1995JFM...284..257H&rft.au=Harley%2C+J.+C.+and+Huang%2C+Y.+and+Bau%2C+H.+H.+and+Zemel%2C+J.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-1d-ee-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-1d-ee_53-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://home.cscamm.umd.edu/centpack/examples/euler1d.htm">"One-dimensional Euler equations"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-01-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=One-dimensional+Euler+equations&rft_id=https%3A%2F%2Fhome.cscamm.umd.edu%2Fcentpack%2Fexamples%2Feuler1d.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-cavazzuti-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-cavazzuti_54-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCavazzuti,_M._and_Corticelli,_M._A._and_Karayiannis,_T._G.2019" class="citation journal cs1">Cavazzuti, M. and Corticelli, M. A. and Karayiannis, T. G. (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tsep.2019.01.003">"Compressible Fanno flows in micro-channels: An enhanced quasi-2D numerical model for laminar flows"</a>. <i>Thermal Science and Engineering Progress</i>. <b>10</b>: 10–26. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019TSEP...10...10C">2019TSEP...10...10C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.tsep.2019.01.003">10.1016/j.tsep.2019.01.003</a></span>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/11392%2F2414220">11392/2414220</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Thermal+Science+and+Engineering+Progress&rft.atitle=Compressible+Fanno+flows+in+micro-channels%3A+An+enhanced+quasi-2D+numerical+model+for+laminar+flows&rft.volume=10&rft.pages=10-26&rft.date=2019&rft_id=info%3Ahdl%2F11392%2F2414220&rft_id=info%3Adoi%2F10.1016%2Fj.tsep.2019.01.003&rft_id=info%3Abibcode%2F2019TSEP...10...10C&rft.au=Cavazzuti%2C+M.+and+Corticelli%2C+M.+A.+and+Karayiannis%2C+T.+G.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.tsep.2019.01.003&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-55"><span class="mw-cite-backlink"><b><a href="#cite_ref-55">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPatankar1980" class="citation book cs1"><a href="/wiki/Suhas_Patankar" title="Suhas Patankar">Patankar, Suhas V.</a> (1980). <i>Numerical Heat Transfer and Fluid FLow</i>. Hemisphere Publishing Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0891165224" title="Special:BookSources/978-0891165224"><bdi>978-0891165224</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Heat+Transfer+and+Fluid+FLow&rft.pub=Hemisphere+Publishing+Corporation&rft.date=1980&rft.isbn=978-0891165224&rft.aulast=Patankar&rft.aufirst=Suhas+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-56"><span class="mw-cite-backlink"><b><a href="#cite_ref-56">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.comsol.com/multiphysics/finite-element-method">"Detailed Explanation of the Finite Element Method (FEM)"</a>. <i>www.comsol.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2022-07-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.comsol.com&rft.atitle=Detailed+Explanation+of+the+Finite+Element+Method+%28FEM%29&rft_id=https%3A%2F%2Fwww.comsol.com%2Fmultiphysics%2Ffinite-element-method&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-:0-57"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_57-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_57-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson1995" class="citation book cs1">Anderson, John David (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=phG_QgAACAAJ"><i>Computational Fluid Dynamics: The Basics with Applications</i></a>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-113210-7" title="Special:BookSources/978-0-07-113210-7"><bdi>978-0-07-113210-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Fluid+Dynamics%3A+The+Basics+with+Applications&rft.pub=McGraw-Hill&rft.date=1995&rft.isbn=978-0-07-113210-7&rft.aulast=Anderson&rft.aufirst=John+David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DphG_QgAACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuranaAlluTenpasReddy2007" class="citation journal cs1">Surana, K.A.; Allu, S.; Tenpas, P.W.; Reddy, J.N. (February 2007). "k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions". <i>International Journal for Numerical Methods in Engineering</i>. <b>69</b> (6): 1109–1157. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007IJNME..69.1109S">2007IJNME..69.1109S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fnme.1801">10.1002/nme.1801</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122551159">122551159</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+for+Numerical+Methods+in+Engineering&rft.atitle=k-version+of+finite+element+method+in+gas+dynamics%3A+higher-order+global+differentiability+numerical+solutions&rft.volume=69&rft.issue=6&rft.pages=1109-1157&rft.date=2007-02&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122551159%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1002%2Fnme.1801&rft_id=info%3Abibcode%2F2007IJNME..69.1109S&rft.aulast=Surana&rft.aufirst=K.A.&rft.au=Allu%2C+S.&rft.au=Tenpas%2C+P.W.&rft.au=Reddy%2C+J.N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuranaAlluTenpasReddy2007" class="citation journal cs1">Surana, KS; Allu, S; Tenpas, PW; Reddy, JN (2007). "k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions". <i>International Journal for Numerical Methods in Engineering</i>. <b>69</b> (6). Wiley Online Library: 1109–1157. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007IJNME..69.1109S">2007IJNME..69.1109S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fnme.1801">10.1002/nme.1801</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+for+Numerical+Methods+in+Engineering&rft.atitle=k-version+of+finite+element+method+in+gas+dynamics%3A+higher-order+global+differentiability+numerical+solutions&rft.volume=69&rft.issue=6&rft.pages=1109-1157&rft.date=2007&rft_id=info%3Adoi%2F10.1002%2Fnme.1801&rft_id=info%3Abibcode%2F2007IJNME..69.1109S&rft.aulast=Surana&rft.aufirst=KS&rft.au=Allu%2C+S&rft.au=Tenpas%2C+PW&rft.au=Reddy%2C+JN&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSuranaAlluTenpasReddy2007" class="citation journal cs1">Surana, KS; Allu, S; Tenpas, PW; Reddy, JN (2007). "k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions". <i>International Journal for Numerical Methods in Engineering</i>. <b>69</b> (6). Wiley Online Library: 1109–1157. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007IJNME..69.1109S">2007IJNME..69.1109S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fnme.1801">10.1002/nme.1801</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+for+Numerical+Methods+in+Engineering&rft.atitle=k-version+of+finite+element+method+in+gas+dynamics%3A+higher-order+global+differentiability+numerical+solutions&rft.volume=69&rft.issue=6&rft.pages=1109-1157&rft.date=2007&rft_id=info%3Adoi%2F10.1002%2Fnme.1801&rft_id=info%3Abibcode%2F2007IJNME..69.1109S&rft.aulast=Surana&rft.aufirst=KS&rft.au=Allu%2C+S&rft.au=Tenpas%2C+PW&rft.au=Reddy%2C+JN&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-61"><span class="mw-cite-backlink"><b><a href="#cite_ref-61">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCottetKoumoutsakos2000" class="citation book cs1">Cottet, Georges-Henri; Koumoutsakos, Petros D. (2000). <i>Vortex Methods: Theory and Practice</i>. Cambridge, UK: Cambridge Univ. Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-62186-0" title="Special:BookSources/0-521-62186-0"><bdi>0-521-62186-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vortex+Methods%3A+Theory+and+Practice&rft.place=Cambridge%2C+UK&rft.pub=Cambridge+Univ.+Press&rft.date=2000&rft.isbn=0-521-62186-0&rft.aulast=Cottet&rft.aufirst=Georges-Henri&rft.au=Koumoutsakos%2C+Petros+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-62"><span class="mw-cite-backlink"><b><a href="#cite_ref-62">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaunderD.B._Spalding1974" class="citation journal cs1">Launder, B.E.; D.B. Spalding (1974). "The Numerical Computation of Turbulent Flows". <i>Computer Methods in Applied Mechanics and Engineering</i>. <b>3</b> (2): 269–289. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974CMAME...3..269L">1974CMAME...3..269L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0045-7825%2874%2990029-2">10.1016/0045-7825(74)90029-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Methods+in+Applied+Mechanics+and+Engineering&rft.atitle=The+Numerical+Computation+of+Turbulent+Flows&rft.volume=3&rft.issue=2&rft.pages=269-289&rft.date=1974&rft_id=info%3Adoi%2F10.1016%2F0045-7825%2874%2990029-2&rft_id=info%3Abibcode%2F1974CMAME...3..269L&rft.aulast=Launder&rft.aufirst=B.E.&rft.au=D.B.+Spalding&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-wilcox-63"><span class="mw-cite-backlink">^ <a href="#cite_ref-wilcox_63-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-wilcox_63-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilcox2006" class="citation book cs1">Wilcox, David C. (2006). <i>Turbulence Modeling for CFD</i> (3 ed.). DCW Industries, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-928729-08-2" title="Special:BookSources/978-1-928729-08-2"><bdi>978-1-928729-08-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Turbulence+Modeling+for+CFD&rft.edition=3&rft.pub=DCW+Industries%2C+Inc.