CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Reynolds number - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"59bb25fc-a815-4ca0-acbe-35c75cbabdd4","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Reynolds_number","wgTitle":"Reynolds number","wgCurRevisionId":1260072694,"wgRevisionId":1260072694,"wgArticleId":23868856,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Pages using multiple image with auto scaled images","Articles needing additional references from February 2014","All articles needing additional references","Articles needing additional references from January 2019","Articles needing additional references from December 2014","All articles with unsourced statements","Articles with unsourced statements from March 2021", "Commons category link is on Wikidata","Aerodynamics","Convection","Dimensionless numbers","Dimensionless numbers of fluid mechanics","Dimensionless numbers of thermodynamics","Fluid dynamics","Piping"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Reynolds_number","wgRelevantArticleId":23868856,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":50000,"wgRelatedArticlesCompat":[], "wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q178932","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage": "ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/0/03/Laminar-turbulent_transition.jpg/1200px-Laminar-turbulent_transition.jpg"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1546"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/0/03/Laminar-turbulent_transition.jpg/800px-Laminar-turbulent_transition.jpg"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="1031"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/0/03/Laminar-turbulent_transition.jpg/640px-Laminar-turbulent_transition.jpg"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="825"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Reynolds number - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Reynolds_number"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Reynolds_number&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Reynolds_number"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Reynolds_number rootpage-Reynolds_number skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Reynolds+number" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Reynolds+number" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Reynolds+number" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Reynolds+number" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definition</span> </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Derivation</span> </div> </a> <button aria-controls="toc-Derivation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Derivation subsection</span> </button> <ul id="toc-Derivation-sublist" class="vector-toc-list"> <li id="toc-Alternative_derivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternative_derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Alternative derivation</span> </div> </a> <ul id="toc-Alternative_derivation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flow_in_a_pipe" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Flow_in_a_pipe"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Flow in a pipe</span> </div> </a> <ul id="toc-Flow_in_a_pipe-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Laminar–turbulent_transition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Laminar–turbulent_transition"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Laminar–turbulent transition</span> </div> </a> <ul id="toc-Laminar–turbulent_transition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flow_in_a_wide_duct" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Flow_in_a_wide_duct"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Flow in a wide duct</span> </div> </a> <ul id="toc-Flow_in_a_wide_duct-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flow_in_an_open_channel" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Flow_in_an_open_channel"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Flow in an open channel</span> </div> </a> <ul id="toc-Flow_in_an_open_channel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Flow_around_airfoils" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Flow_around_airfoils"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Flow around airfoils</span> </div> </a> <ul id="toc-Flow_around_airfoils-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Object_in_a_fluid" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Object_in_a_fluid"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Object in a fluid</span> </div> </a> <button aria-controls="toc-Object_in_a_fluid-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Object in a fluid subsection</span> </button> <ul id="toc-Object_in_a_fluid-sublist" class="vector-toc-list"> <li id="toc-In_viscous_fluids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_viscous_fluids"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>In viscous fluids</span> </div> </a> <ul id="toc-In_viscous_fluids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sphere_in_a_fluid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sphere_in_a_fluid"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Sphere in a fluid</span> </div> </a> <ul id="toc-Sphere_in_a_fluid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rectangular_object_in_a_fluid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rectangular_object_in_a_fluid"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.3</span> <span>Rectangular object in a fluid</span> </div> </a> <ul id="toc-Rectangular_object_in_a_fluid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fall_velocity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fall_velocity"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.4</span> <span>Fall velocity</span> </div> </a> <ul id="toc-Fall_velocity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Packed_bed" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Packed_bed"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.5</span> <span>Packed bed</span> </div> </a> <ul id="toc-Packed_bed-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stirred_vessel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stirred_vessel"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.6</span> <span>Stirred vessel</span> </div> </a> <ul id="toc-Stirred_vessel-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Pipe_friction" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Pipe_friction"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Pipe friction</span> </div> </a> <ul id="toc-Pipe_friction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Similarity_of_flows" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Similarity_of_flows"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Similarity of flows</span> </div> </a> <ul id="toc-Similarity_of_flows-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Smallest_scales_of_turbulent_motion" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Smallest_scales_of_turbulent_motion"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Smallest scales of turbulent motion</span> </div> </a> <ul id="toc-Smallest_scales_of_turbulent_motion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_physiology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#In_physiology"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>In physiology</span> </div> </a> <ul id="toc-In_physiology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_systems" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Complex_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Complex systems</span> </div> </a> <ul id="toc-Complex_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationship_to_other_dimensionless_parameters" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relationship_to_other_dimensionless_parameters"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Relationship to other dimensionless parameters</span> </div> </a> <ul id="toc-Relationship_to_other_dimensionless_parameters-sublist" class="vector-toc-list"> </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">16</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">17</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">17.1</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">17.2</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">17.3</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">17.4</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </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">18</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">Reynolds number</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 61 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-61" 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">61 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Reynoldsgetal" title="Reynoldsgetal – Afrikaans" lang="af" hreflang="af" data-title="Reynoldsgetal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%B1%D9%8A%D9%86%D9%88%D9%84%D8%AF%D8%B2" title="عدد رينولدز – Arabic" lang="ar" hreflang="ar" data-title="عدد رينولدز" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/N%C3%BAmberu_de_Reynolds" title="Númberu de Reynolds – Asturian" lang="ast" hreflang="ast" data-title="Númberu de Reynolds" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Reynolds_%C9%99d%C9%99di" title="Reynolds ədədi – Azerbaijani" lang="az" hreflang="az" data-title="Reynolds ədədi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B0%E0%A7%87%E0%A6%A8%E0%A6%B2%E0%A7%8D%E0%A6%A1%E0%A6%B8_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE" title="রেনল্ডস সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="রেনল্ডস সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Reynolds_s%C3%B2%CD%98" title="Reynolds sò͘ – Minnan" lang="nan" hreflang="nan" data-title="Reynolds sò͘" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9B%D1%96%D0%BA_%D0%A0%D1%8D%D0%B9%D0%BD%D0%B0%D0%BB%D1%8C%D0%B4%D1%81%D0%B0" title="Лік Рэйнальдса – Belarusian" lang="be" hreflang="be" data-title="Лік Рэйнальдса" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9B%D1%96%D0%BA_%D0%A0%D1%8D%D0%B9%D0%BD%D0%B0%D0%BB%D1%8C%D0%B4%D1%81%D0%B0" title="Лік Рэйнальдса – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Лік Рэйнальдса" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D1%80%D0%B8%D1%82%D0%B5%D1%80%D0%B8%D0%B9_%D0%BD%D0%B0_%D0%A0%D0%B5%D0%B9%D0%BD%D0%BE%D0%BB%D0%B4%D1%81" title="Критерий на Рейнолдс – Bulgarian" lang="bg" hreflang="bg" data-title="Критерий на Рейнолдс" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Reynoldsov_broj" title="Reynoldsov broj – Bosnian" lang="bs" hreflang="bs" data-title="Reynoldsov broj" 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/Nombre_de_Reynolds" title="Nombre de Reynolds – Catalan" lang="ca" hreflang="ca" data-title="Nombre de Reynolds" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B9%D0%BD%D0%BE%D0%BB%D1%8C%D0%B4%D1%81_%D1%85%D0%B8%D1%81%D0%B5%D0%BF%C4%95" title="Рейнольдс хисепĕ – Chuvash" lang="cv" hreflang="cv" data-title="Рейнольдс хисепĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Reynoldsovo_%C4%8D%C3%ADslo" title="Reynoldsovo číslo – Czech" lang="cs" hreflang="cs" data-title="Reynoldsovo číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Reynolds_tal" title="Reynolds tal – Danish" lang="da" hreflang="da" data-title="Reynolds tal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Reynolds-Zahl" title="Reynolds-Zahl – German" lang="de" hreflang="de" data-title="Reynolds-Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Reynoldsi_arv" title="Reynoldsi arv – Estonian" lang="et" hreflang="et" data-title="Reynoldsi arv" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82_%CE%A1%CE%AD%CE%B9%CE%BD%CE%BF%CE%BB%CE%BD%CF%84%CF%82" title="Αριθμός Ρέινολντς – Greek" lang="el" hreflang="el" data-title="Αριθμός Ρέινολντς" 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/N%C3%BAmero_de_Reynolds" title="Número de Reynolds – Spanish" lang="es" hreflang="es" data-title="Número de Reynolds" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Nombro_de_Reynolds" title="Nombro de Reynolds – Esperanto" lang="eo" hreflang="eo" data-title="Nombro de Reynolds" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Reynolds_zenbakia" title="Reynolds zenbakia – Basque" lang="eu" hreflang="eu" data-title="Reynolds zenbakia" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%B1%DB%8C%D9%86%D9%88%D9%84%D8%AF%D8%B2" title="عدد رینولدز – Persian" lang="fa" hreflang="fa" data-title="عدد رینولدز" 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/Nombre_de_Reynolds" title="Nombre de Reynolds – French" lang="fr" hreflang="fr" data-title="Nombre de Reynolds" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Uimhir_Reynolds" title="Uimhir Reynolds – Irish" lang="ga" hreflang="ga" data-title="Uimhir Reynolds" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/N%C3%BAmero_de_Reynolds" title="Número de Reynolds – Galician" lang="gl" hreflang="gl" data-title="Número de Reynolds" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A0%88%EC%9D%B4%EB%86%80%EC%A6%88_%EC%88%98" title="레이놀즈 수 – Korean" lang="ko" hreflang="ko" data-title="레이놀즈 수" 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%B0%E0%A5%87%E0%A4%A8%E0%A4%BE%E0%A4%B2%E0%A5%8D%E0%A4%A1_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="रेनाल्ड संख्या – Hindi" lang="hi" hreflang="hi" data-title="रेनाल्ड संख्या" 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/Reynoldsov_broj" title="Reynoldsov broj – Croatian" lang="hr" hreflang="hr" data-title="Reynoldsov broj" 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/Bilangan_Reynolds" title="Bilangan Reynolds – Indonesian" lang="id" hreflang="id" data-title="Bilangan Reynolds" 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/Numero_di_Reynolds" title="Numero di Reynolds – Italian" lang="it" hreflang="it" data-title="Numero di Reynolds" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_%D7%A8%D7%99%D7%99%D7%A0%D7%95%D7%9C%D7%93%D7%A1" title="מספר ריינולדס – Hebrew" lang="he" hreflang="he" data-title="מספר ריינולדס" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A0%E1%83%94%E1%83%98%E1%83%9C%E1%83%9D%E1%83%9A%E1%83%93%E1%83%A1%E1%83%98%E1%83%A1_%E1%83%A0%E1%83%98%E1%83%AA%E1%83%AE%E1%83%95%E1%83%98" title="რეინოლდსის რიცხვი – Georgian" lang="ka" hreflang="ka" data-title="რეინოლდსის რიცხვი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A0%D0%B5%D0%B9%D0%BD%D0%BE%D0%BB%D1%8C%D0%B4%D1%81_%D1%81%D0%B0%D0%BD%D1%8B" title="Рейнольдс саны – Kazakh" lang="kk" hreflang="kk" data-title="Рейнольдс саны" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Jimara_Reynolds" title="Jimara Reynolds – Kurdish" lang="ku" hreflang="ku" data-title="Jimara Reynolds" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Reinoldsa_skaitlis" title="Reinoldsa skaitlis – Latvian" lang="lv" hreflang="lv" data-title="Reinoldsa skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Reinoldso_skai%C4%8Dius" title="Reinoldso skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Reinoldso skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Numer_del_Reynolds" title="Numer del Reynolds – Lombard" lang="lmo" hreflang="lmo" data-title="Numer del Reynolds" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Reynolds-sz%C3%A1m" title="Reynolds-szám – Hungarian" lang="hu" hreflang="hu" data-title="Reynolds-szám" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B1%E0%B5%86%E0%B4%AF%E0%B5%8D%E0%B4%A8%E0%B5%8B%E0%B5%BE%E0%B4%A1%E0%B5%8D%E0%B4%B8%E0%B5%8D_%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF" title="റെയ്നോൾഡ്സ് സംഖ്യ – Malayalam" lang="ml" hreflang="ml" data-title="റെയ്നോൾഡ്സ് സംഖ്യ" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Reynoldsgetal" title="Reynoldsgetal – Dutch" lang="nl" hreflang="nl" data-title="Reynoldsgetal" 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/%E3%83%AC%E3%82%A4%E3%83%8E%E3%83%AB%E3%82%BA%E6%95%B0" title="レイノルズ数 – Japanese" lang="ja" hreflang="ja" data-title="レイノルズ数" 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/Reynoldstall" title="Reynoldstall – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Reynoldstall" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Reynoldstalet" title="Reynoldstalet – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Reynoldstalet" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_de_Reynolds" title="Nombre de Reynolds – Occitan" lang="oc" hreflang="oc" data-title="Nombre de Reynolds" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Reynolds_soni_va_kelib_chiqishi" title="Reynolds soni va kelib chiqishi – Uzbek" lang="uz" hreflang="uz" data-title="Reynolds soni va kelib chiqishi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczba_Reynoldsa" title="Liczba Reynoldsa – Polish" lang="pl" hreflang="pl" data-title="Liczba Reynoldsa" 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/Coeficiente_de_Reynolds" title="Coeficiente de Reynolds – Portuguese" lang="pt" hreflang="pt" data-title="Coeficiente de Reynolds" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_Reynolds" title="Număr Reynolds – Romanian" lang="ro" hreflang="ro" data-title="Număr Reynolds" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%A0%D0%B5%D0%B9%D0%BD%D0%BE%D0%BB%D1%8C%D0%B4%D1%81%D0%B0" title="Число Рейнольдса – Russian" lang="ru" hreflang="ru" data-title="Число Рейнольдса" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Reynoldsovo_%C4%8D%C3%ADslo" title="Reynoldsovo číslo – Slovak" lang="sk" hreflang="sk" data-title="Reynoldsovo číslo" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Reynoldsovo_%C5%A1tevilo" title="Reynoldsovo število – Slovenian" lang="sl" hreflang="sl" data-title="Reynoldsovo število" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A0%D0%B5%D1%98%D0%BD%D0%BE%D0%BB%D0%B4%D1%81%D0%BE%D0%B2_%D0%B1%D1%80%D0%BE%D1%98" title="Рејнолдсов број – Serbian" lang="sr" hreflang="sr" data-title="Рејнолдсов број" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Reynoldsov_broj" title="Reynoldsov broj – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Reynoldsov broj" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Reynoldsin_luku" title="Reynoldsin luku – Finnish" lang="fi" hreflang="fi" data-title="Reynoldsin luku" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Reynoldstal" title="Reynoldstal – Swedish" lang="sv" hreflang="sv" data-title="Reynoldstal" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%B0%E0%AF%87%E0%AE%A9%E0%AE%BE%E0%AE%B2%E0%AF%8D%E0%AE%9F%E0%AF%8D%E0%AE%9A%E0%AF%81_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="இரேனால்ட்சு எண் – Tamil" lang="ta" hreflang="ta" data-title="இரேனால்ட்சு