&rft.date=2006&rft.isbn=978-1-928729-08-2&rft.aulast=Wilcox&rft.aufirst=David+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-Pope_2000-64"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pope_2000_64-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPope,_S.B.2000" class="citation book cs1">Pope, S.B. (2000). <i>Turbulent Flows</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-59886-6" title="Special:BookSources/978-0-521-59886-6"><bdi>978-0-521-59886-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Turbulent+Flows&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-59886-6&rft.au=Pope%2C+S.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-Farge_2001-65"><span class="mw-cite-backlink"><b><a href="#cite_ref-Farge_2001_65-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFargeSchneider,_Kai2001" class="citation journal cs1"><a href="/wiki/Marie_Farge" title="Marie Farge">Farge, Marie</a>; Schneider, Kai (2001). "Coherent Vortex Simulation (CVS), A Semi-Deterministic Turbulence Model Using Wavelets". <i>Flow, Turbulence and Combustion</i>. <b>66</b> (4): 393–426. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2001FTC....66..393F">2001FTC....66..393F</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1013512726409">10.1023/A:1013512726409</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53464243">53464243</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Flow%2C+Turbulence+and+Combustion&rft.atitle=Coherent+Vortex+Simulation+%28CVS%29%2C+A+Semi-Deterministic+Turbulence+Model+Using+Wavelets&rft.volume=66&rft.issue=4&rft.pages=393-426&rft.date=2001&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53464243%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FA%3A1013512726409&rft_id=info%3Abibcode%2F2001FTC....66..393F&rft.aulast=Farge&rft.aufirst=Marie&rft.au=Schneider%2C+Kai&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-Goldstein_2004-66"><span class="mw-cite-backlink"><b><a href="#cite_ref-Goldstein_2004_66-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGoldstein,_DanielVasilyev,_Oleg1995" class="citation journal cs1">Goldstein, Daniel; Vasilyev, Oleg (1995). "Stochastic coherent adaptive large eddy simulation method". <i>Physics of Fluids A</i>. <b>24</b> (7): 2497. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004PhFl...16.2497G">2004PhFl...16.2497G</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.415.6540">10.1.1.415.6540</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1736671">10.1063/1.1736671</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+Fluids+A&rft.atitle=Stochastic+coherent+adaptive+large+eddy+simulation+method&rft.volume=24&rft.issue=7&rft.pages=2497&rft.date=1995&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.415.6540%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1063%2F1.1736671&rft_id=info%3Abibcode%2F2004PhFl...16.2497G&rft.au=Goldstein%2C+Daniel&rft.au=Vasilyev%2C+Oleg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-Lundgren_1969-67"><span class="mw-cite-backlink"><b><a href="#cite_ref-Lundgren_1969_67-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLundgren,_T.S.1969" class="citation journal cs1">Lundgren, T.S. (1969). "Model equation for nonhomogeneous turbulence". <i>Physics of Fluids A</i>. <b>12</b> (3): 485–497. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1969PhFl...12..485L">1969PhFl...12..485L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1692511">10.1063/1.1692511</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+Fluids+A&rft.atitle=Model+equation+for+nonhomogeneous+turbulence&rft.volume=12&rft.issue=3&rft.pages=485-497&rft.date=1969&rft_id=info%3Adoi%2F10.1063%2F1.1692511&rft_id=info%3Abibcode%2F1969PhFl...12..485L&rft.au=Lundgren%2C+T.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-Colucci_1998-68"><span class="mw-cite-backlink"><b><a href="#cite_ref-Colucci_1998_68-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFColucci,_P.J.Jaberi,_F.AGivi,_P.Pope,_S.B.1998" class="citation journal cs1">Colucci, P.J.; Jaberi, F.A; Givi, P.; Pope, S.B. (1998). "Filtered density function for large eddy simulation of turbulent reacting flows". <i>Physics of Fluids A</i>. <b>10</b> (2): 499–515. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998PhFl...10..499C">1998PhFl...10..499C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.869537">10.1063/1.869537</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+Fluids+A&rft.atitle=Filtered+density+function+for+large+eddy+simulation+of+turbulent+reacting+flows&rft.volume=10&rft.issue=2&rft.pages=499-515&rft.date=1998&rft_id=info%3Adoi%2F10.1063%2F1.869537&rft_id=info%3Abibcode%2F1998PhFl...10..499C&rft.au=Colucci%2C+P.J.