எண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%90%D0%B4%D0%B0%D0%B4%D0%B8_%D0%A0%D0%B5%D0%B9%D0%BD%D0%BE%D0%BB%D0%B4%D1%81" title="Адади Рейнолдс – Tajik" lang="tg" hreflang="tg" data-title="Адади Рейнолдс" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Reynolds_say%C4%B1s%C4%B1" title="Reynolds sayısı – Turkish" lang="tr" hreflang="tr" data-title="Reynolds sayısı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%A0%D0%B5%D0%B9%D0%BD%D0%BE%D0%BB%D1%8C%D0%B4%D1%81%D0%B0" title="Число Рейнольдса – Ukrainian" lang="uk" hreflang="uk" data-title="Число Рейнольдса" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/S%E1%BB%91_Reynolds" title="Số Reynolds – Vietnamese" lang="vi" hreflang="vi" data-title="Số Reynolds" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%9B%B7%E8%AF%BA%E6%95%B0" title="雷诺数 – Wu" lang="wuu" hreflang="wuu" data-title="雷诺数" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9B%B7%E8%AF%BA%E6%95%B0" title="雷诺数 – Chinese" lang="zh" hreflang="zh" data-title="雷诺数" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q178932#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Reynolds_number" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Reynolds_number" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Reynolds_number"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Reynolds_number&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Reynolds_number&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Reynolds_number"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Reynolds_number&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Reynolds_number&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Reynolds_number" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Reynolds_number" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Reynolds_number&oldid=1260072694" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Reynolds_number&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Reynolds_number&id=1260072694&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FReynolds_number"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FReynolds_number"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Reynolds_number&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Reynolds_number&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Reynolds_number" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q178932" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Ratio of inertial to viscous forces acting on a liquid</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Laminar-turbulent_transition.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Laminar-turbulent_transition.jpg/220px-Laminar-turbulent_transition.jpg" decoding="async" width="220" height="283" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Laminar-turbulent_transition.jpg/330px-Laminar-turbulent_transition.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Laminar-turbulent_transition.jpg/440px-Laminar-turbulent_transition.jpg 2x" data-file-width="2152" data-file-height="2773" /></a><figcaption>The plume from this candle flame transitions from laminar to turbulent.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:242px;max-width:242px"><div class="trow"><div class="tsingle" style="width:132px;max-width:132px"><div class="thumbimage" style="height:174px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Tap-gb0fe5d2cc_1920.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Tap-gb0fe5d2cc_1920.jpg/130px-Tap-gb0fe5d2cc_1920.jpg" decoding="async" width="130" height="175" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Tap-gb0fe5d2cc_1920.jpg/195px-Tap-gb0fe5d2cc_1920.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Tap-gb0fe5d2cc_1920.jpg/260px-Tap-gb0fe5d2cc_1920.jpg 2x" data-file-width="1428" data-file-height="1920" /></a></span></div></div><div class="tsingle" style="width:106px;max-width:106px"><div class="thumbimage" style="height:174px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Clean_water_for_a_village_in_West_Lombok_(10686572086).jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Clean_water_for_a_village_in_West_Lombok_%2810686572086%29.jpg/104px-Clean_water_for_a_village_in_West_Lombok_%2810686572086%29.jpg" decoding="async" width="104" height="174" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Clean_water_for_a_village_in_West_Lombok_%2810686572086%29.jpg/156px-Clean_water_for_a_village_in_West_Lombok_%2810686572086%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Clean_water_for_a_village_in_West_Lombok_%2810686572086%29.jpg/208px-Clean_water_for_a_village_in_West_Lombok_%2810686572086%29.jpg 2x" data-file-width="896" data-file-height="1500" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">The transition from laminar (left) to turbulent (right) flow of water from a tap occurs as the Reynolds number increases.</div></div></div></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Vortex-street-animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Vortex-street-animation.gif/220px-Vortex-street-animation.gif" decoding="async" width="220" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Vortex-street-animation.gif/330px-Vortex-street-animation.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b4/Vortex-street-animation.gif 2x" data-file-width="400" data-file-height="200" /></a><figcaption>A <a href="/wiki/Vortex_street" class="mw-redirect" title="Vortex street">vortex street</a> around a cylinder. This can occur around cylinders and spheres, for any fluid, cylinder size, and fluid speed provided that it has a Reynolds number between roughly 40 and 1000.<sup id="cite_ref-FOOTNOTETansleyMarshall20013274–3283_1-0" class="reference"><a href="#cite_note-FOOTNOTETansleyMarshall20013274–3283-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Ggstokes.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Ggstokes.jpg/220px-Ggstokes.jpg" decoding="async" width="220" height="299" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/a/ad/Ggstokes.jpg 1.5x" data-file-width="329" data-file-height="447" /></a><figcaption><a href="/wiki/Sir_George_Stokes,_1st_Baronet" title="Sir George Stokes, 1st Baronet">George Stokes</a> introduced Reynolds numbers.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Osborne_Reynolds.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/d/df/Osborne_Reynolds.jpg" decoding="async" width="120" height="159" class="mw-file-element" data-file-width="120" data-file-height="159" /></a><figcaption><a href="/wiki/Osborne_Reynolds" title="Osborne Reynolds">Osborne Reynolds</a> popularised the concept.</figcaption></figure> <p>In <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, the <b>Reynolds number</b> (<b><span class="texhtml">Re</span></b>) is a <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless quantity</a> that helps predict <a href="/wiki/Fluid_flow" class="mw-redirect" title="Fluid flow">fluid flow</a> patterns in different situations by measuring the ratio between <a href="/wiki/Inertia" title="Inertia">inertial</a> and <a href="/wiki/Viscous" class="mw-redirect" title="Viscous">viscous</a> forces.<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> At low Reynolds numbers, flows tend to be dominated by <a href="/wiki/Laminar_flow" title="Laminar flow">laminar (sheet-like) flow</a>, while at high Reynolds numbers, flows tend to be <a href="/wiki/Turbulence" title="Turbulence">turbulent</a>. The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow (<a href="/wiki/Eddy_(fluid_dynamics)" title="Eddy (fluid dynamics)">eddy currents</a>). These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of <a href="/wiki/Cavitation" title="Cavitation">cavitation</a>. </p><p>The Reynolds number has wide applications, ranging from liquid flow in a pipe to the passage of air over an aircraft wing. It is used to predict the transition from <a href="/wiki/Laminar%E2%80%93turbulent_transition" title="Laminar–turbulent transition">laminar to turbulent</a> flow and is used in the scaling of similar but different-sized flow situations, such as between an aircraft model in a wind tunnel and the full-size version. The predictions of the onset of turbulence and the ability to calculate scaling effects can be used to help predict fluid behavior on a larger scale, such as in local or global air or water movement, and thereby the associated meteorological and climatological effects. </p><p>The concept was introduced by <a href="/wiki/Sir_George_Stokes,_1st_Baronet" title="Sir George Stokes, 1st Baronet">George Stokes</a> in 1851,<sup id="cite_ref-FOOTNOTEStokes18518–106_3-0" class="reference"><a href="#cite_note-FOOTNOTEStokes18518–106-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> but the Reynolds number was named by <a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Arnold Sommerfeld</a> in 1908<sup id="cite_ref-FOOTNOTESommerfeld1908116–124_4-0" class="reference"><a href="#cite_note-FOOTNOTESommerfeld1908116–124-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> after <a href="/wiki/Osborne_Reynolds" title="Osborne Reynolds">Osborne Reynolds</a> who popularized its use in 1883.