&rft.au=Jaberi%2C+F.A&rft.au=Givi%2C+P.&rft.au=Pope%2C+S.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-Fox_2003-69"><span class="mw-cite-backlink"><b><a href="#cite_ref-Fox_2003_69-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFox,_Rodney2003" class="citation book cs1">Fox, Rodney (2003). <i>Computational models for turbulent reacting flows</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-65049-6" title="Special:BookSources/978-0-521-65049-6"><bdi>978-0-521-65049-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+models+for+turbulent+reacting+flows&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-65049-6&rft.au=Fox%2C+Rodney&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-Pope_1985-70"><span class="mw-cite-backlink"><b><a href="#cite_ref-Pope_1985_70-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPope,_S.B.1985" class="citation journal cs1">Pope, S.B. (1985). "PDF methods for turbulent reactive flows". <i>Progress in Energy and Combustion Science</i>. <b>11</b> (2): 119–192. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985PrECS..11..119P">1985PrECS..11..119P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0360-1285%2885%2990002-4">10.1016/0360-1285(85)90002-4</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Progress+in+Energy+and+Combustion+Science&rft.atitle=PDF+methods+for+turbulent+reactive+flows&rft.volume=11&rft.issue=2&rft.pages=119-192&rft.date=1985&rft_id=info%3Adoi%2F10.1016%2F0360-1285%2885%2990002-4&rft_id=info%3Abibcode%2F1985PrECS..11..119P&rft.au=Pope%2C+S.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-71"><span class="mw-cite-backlink"><b><a href="#cite_ref-71">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKrueger1993" class="citation journal cs1">Krueger, Steven K. (1993). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1232081">"Linear Eddy Simulations Of Mixing In A Homogeneous Turbulent Flow"</a>. <i>Physics of Fluids</i>. <b>5</b> (4): 1023–1034. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1993PhFlA...5.1023M">1993PhFlA...5.1023M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.858667">10.1063/1.858667</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+Fluids&rft.atitle=Linear+Eddy+Simulations+Of+Mixing+In+A+Homogeneous+Turbulent+Flow&rft.volume=5&rft.issue=4&rft.pages=1023-1034&rft.date=1993&rft_id=info%3Adoi%2F10.1063%2F1.858667&rft_id=info%3Abibcode%2F1993PhFlA...5.1023M&rft.aulast=Krueger&rft.aufirst=Steven+K.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1232081&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-72"><span class="mw-cite-backlink"><b><a href="#cite_ref-72">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHirtNichols1981" class="citation journal cs1">Hirt, C.W; Nichols, B.D (January 1981). "Volume of fluid (VOF) method for the dynamics of free boundaries". <i>Journal of Computational Physics</i>. <b>39</b> (1): 201–225. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981JCoPh..39..201H">1981JCoPh..39..201H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2881%2990145-5">10.1016/0021-9991(81)90145-5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Physics&rft.atitle=Volume+of+fluid+%28VOF%29+method+for+the+dynamics+of+free+boundaries&rft.volume=39&rft.issue=1&rft.pages=201-225&rft.date=1981-01&rft_id=info%3Adoi%2F10.1016%2F0021-9991%2881%2990145-5&rft_id=info%3Abibcode%2F1981JCoPh..39..201H&rft.aulast=Hirt&rft.aufirst=C.W&rft.au=Nichols%2C+B.D&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-73"><span class="mw-cite-backlink"><b><a href="#cite_ref-73">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUnverdiTryggvason1992" class="citation journal cs1">Unverdi, Salih Ozen; Tryggvason, Grétar (May 1992). "A front-tracking method for viscous, incompressible, multi-fluid flows". <i>Journal of Computational Physics</i>. <b>100</b> (1): 25–37. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992JCoPh.100...25U">1992JCoPh.100...25U</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9991%2892%2990307-K">10.1016/0021-9991(92)90307-K</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/2027.42%2F30059">2027.42/30059</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Physics&rft.atitle=A+front-tracking+method+for+viscous%2C+incompressible%2C+multi-fluid+flows&rft.volume=100&rft.issue=1&rft.pages=25-37&rft.date=1992-05&rft_id=info%3Ahdl%2F2027.42%2F30059&rft_id=info%3Adoi%2F10.1016%2F0021-9991%2892%2990307-K&rft_id=info%3Abibcode%2F1992JCoPh.100...25U&rft.aulast=Unverdi&rft.aufirst=Salih+Ozen&rft.au=Tryggvason%2C+Gr%C3%A9tar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-74"><span class="mw-cite-backlink"><b><a href="#cite_ref-74">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenziGolubLiesen2005" class="citation journal cs1">Benzi, Michele; Golub, Gene H.; Liesen, Jörg (May 2005). "Numerical solution of saddle point problems". <i>Acta Numerica</i>. <b>14</b>: 1–137. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005AcNum..14....1B">2005AcNum..14....1B</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.409.4160">10.1.1.409.4160</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0962492904000212">10.1017/S0962492904000212</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122717775">122717775</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Numerica&rft.atitle=Numerical+solution+of+saddle+point+problems&rft.volume=14&rft.pages=1-137&rft.date=2005-05&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.409.4160%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122717775%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0962492904000212&rft_id=info%3Abibcode%2F2005AcNum..14....1B&rft.aulast=Benzi&rft.aufirst=Michele&rft.au=Golub%2C+Gene+H.&rft.au=Liesen%2C+J%C3%B6rg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-75"><span class="mw-cite-backlink"><b><a href="#cite_ref-75">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElmanHowleShadidShuttleworth2008" class="citation journal cs1">Elman, Howard; Howle, V.E.; Shadid, John; Shuttleworth, Robert; Tuminaro, Ray (January 2008). <a rel="nofollow" class="external text" href="https://digital.library.unt.edu/ark:/67531/metadc902332/">"A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations"</a>. <i>Journal of Computational Physics</i>. <b>227</b> (3): 1790–1808. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008JCoPh.227.1790E">2008JCoPh.227.1790E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jcp.2007.09.026">10.1016/j.jcp.2007.09.026</a>. <a href="/wiki/OSTI_(identifier)" class="mw-redirect" title="OSTI (identifier)">OSTI</a> <a rel="nofollow" class="external text" href="https://www.osti.gov/biblio/920807">920807</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16365489">16365489</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+Physics&rft.atitle=A+taxonomy+and+comparison+of+parallel+block+multi-level+preconditioners+for+the+incompressible+Navier%E2%80%93Stokes+equations&rft.volume=227&rft.issue=3&rft.pages=1790-1808&rft.date=2008-01&rft_id=info%3Adoi%2F10.1016%2Fj.jcp.2007.09.026&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16365489%23id-name%3DS2CID&rft_id=https%3A%2F%2Fwww.osti.gov%2Fbiblio%2F920807%23id-name%3DOSTI&rft_id=info%3Abibcode%2F2008JCoPh.227.1790E&rft.aulast=Elman&rft.aufirst=Howard&rft.au=Howle%2C+V.E.&rft.au=Shadid%2C+John&rft.au=Shuttleworth%2C+Robert&rft.au=Tuminaro%2C+Ray&rft_id=https%3A%2F%2Fdigital.library.unt.edu%2Fark%3A%2F67531%2Fmetadc902332%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-76"><span class="mw-cite-backlink"><b><a href="#cite_ref-76">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdamson2006" class="citation journal cs1">Adamson, M.R. (January 2006). "Biographies". <i>IEEE Annals of the History of Computing</i>. <b>28</b> (1): 99–103. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FMAHC.2006.5">10.1109/MAHC.2006.5</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Annals+of+the+History+of+Computing&rft.atitle=Biographies&rft.volume=28&rft.issue=1&rft.pages=99-103&rft.date=2006-01&rft_id=info%3Adoi%2F10.1109%2FMAHC.2006.5&rft.aulast=Adamson&rft.aufirst=M.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-77"><span class="mw-cite-backlink"><b><a href="#cite_ref-77">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJameson1974" class="citation journal cs1">Jameson, Antony (May 1974). "Iterative solution of transonic flows over airfoils and wings, including flows at mach 1". <i>Communications on Pure and Applied Mathematics</i>. <b>27</b> (3): 283–309. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fcpa.3160270302">10.1002/cpa.3160270302</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+on+Pure+and+Applied+Mathematics&rft.atitle=Iterative+solution+of+transonic+flows+over+airfoils+and+wings%2C+including+flows+at+mach+1&rft.volume=27&rft.issue=3&rft.pages=283-309&rft.date=1974-05&rft_id=info%3Adoi%2F10.1002%2Fcpa.3160270302&rft.aulast=Jameson&rft.aufirst=Antony&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-78"><span class="mw-cite-backlink"><b><a href="#cite_ref-78">^</a></b></span> <span class="reference-text">Borland, C.J., "XTRAN3S - Transonic Steady and Unsteady Aerodynamics for Aeroelastic Applications,"AFWAL-TR-85-3214, Air Force Wright Aeronautical Laboratories, Wright-Patterson AFB, OH, January, 1986</span> </li> <li id="cite_note-79"><span class="mw-cite-backlink"><b><a href="#cite_ref-79">^</a></b></span> <span class="reference-text">Kaufmann, T.A.S., Graefe, R., Hormes, M., Schmitz-Rode, T. and Steinseiferand, U., "Computational Fluid Dynamics in Biomedical Engineering", Computational Fluid Dynamics: Theory, Analysis and Applications, pp. 109–136</span> </li> <li id="cite_note-80"><span class="mw-cite-backlink"><b><a href="#cite_ref-80">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLaoHoltVaidhynathanSitaraman2021" class="citation arxiv cs1">Lao, Shandong; Holt, Aaron; Vaidhynathan, Deepthi; Sitaraman, Hariswaran; Hrenya, Christine M.; Hauser, Thomas (2021). "Performance comparison of CFD-DEM solver MFiX-Exa, on GPUs and CPUs". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2108.08821">2108.08821</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/cs.DC">cs.DC</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Performance+comparison+of+CFD-DEM+solver+MFiX-Exa%2C+on+GPUs+and+CPUs&rft.date=2021&rft_id=info%3Aarxiv%2F2108.08821&rft.aulast=Lao&rft.aufirst=Shandong&rft.au=Holt%2C+Aaron&rft.au=Vaidhynathan%2C+Deepthi&rft.au=Sitaraman%2C+Hariswaran&rft.au=Hrenya%2C+Christine+M.&rft.au=Hauser%2C+Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-81"><span class="mw-cite-backlink"><b><a href="#cite_ref-81">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWuTruongYukselHoetzlein2018" class="citation journal cs1">Wu, Kui; Truong, Nghia; Yuksel, Cem; Hoetzlein, Rama (May 2018). "Fast Fluid Simulations with Sparse Volumes on the GPU". <i>Computer Graphics Forum</i>. <b>37</b> (2): 157–167. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fcgf.13350">10.1111/cgf.13350</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:43945038">43945038</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computer+Graphics+Forum&rft.atitle=Fast+Fluid+Simulations+with+Sparse+Volumes+on+the+GPU&rft.volume=37&rft.issue=2&rft.pages=157-167&rft.date=2018-05&rft_id=info%3Adoi%2F10.1111%2Fcgf.13350&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A43945038%23id-name%3DS2CID&rft.aulast=Wu&rft.aufirst=Kui&rft.au=Truong%2C+Nghia&rft.au=Yuksel%2C+Cem&rft.au=Hoetzlein%2C+Rama&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> <li id="cite_note-82"><span class="mw-cite-backlink"><b><a href="#cite_ref-82">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.nvidia.com/content/intersect-360-HPC-application-support.pdf">"Intersect 360 HPC application Support"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Intersect+360+HPC+application+Support&rft_id=http%3A%2F%2Fwww.nvidia.com%2Fcontent%2Fintersect-360-HPC-application-support.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=29" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson1995" class="citation book cs1">Anderson, John D. (1995). <i>Computational Fluid Dynamics: The Basics With Applications</i>. Science/Engineering/Math. McGraw-Hill Science. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-001685-9" title="Special:BookSources/978-0-07-001685-9"><bdi>978-0-07-001685-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Fluid+Dynamics%3A+The+Basics+With+Applications&rft.series=Science%2FEngineering%2FMath&rft.pub=McGraw-Hill+Science&rft.date=1995&rft.isbn=978-0-07-001685-9&rft.aulast=Anderson&rft.aufirst=John+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPatankar1980" class="citation book cs1">Patankar, Suhas (1980). <i>Numerical Heat Transfer and Fluid Flow</i>. Hemisphere Series on Computational Methods in Mechanics and Thermal Science. Taylor & Francis. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89116-522-4" title="Special:BookSources/978-0-89116-522-4"><bdi>978-0-89116-522-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+Heat+Transfer+and+Fluid+Flow&rft.series=Hemisphere+Series+on+Computational+Methods+in+Mechanics+and+Thermal+Science&rft.pub=Taylor+%26+Francis&rft.date=1980&rft.isbn=978-0-89116-522-4&rft.aulast=Patankar&rft.aufirst=Suhas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComputational+fluid+dynamics" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Computational_fluid_dynamics&action=edit&section=30" title="Edit section: External links"><span>edit</span></a><span 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