<sup id="cite_ref-FOOTNOTEReynolds1883935–982_5-0" class="reference"><a href="#cite_note-FOOTNOTEReynolds1883935–982-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTERott19901–11_6-0" class="reference"><a href="#cite_note-FOOTNOTERott19901–11-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> (cf. <a href="/wiki/List_of_examples_of_Stigler%27s_law" title="List of examples of Stigler's law">this list</a>) </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Reynolds number is the <a href="/wiki/Ratio" title="Ratio">ratio</a> of <a href="/wiki/Inertia" title="Inertia">inertial</a> forces to <a href="/wiki/Viscous" class="mw-redirect" title="Viscous">viscous</a> forces within a fluid that is subjected to relative internal movement due to different fluid velocities. A region where these forces change behavior is known as a <a href="/wiki/Boundary_layer" title="Boundary layer">boundary layer</a>, such as the bounding surface in the interior of a pipe. A similar effect is created by the introduction of a stream of high-velocity fluid into a low-velocity fluid, such as the hot gases emitted from a flame in air. This relative movement generates fluid friction, which is a factor in developing turbulent flow. Counteracting this effect is the <a href="/wiki/Viscosity" title="Viscosity">viscosity</a> of the fluid, which tends to inhibit turbulence. The Reynolds number quantifies the relative importance of these two types of forces for given flow conditions and is a guide to when turbulent flow will occur in a particular situation.<sup id="cite_ref-FOOTNOTEFalkovich2018_7-0" class="reference"><a href="#cite_note-FOOTNOTEFalkovich2018-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>This ability to predict the onset of turbulent flow is an important design tool for equipment such as piping systems or aircraft wings, but the Reynolds number is also used in scaling of fluid dynamics problems and is used to determine <a href="/wiki/Dynamic_similitude" class="mw-redirect" title="Dynamic similitude">dynamic similitude</a> between two different cases of fluid flow, such as between a model aircraft, and its full-size version. Such scaling is not linear and the application of Reynolds numbers to both situations allows scaling factors to be developed. </p><p>With respect to <a href="/wiki/Laminar_flow" title="Laminar flow">laminar</a> and <a href="/wiki/Turbulence" title="Turbulence">turbulent flow</a> regimes: </p> <ul><li>laminar flow occurs at low Reynolds numbers, where viscous forces are dominant, and is characterized by smooth, constant fluid motion;</li> <li>turbulent flow occurs at high Reynolds numbers and is dominated by inertial forces, which tend to produce chaotic <a href="/wiki/Eddy_(fluid_dynamics)" title="Eddy (fluid dynamics)">eddies</a>, <a href="/wiki/Vortex" title="Vortex">vortices</a> and other flow instabilities.<sup id="cite_ref-Hall2015_8-0" class="reference"><a href="#cite_note-Hall2015-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li></ul><p> The Reynolds number is defined as:<sup id="cite_ref-FOOTNOTESommerfeld1908116–124_4-1" class="reference"><a href="#cite_note-FOOTNOTESommerfeld1908116–124-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></p><div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <p><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 \mathrm {Re} ={\frac {uL}{\nu }}={\frac {\rho uL}{\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mi>L</mi> </mrow> <mi>ν</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mi>u</mi> <mi>L</mi> </mrow> <mi>μ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ={\frac {uL}{\nu }}={\frac {\rho uL}{\mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0daa369161ae16e9b6c9ff0c7ee13696e1a99da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.639ex; height:5.843ex;" alt="{\displaystyle \mathrm {Re} ={\frac {uL}{\nu }}={\frac {\rho uL}{\mu }}}"></span> </p> </div><p>where: </p><ul><li><span class="texhtml"><i>ρ</i></span> is the <a href="/wiki/Density" title="Density">density</a> of the fluid (<a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI units</a>: kg/m<sup>3</sup>)</li> <li><span class="texhtml"><i>u</i></span> is the <a href="/wiki/Flow_speed" class="mw-redirect" title="Flow speed">flow speed</a> (m/s)</li> <li><span class="texhtml"><i>L</i></span> is a <a href="/wiki/Characteristic_length" title="Characteristic length">characteristic length</a> (m)</li> <li><span class="texhtml"><i>μ</i></span> is the <a href="/wiki/Dynamic_viscosity" class="mw-redirect" title="Dynamic viscosity">dynamic viscosity</a> of the <a href="/wiki/Fluid" title="Fluid">fluid</a> (Pa·s or N·s/m<sup>2</sup> or kg/(m·s))</li> <li><span class="texhtml"><i>ν</i></span> is the <a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a> of the fluid (m<sup>2</sup>/s).</li></ul> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Brezina_equation.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Brezina_equation.jpg/220px-Brezina_equation.jpg" decoding="async" width="220" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Brezina_equation.jpg/330px-Brezina_equation.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Brezina_equation.jpg/440px-Brezina_equation.jpg 2x" data-file-width="1024" data-file-height="480" /></a><figcaption>The Brezina equation</figcaption></figure> <p>The Reynolds number can be defined for several different situations where a fluid is in relative motion to a surface.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>n 1<span class="cite-bracket">]</span></a></sup> These definitions generally include the fluid properties of density and viscosity, plus a velocity and a <a href="/wiki/Characteristic_length" title="Characteristic length">characteristic length</a> or characteristic dimension (L in the above equation). This dimension is a matter of convention—for example radius and diameter are equally valid to describe spheres or circles, but one is chosen by convention. For aircraft or ships, the length or width can be used. For flow in a pipe, or for a sphere moving in a fluid, the internal diameter is generally used today. Other shapes such as rectangular pipes or non-spherical objects have an <i><a href="/wiki/Equivalent_spherical_diameter" title="Equivalent spherical diameter">equivalent diameter</a></i> defined. For fluids of variable density such as compressible gases or fluids of variable viscosity such as <a href="/wiki/Non-Newtonian_fluid" title="Non-Newtonian fluid">non-Newtonian fluids</a>, special rules apply. The velocity may also be a matter of convention in some circumstances, notably stirred vessels. </p><p>In practice, matching the Reynolds number is not on its own sufficient to guarantee similitude. Fluid flow is generally chaotic, and very small changes to shape and surface roughness of bounding surfaces can result in very different flows. Nevertheless, Reynolds numbers are a very important guide and are widely used. </p> <div class="mw-heading mw-heading2"><h2 id="Derivation">Derivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=2" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If we know that the relevant physical quantities in a physical system are only <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 ,u,L,\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>L</mi> <mo>,</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ,u,L,\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135b950683f1779c22fb25793a2c128bbd209727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.618ex; height:2.676ex;" alt="{\displaystyle \rho ,u,L,\mu }"></span>, then the Reynolds number is essentially fixed by the <a href="/wiki/Buckingham_%CF%80_theorem" title="Buckingham π theorem">Buckingham π theorem</a>. </p><p>In detail, since there are 4 quantities <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 ,u,L,\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>L</mi> <mo>,</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ,u,L,\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135b950683f1779c22fb25793a2c128bbd209727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.618ex; height:2.676ex;" alt="{\displaystyle \rho ,u,L,\mu }"></span>, but they have only 3 dimensions (length, time, mass), we can consider <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 ^{x_{1}}u^{x_{2}}L^{x_{3}}\mu ^{x_{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{x_{1}}u^{x_{2}}L^{x_{3}}\mu ^{x_{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b02e8b8b167b5e3bb926643fcaf59f943ab34129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.532ex; height:2.843ex;" alt="{\displaystyle \rho ^{x_{1}}u^{x_{2}}L^{x_{3}}\mu ^{x_{4}}}"></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 x_{1},...,x_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},...,x_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36b543583ff7b9b545d05a2aaf2c95ac87424cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.938ex; height:2.009ex;" alt="{\displaystyle x_{1},...,x_{4}}"></span> are real numbers. Setting the three dimensions 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 \rho ^{x_{1}}u^{x_{2}}L^{x_{3}}\mu ^{x_{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msup> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ^{x_{1}}u^{x_{2}}L^{x_{3}}\mu ^{x_{4}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b02e8b8b167b5e3bb926643fcaf59f943ab34129" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.532ex; height:2.843ex;" alt="{\displaystyle \rho ^{x_{1}}u^{x_{2}}L^{x_{3}}\mu ^{x_{4}}}"></span> to zero, we obtain 3 independent linear constraints, so the solution space has 1 dimension, and it is spanned by the vector <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 (1,1,1,-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,1,1,-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57a61f6044dbc85ae3526b3336376b62eb2d30a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.369ex; height:2.843ex;" alt="{\displaystyle (1,1,1,-1)}"></span>. </p><p>Thus, any dimensionless quantity constructed out 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 \rho ,u,L,\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>,</mo> <mi>u</mi> <mo>,</mo> <mi>L</mi> <mo>,</mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ,u,L,\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/135b950683f1779c22fb25793a2c128bbd209727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.618ex; height:2.676ex;" alt="{\displaystyle \rho ,u,L,\mu }"></span> is a function 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 \rho uL\mu ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mi>u</mi> <mi>L</mi> <msup> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho uL\mu ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ec585d97aa16727824c3d07bf42650f60658e41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.849ex; height:3.176ex;" alt="{\displaystyle \rho uL\mu ^{-1}}"></span>, the Reynolds number. </p><p>Alternatively, we can take the <a href="/wiki/Navier%E2%80%93Stokes_equations#Incompressible_flow" title="Navier–Stokes equations">incompressible Navier–Stokes equations (convective form)</a>:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\mathbf {g} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>−</mo> <mi>ν</mi> <mspace width="thinmathspace" /> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\mathbf {g} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38ded2e85fb75e28745a8007230e52fde357a07" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.037ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-{\frac {1}{\rho }}\nabla p+\mathbf {g} }"></span>Remove the gravity term <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 {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9129915a2c7563a0aa21b75e6d87f3b62328b55c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.337ex; height:2.009ex;" alt="{\displaystyle {\mathbf {g}}}"></span>, then the left side consists of inertial force <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 \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f773bada7a2887b3b62e2d7ea2340f546d3abc64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.875ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} }"></span>, and viscous force <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 \nu \,\nabla ^{2}\mathbf {u} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mspace width="thinmathspace" /> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu \,\nabla ^{2}\mathbf {u} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662e8fde35a8dd36b5b4f0366b9c574b044c10ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.095ex; height:2.676ex;" alt="{\displaystyle \nu \,\nabla ^{2}\mathbf {u} }"></span>. Their ratio has the order 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 {\frac {(\mathbf {u} \cdot \nabla )\mathbf {u} }{\nu \,\nabla ^{2}\mathbf {u} }}\sim {\frac {u^{2}/L}{\nu u/L^{2}}}={\frac {uL}{\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> <mrow> <mi>ν</mi> <mspace width="thinmathspace" /> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mrow> </mfrac> </mrow> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mrow> <mrow> <mi>ν</mi> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <mi>L</mi> </mrow> <mi>ν</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(\mathbf {u} \cdot \nabla )\mathbf {u} }{\nu \,\nabla ^{2}\mathbf {u} }}\sim {\frac {u^{2}/L}{\nu u/L^{2}}}={\frac {uL}{\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13197ba6ff5c6f899fc2572ccd2aac1a3e074b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.374ex; height:6.676ex;" alt="{\displaystyle {\frac {(\mathbf {u} \cdot \nabla )\mathbf {u} }{\nu \,\nabla ^{2}\mathbf {u} }}\sim {\frac {u^{2}/L}{\nu u/L^{2}}}={\frac {uL}{\nu }}}"></span>, the Reynolds number. This argument is written out in detail on the <a href="/wiki/Scallop_theorem" title="Scallop theorem">Scallop theorem</a> page. </p> <div class="mw-heading mw-heading3"><h3 id="Alternative_derivation">Alternative derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=3" title="Edit section: Alternative derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Reynolds number can be obtained when one uses the <a href="/wiki/Nondimensional_number" class="mw-redirect" title="Nondimensional number">nondimensional</a> form of the incompressible <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a> for a newtonian fluid expressed in terms of the <a href="/wiki/Lagrangian_derivative" class="mw-redirect" title="Lagrangian derivative">Lagrangian derivative</a>: </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 \rho {\frac {D\mathbf {v} }{Dt}}=-\nabla p+\mu \nabla ^{2}\mathbf {v} +\rho \mathbf {f} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mi>μ</mi> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-\nabla p+\mu \nabla ^{2}\mathbf {v} +\rho \mathbf {f} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbe58ab56319bf38ed34fad5a8d2d6ceafbdc526" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.771ex; height:5.176ex;" alt="{\displaystyle \rho {\frac {D\mathbf {v} }{Dt}}=-\nabla p+\mu \nabla ^{2}\mathbf {v} +\rho \mathbf {f} .}"></span></dd></dl> <p>Each term in the above equation has the units of a "body force" (force per unit volume) with the same dimensions of a density times an acceleration. Each term is thus dependent on the exact measurements of a flow. When one renders the equation nondimensional, that is when we multiply it by a factor with inverse units of the base equation, we obtain a form that does not depend directly on the physical sizes. One possible way to obtain a nondimensional equation is to multiply the whole equation by the factor </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 {L}{\rho V^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mrow> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {L}{\rho V^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/112605e76f10269a0ddab5134afac99d59599b93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:5.656ex; height:5.843ex;" alt="{\displaystyle {\frac {L}{\rho V^{2}}},}"></span></dd></dl> <p>where </p> <ul><li><span class="texhtml"><i>V</i></span> is the mean velocity, <span class="texhtml"><i>v</i></span> or <span class="texhtml"><span style="text-decoration:overline;"><b>v</b></span></span>, relative to the fluid (m/s),</li> <li><span class="texhtml"><i>L</i></span> is the characteristic length (m),</li> <li><span class="texhtml"><i>ρ</i></span> is the fluid density (kg/m<sup>3</sup>).</li></ul> <p>If we now set </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 {\begin{aligned}\mathbf {v} '&={\frac {\mathbf {v} }{V}},&p'&=p{\frac {1}{\rho V^{2}}},&\mathbf {f} '&=\mathbf {f} {\frac {L}{V^{2}}},&{\frac {\partial }{\partial t'}}&={\frac {L}{V}}{\frac {\partial }{\partial t}},&\nabla '&=L\nabla ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mi>V</mi> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <msup> <mi>p</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ρ</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>V</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> <mtd> <msup> <mi mathvariant="normal">∇</mi> <mo>′</mo> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>L</mi> <mi mathvariant="normal">∇</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {v} '&={\frac {\mathbf {v} }{V}},&p'&=p{\frac {1}{\rho V^{2}}},&\mathbf {f} '&=\mathbf {f} {\frac {L}{V^{2}}},&{\frac {\partial }{\partial t'}}&={\frac {L}{V}}{\frac {\partial }{\partial t}},&\nabla '&=L\nabla ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7926d41073b9ed8f00f6603d7b0a756cee84650" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:72.915ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {v} '&={\frac {\mathbf {v} }{V}},&p'&=p{\frac {1}{\rho V^{2}}},&\mathbf {f} '&=\mathbf {f} {\frac {L}{V^{2}}},&{\frac {\partial }{\partial t'}}&={\frac {L}{V}}{\frac {\partial }{\partial t}},&\nabla '&=L\nabla ,\end{aligned}}}"></span></dd></dl> <p>we can rewrite the Navier–Stokes equation without dimensions: </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 {D\mathbf {v} '}{Dt'}}=-\nabla 'p'+{\frac {\mu }{\rho LV}}\nabla '^{2}\mathbf {v} '+\mathbf {f} ',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>′</mo> </msup> </mrow> <mrow> <mi>D</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mo>′</mo> </msup> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <mrow> <mi>ρ</mi> <mi>L</mi> <mi>V</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow> <mo class="MJX-variant">′</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>′</mo> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D\mathbf {v} '}{Dt'}}=-\nabla 'p'+{\frac {\mu }{\rho LV}}\nabla '^{2}\mathbf {v} '+\mathbf {f} ',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6929d88f40fe74544cc86f8f21a1d6bde2e941" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.321ex; height:6.009ex;" alt="{\displaystyle {\frac {D\mathbf {v} '}{Dt'}}=-\nabla 'p'+{\frac {\mu }{\rho LV}}\nabla '^{2}\mathbf {v} '+\mathbf {f} ',}"></span></dd></dl> <p>where the term <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>μ</i></span><span class="sr-only">/</span><span class="den"><i>ρLV</i></span></span>⁠</span> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">Re</span></span>⁠</span></span>. </p><p>Finally, dropping the primes for ease of reading: </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 {D\mathbf {v} }{Dt}}=-\nabla p+{\frac {1}{\mathrm {Re} }}\nabla ^{2}\mathbf {v} +\mathbf {f} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>D</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>p</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {D\mathbf {v} }{Dt}}=-\nabla p+{\frac {1}{\mathrm {Re} }}\nabla ^{2}\mathbf {v} +\mathbf {f} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d785f859e36e4d33c9f110e5ae34292a4949b84d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:27.544ex; height:5.343ex;" alt="{\displaystyle {\frac {D\mathbf {v} }{Dt}}=-\nabla p+{\frac {1}{\mathrm {Re} }}\nabla ^{2}\mathbf {v} +\mathbf {f} .}"></span></dd></dl> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SEDequation1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/SEDequation1.jpg/220px-SEDequation1.jpg" decoding="async" width="220" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/SEDequation1.jpg/330px-SEDequation1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/SEDequation1.jpg/440px-SEDequation1.jpg 2x" data-file-width="1024" data-file-height="759" /></a><figcaption>Universal sedimentation equation — drag coefficient, a function of Reynolds number and shape factor, 2D diagram</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:SEDequation3.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/SEDequation3.jpg/220px-SEDequation3.jpg" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/SEDequation3.jpg/330px-SEDequation3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/SEDequation3.jpg/440px-SEDequation3.jpg 2x" data-file-width="1024" data-file-height="738" /></a><figcaption>Universal sedimentation equation — drag coefficient, a function of Reynolds number and shape factor, 3D diagram</figcaption></figure> <p>This is why mathematically all Newtonian, incompressible flows with the same Reynolds number are comparable. Notice also that in the above equation, the viscous terms vanish for <span class="texhtml">Re → ∞</span>. Thus flows with high Reynolds numbers are approximately inviscid in the free stream. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=4" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Reynolds_fluid_turbulence_experiment_1883.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Reynolds_fluid_turbulence_experiment_1883.jpg/220px-Reynolds_fluid_turbulence_experiment_1883.jpg" decoding="async" width="220" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Reynolds_fluid_turbulence_experiment_1883.jpg/330px-Reynolds_fluid_turbulence_experiment_1883.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Reynolds_fluid_turbulence_experiment_1883.jpg/440px-Reynolds_fluid_turbulence_experiment_1883.jpg 2x" data-file-width="2100" data-file-height="2031" /></a><figcaption>Osborne Reynolds's apparatus of 1883 demonstrating the onset of turbulent flow. The apparatus is still at the University of Manchester.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Flows_from_Reynolds_1883_paper.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Flows_from_Reynolds_1883_paper.jpg/220px-Flows_from_Reynolds_1883_paper.jpg" decoding="async" width="220" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Flows_from_Reynolds_1883_paper.jpg/330px-Flows_from_Reynolds_1883_paper.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d3/Flows_from_Reynolds_1883_paper.jpg/440px-Flows_from_Reynolds_1883_paper.jpg 2x" data-file-width="1253" data-file-height="645" /></a><figcaption>Diagram from Reynolds's 1883 paper showing onset of turbulent flow.</figcaption></figure> <p>Osborne Reynolds famously studied the conditions in which the flow of fluid in pipes <a href="/wiki/Laminar%E2%80%93turbulent_transition" title="Laminar–turbulent transition">transitioned</a> from <a href="/wiki/Laminar_flow" title="Laminar flow">laminar flow</a> to <a href="/wiki/Turbulent_flow" class="mw-redirect" title="Turbulent flow">turbulent flow</a>. In his 1883 paper Reynolds described the transition from laminar to turbulent flow in a classic experiment in which he examined the behaviour of water flow under different flow velocities using a small stream of dyed water introduced into the centre of clear water flow in a larger pipe. </p><p>The larger pipe was glass so the behaviour of the layer of the dyed stream could be observed. At the end of this pipe, there was a flow control valve used to vary the water velocity inside the tube. When the velocity was low, the dyed layer remained distinct throughout the entire length of the large tube. When the velocity was increased, the layer broke up at a given point and diffused throughout the fluid's cross-section. The point at which this happened was the transition point from laminar to turbulent flow. </p><p>From these experiments came the dimensionless Reynolds number for dynamic similarity—the ratio of <a href="/wiki/Inertia" title="Inertia">inertial</a> forces to <a href="/wiki/Viscosity" title="Viscosity">viscous</a> forces. Reynolds also proposed what is now known as the <a href="/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equations" title="Reynolds-averaged Navier–Stokes equations">Reynolds averaging</a> of turbulent flows, where quantities such as <a href="/wiki/Velocity" title="Velocity">velocity</a> are expressed as the sum of mean and fluctuating components. Such averaging allows for 'bulk' description of turbulent flow, for example using the <a href="/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equations" title="Reynolds-averaged Navier–Stokes equations">Reynolds-averaged Navier–Stokes equations</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Flow_in_a_pipe">Flow in a pipe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=5" title="Edit section: Flow in a pipe"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For <a href="/wiki/Flow_conditioning" title="Flow conditioning">flow in a pipe</a> or tube, the Reynolds number is generally defined as<sup id="cite_ref-Engineeringtoolbox_10-0" class="reference"><a href="#cite_note-Engineeringtoolbox-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></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 \mathrm {Re} ={\frac {uD_{\text{H}}}{\nu }}={\frac {\rho uD_{\text{H}}}{\mu }}={\frac {\rho QD_{\text{H}}}{\mu A}}={\frac {WD_{\text{H}}}{\mu A}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>u</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mrow> <mi>ν</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mi>u</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mrow> <mi>μ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mi>Q</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mrow> <mrow> <mi>μ</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>W</mi> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> </mrow> <mrow> <mi>μ</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ={\frac {uD_{\text{H}}}{\nu }}={\frac {\rho uD_{\text{H}}}{\mu }}={\frac {\rho QD_{\text{H}}}{\mu A}}={\frac {WD_{\text{H}}}{\mu A}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7d2eec49251e8bd37552d8a54a43f34335d9753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.021ex; height:6.176ex;" alt="{\displaystyle \mathrm {Re} ={\frac {uD_{\text{H}}}{\nu }}={\frac {\rho uD_{\text{H}}}{\mu }}={\frac {\rho QD_{\text{H}}}{\mu A}}={\frac {WD_{\text{H}}}{\mu A}},}"></span></dd></dl> <p>where </p> <ul><li><span class="texhtml"><i>D</i><sub>H</sub></span> is the <a href="/wiki/Hydraulic_diameter" title="Hydraulic diameter">hydraulic diameter</a> of the pipe (the inside diameter if the pipe is circular) (m),</li> <li><span class="texhtml"><i>Q</i></span> is the <a href="/wiki/Volumetric_flow_rate" title="Volumetric flow rate">volumetric flow rate</a> (m<sup>3</sup>/s),</li> <li><span class="texhtml"><i>A</i></span> is the pipe's <i>cross-sectional</i> area (<span class="texhtml"><i>A</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>πD<sup>2</sup></i></span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span></span>) (m<sup>2</sup>),</li> <li><span class="texhtml"><i>u</i></span> is the mean velocity of the fluid (m/s),</li> <li><span class="texhtml"><i>μ</i></span> (mu) is the <a href="/wiki/Dynamic_viscosity" class="mw-redirect" title="Dynamic viscosity">dynamic viscosity</a> of the <a href="/wiki/Fluid" title="Fluid">fluid</a> (Pa·s = N·s/m<sup>2</sup> = kg/(m·s)),</li> <li><span class="texhtml"><i>ν</i></span> (nu) is the <a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a> (<span class="texhtml"><i>ν</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>μ</i></span><span class="sr-only">/</span><span class="den"><i>ρ</i></span></span>⁠</span></span>) (m<sup>2</sup>/s),</li> <li><span class="texhtml"><i>ρ</i></span> (rho) is the <a href="/wiki/Density" title="Density">density</a> of the fluid (kg/m<sup>3</sup>),</li> <li><span class="texhtml"><i>W</i></span> is the mass flowrate of the fluid (kg/s).</li></ul> <p>For shapes such as squares, rectangular or annular ducts where the height and width are comparable, the characteristic dimension for internal-flow situations is taken to be the <a href="/wiki/Hydraulic_diameter" title="Hydraulic diameter">hydraulic diameter</a>, <span class="texhtml"><i>D</i><sub>H</sub></span>, defined as </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 D_{\text{H}}={\frac {4A}{P}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>A</mi> </mrow> <mi>P</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\text{H}}={\frac {4A}{P}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0cc89adadccfc156c8327d148af4ac533ea4c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.876ex; height:5.343ex;" alt="{\displaystyle D_{\text{H}}={\frac {4A}{P}},}"></span></dd></dl> <p>where <span class="texhtml"><i>A</i></span> is the cross-sectional area, and <span class="texhtml"><i>P</i></span> is the <a href="/wiki/Wetted_perimeter" title="Wetted perimeter">wetted perimeter</a>. The wetted perimeter for a channel is the total perimeter of all channel walls that are in contact with the flow.<sup id="cite_ref-FOOTNOTEHolman2002_11-0" class="reference"><a href="#cite_note-FOOTNOTEHolman2002-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This means that the length of the channel exposed to air is <i>not</i> included in the wetted perimeter. </p><p>For a circular pipe, the hydraulic diameter is exactly equal to the inside pipe diameter: </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 D_{\text{H}}=D.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H</mtext> </mrow> </msub> <mo>=</mo> <mi>D</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\text{H}}=D.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b98c974da428f7b43456f2e10b7be1174510293c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.059ex; height:2.509ex;" alt="{\displaystyle D_{\text{H}}=D.}"></span></dd></dl> <p>For an annular duct, such as the outer channel in a tube-in-tube <a href="/wiki/Heat_exchanger" title="Heat exchanger">heat exchanger</a>, the hydraulic diameter can be shown algebraically to reduce to </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 D_{\text{H,annulus}}=D_{\text{o}}-D_{\text{i}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>H,annulus</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\text{H,annulus}}=D_{\text{o}}-D_{\text{i}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/803d73fc28157d03afba2b8b23e57a5b09a7b3e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.608ex; height:2.843ex;" alt="{\displaystyle D_{\text{H,annulus}}=D_{\text{o}}-D_{\text{i}},}"></span></dd></dl> <p>where </p> <ul><li><span class="texhtml"><i>D</i><sub>o</sub></span> is the inside diameter of the outer pipe,</li> <li><span class="texhtml"><i>D</i><sub>i</sub></span> is the outside diameter of the inner pipe.</li></ul> <p>For calculation involving flow in non-circular ducts, the hydraulic diameter can be substituted for the diameter of a circular duct, with reasonable accuracy, if the aspect ratio AR of the duct cross-section remains in the range <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span> < AR < 4.<sup id="cite_ref-FOOTNOTEFoxMcDonaldPritchard2004348_12-0" class="reference"><a href="#cite_note-FOOTNOTEFoxMcDonaldPritchard2004348-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Laminar–turbulent_transition"><span id="Laminar.E2.80.93turbulent_transition"></span>Laminar–turbulent transition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=6" title="Edit section: Laminar–turbulent transition"><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">Main article: <a href="/wiki/Laminar%E2%80%93turbulent_transition" title="Laminar–turbulent transition">Laminar–turbulent transition</a></div> <p>In <a href="/wiki/Boundary_layer" title="Boundary layer">boundary layer</a> flow over a flat plate, experiments confirm that, after a certain length of flow, a laminar boundary layer will become unstable and turbulent. This instability occurs across different scales and with different fluids, usually when <span class="texhtml">Re<sub><i>x</i></sub></span> ≈ <span class="nowrap"><span data-sort-value="7005500000000000000♠"></span>5<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>5</sup></span>,<sup id="cite_ref-FOOTNOTEIncroperaDeWitt1981_13-0" class="reference"><a href="#cite_note-FOOTNOTEIncroperaDeWitt1981-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> where <span class="texhtml"><i>x</i></span> is the distance from the leading edge of the flat plate, and the flow velocity is the <a href="/wiki/Freestream" title="Freestream">freestream</a> velocity of the fluid outside the boundary layer. </p><p>For flow in a pipe of diameter <span class="texhtml"><i>D</i></span>, experimental observations show that for "fully developed" flow,<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>n 2<span class="cite-bracket">]</span></a></sup> laminar flow occurs when <span class="texhtml">Re<sub><i>D</i></sub></span> < 2300 and turbulent flow occurs when <span class="texhtml">Re<sub><i>D</i></sub></span> > 2900.<sup id="cite_ref-FOOTNOTESchlichtingGersten2017416–419_15-0" class="reference"><a href="#cite_note-FOOTNOTESchlichtingGersten2017416–419-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEHolman2002207_16-0" class="reference"><a href="#cite_note-FOOTNOTEHolman2002207-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> At the lower end of this range, a continuous turbulent-flow will form, but only at a very long distance from the inlet of the pipe. The flow in between will begin to transition from laminar to turbulent and then back to laminar at irregular intervals, called intermittent flow. This is due to the different speeds and conditions of the fluid in different areas of the pipe's cross-section, depending on other factors such as pipe roughness and flow uniformity. Laminar flow tends to dominate in the fast-moving center of the pipe while slower-moving turbulent flow dominates near the wall. As the Reynolds number increases, the continuous turbulent-flow moves closer to the inlet and the intermittency in between increases, until the flow becomes fully turbulent at <span class="texhtml">Re<sub><i>D</i></sub></span> > 2900.<sup id="cite_ref-FOOTNOTESchlichtingGersten2017416–419_15-1" class="reference"><a href="#cite_note-FOOTNOTESchlichtingGersten2017416–419-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> This result is generalized to non-circular channels using the <a href="/wiki/Hydraulic_diameter" title="Hydraulic diameter">hydraulic diameter</a>, allowing a transition Reynolds number to be calculated for other shapes of channel.<sup id="cite_ref-FOOTNOTESchlichtingGersten2017416–419_15-2" class="reference"><a href="#cite_note-FOOTNOTESchlichtingGersten2017416–419-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>These <a href="/wiki/Laminar%E2%80%93turbulent_transition" title="Laminar–turbulent transition">transition</a> Reynolds numbers are also called <i>critical Reynolds numbers</i>, and were studied by Osborne Reynolds around 1895.<sup id="cite_ref-FOOTNOTERott19901–11_6-1" class="reference"><a href="#cite_note-FOOTNOTERott19901–11-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The critical Reynolds number is different for every geometry.<sup id="cite_ref-FOOTNOTEPotterWiggertRamadan2012105_17-0" class="reference"><a href="#cite_note-FOOTNOTEPotterWiggertRamadan2012105-17"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Flow_in_a_wide_duct">Flow in a wide duct</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=7" title="Edit section: Flow in a wide duct"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For a fluid moving between two plane parallel surfaces—where the width is much greater than the space between the plates—then the characteristic dimension is equal to the distance between the plates.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> This is consistent with the annular duct and rectangular duct cases above, taken to a limiting aspect ratio. </p> <div class="mw-heading mw-heading2"><h2 id="Flow_in_an_open_channel">Flow in an open channel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=8" title="Edit section: Flow in an open channel"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For calculating the flow of liquid with a free surface, the <a href="/wiki/Hydraulic_radius" class="mw-redirect" title="Hydraulic radius">hydraulic radius</a> must be determined. This is the cross-sectional area of the channel divided by the wetted perimeter. For a semi-circular channel, it is a quarter of the diameter (in case of full pipe flow). For a rectangular channel, the hydraulic radius is the cross-sectional area divided by the wetted perimeter. Some texts then use a characteristic dimension that is four times the hydraulic radius, chosen because it gives the same value of <span class="texhtml">Re</span> for the onset of turbulence as in pipe flow,<sup id="cite_ref-FOOTNOTEStreeter1965_19-0" class="reference"><a href="#cite_note-FOOTNOTEStreeter1965-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> while others use the hydraulic radius as the characteristic length-scale with consequently different values of <span class="texhtml">Re</span> for transition and turbulent flow. </p> <div class="mw-heading mw-heading2"><h2 id="Flow_around_airfoils">Flow around airfoils</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=9" title="Edit section: Flow around airfoils"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Reynolds numbers are used in <a href="/wiki/Airfoil" title="Airfoil">airfoil</a> design to (among other things) manage "scale effect" when computing/comparing characteristics (a tiny wing, scaled to be huge, will perform differently).<sup id="cite_ref-FOOTNOTELissaman1983223–239_20-0" class="reference"><a href="#cite_note-FOOTNOTELissaman1983223–239-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Fluid dynamicists define the <i>chord Reynolds number</i> <span class="texhtml"><i>R</i> = <i>Vc</i>/<i>ν</i></span>, where <span class="texhtml"><i>V</i></span> is the flight speed, <span class="texhtml"><i>c</i></span> is the chord length, and <span class="texhtml"><i>ν</i></span> is the kinematic viscosity of the fluid in which the airfoil operates, which is <span class="nowrap"><span data-sort-value="6995146000000000000♠"></span>1.460<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−5</sup> m<sup>2</sup>/s</span> for the atmosphere at <a href="/wiki/Sea_level" title="Sea level">sea level</a>.<sup id="cite_ref-eng.cam.ac.uk_21-0" class="reference"><a href="#cite_note-eng.cam.ac.uk-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> In some special studies a characteristic length other than chord may be used; rare is the "span Reynolds number", which is not to be confused with span-wise stations on a wing, where chord is still used.<sup id="cite_ref-FOOTNOTEEhrensteinEloy2013321–346_22-0" class="reference"><a href="#cite_note-FOOTNOTEEhrensteinEloy2013321–346-22"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Object_in_a_fluid">Object in a fluid</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=10" title="Edit section: Object in a fluid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Honey-miel.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Honey-miel.jpg/220px-Honey-miel.jpg" decoding="async" width="220" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Honey-miel.jpg/330px-Honey-miel.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/f/f3/Honey-miel.jpg 2x" data-file-width="416" data-file-height="339" /></a><figcaption>The high viscosity of honey results in perfectly laminar flow when poured from a bucket, while the low surface tension allows it to remain sheet-like even after reaching the fluid below. Analogous to turbulence, when the flow meets resistance it slows and begins oscillating back and forth, piling upon itself.</figcaption></figure> <p>The Reynolds number for an object moving in a fluid, called the particle Reynolds number and often denoted <span class="texhtml">Re<sub>p</sub></span>, characterizes the nature of the surrounding flow and its fall velocity. </p> <div class="mw-heading mw-heading3"><h3 id="In_viscous_fluids">In viscous fluids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=11" title="Edit section: In viscous fluids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Reynolds_number" title="Special:EditPage/Reynolds number">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">February 2014</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stokes_sphere.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Stokes_sphere.svg/200px-Stokes_sphere.svg.png" decoding="async" width="200" height="288" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Stokes_sphere.svg/300px-Stokes_sphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Stokes_sphere.svg/400px-Stokes_sphere.svg.png 2x" data-file-width="403" data-file-height="580" /></a><figcaption>Creeping flow past a falling sphere: <a href="/wiki/Streamlines,_streaklines,_and_pathlines" title="Streamlines, streaklines, and pathlines">streamlines</a>, drag force <span class="texhtml"><i>F</i><sub>d</sub></span> and force by gravity <span class="texhtml"><i>F</i><sub>g</sub></span>.</figcaption></figure> <p>Where the viscosity is naturally high, such as polymer solutions and polymer melts, flow is normally laminar. The Reynolds number is very small and <a href="/wiki/Stokes%27_law" title="Stokes' law">Stokes' law</a> can be used to measure the <a href="/wiki/Viscosity" title="Viscosity">viscosity</a> of the fluid. Spheres are allowed to fall through the fluid and they reach the <a href="/wiki/Terminal_velocity" title="Terminal velocity">terminal velocity</a> quickly, from which the viscosity can be determined.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>The laminar flow of polymer solutions is exploited by animals such as fish and dolphins, who exude viscous solutions from their skin to aid flow over their bodies while swimming.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> It has been used in yacht racing by owners who want to gain a speed advantage by pumping a polymer solution such as low molecular weight <a href="/wiki/Polyoxyethylene" class="mw-redirect" title="Polyoxyethylene">polyoxyethylene</a> in water, over the wetted surface of the hull. </p><p>It is, however, a problem for mixing polymers, because turbulence is needed to distribute fine filler (for example) through the material. Inventions such as the "cavity transfer mixer" have been developed to produce multiple folds into a moving melt so as to improve <a href="/wiki/Mixture" title="Mixture">mixing</a> efficiency. The device can be fitted onto <a href="/wiki/Extruder" class="mw-redirect" title="Extruder">extruders</a> to aid mixing. </p> <div class="mw-heading mw-heading3"><h3 id="Sphere_in_a_fluid">Sphere in a fluid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=12" title="Edit section: Sphere in a fluid"><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:Drag_coefficient_on_a_sphere_vs._Reynolds_number_-_main_trends.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Drag_coefficient_on_a_sphere_vs._Reynolds_number_-_main_trends.svg/390px-Drag_coefficient_on_a_sphere_vs._Reynolds_number_-_main_trends.svg.png" decoding="async" width="390" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Drag_coefficient_on_a_sphere_vs._Reynolds_number_-_main_trends.svg/585px-Drag_coefficient_on_a_sphere_vs._Reynolds_number_-_main_trends.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Drag_coefficient_on_a_sphere_vs._Reynolds_number_-_main_trends.svg/780px-Drag_coefficient_on_a_sphere_vs._Reynolds_number_-_main_trends.svg.png 2x" data-file-width="655" data-file-height="341" /></a><figcaption><a href="/wiki/Drag_coefficient" title="Drag coefficient">Drag coefficient</a> <i>C</i><sub>d</sub> for a sphere as a function of Reynolds number <i>Re</i>, as obtained from laboratory experiments. The dark line is for a sphere with a smooth surface, while the lighter line is for the case of a rough surface (e.g. with small dimples). There is a range of fluid velocities where a rough-surfaced golf ball experiences less drag than a smooth ball. The numbers along the line indicate several flow regimes and associated changes in the drag coefficient:<div><ol start="2" style="margin-left:0;"><li style="list-style-position:inside;">attached flow (<a href="/wiki/Stokes_flow" title="Stokes flow">Stokes flow</a>) and <a href="/wiki/Steady_flow" class="mw-redirect" title="Steady flow">steady</a> <a href="/wiki/Flow_separation" title="Flow separation">separated flow</a>,</li><li style="list-style-position:inside;">separated unsteady flow, having a <a href="/wiki/Laminar_flow" title="Laminar flow">laminar flow</a> <a href="/wiki/Boundary_layer" title="Boundary layer">boundary layer</a> upstream of the separation, and producing a <a href="/wiki/Vortex_street" class="mw-redirect" title="Vortex street">vortex street</a>,</li><li style="list-style-position:inside;">separated unsteady flow with a laminar boundary layer at the upstream side, before flow separation, with downstream of the sphere a chaotic <a href="/wiki/Turbulence" title="Turbulence">turbulent</a> <a href="/wiki/Wake_(physics)" title="Wake (physics)">wake</a>,</li><li style="list-style-position:inside;">post-critical separated flow, with a turbulent boundary layer.</li></ol></div></figcaption></figure> <p>For a sphere in a fluid, the characteristic length-scale is the diameter of the sphere and the characteristic velocity is that of the sphere relative to the fluid some distance away from the sphere, such that the motion of the sphere does not disturb that reference parcel of fluid. The density and viscosity are those belonging to the fluid.<sup id="cite_ref-FOOTNOTERhodes198929_25-0" class="reference"><a href="#cite_note-FOOTNOTERhodes198929-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Note that purely laminar flow only exists up to <span class="texhtml">Re</span> = 10 under this definition. </p><p>Under the condition of low <span class="texhtml">Re</span>, the relationship between force and speed of motion is given by <a href="/wiki/Stokes%27_law" title="Stokes' law">Stokes' law</a>.<sup id="cite_ref-FOOTNOTEDusenbery200949_26-0" class="reference"><a href="#cite_note-FOOTNOTEDusenbery200949-26"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p><p>At higher Reynolds numbers the drag on a sphere depends on surface roughness. Thus, for example, adding dimples on the surface of a <a href="/wiki/Golf_ball" title="Golf ball">golf ball</a> causes the boundary layer on the upstream side of the ball to transition from laminar to turbulent. The turbulent boundary layer is able to remain attached to the surface of the ball much longer than a laminar boundary and so creates a narrower low-pressure wake and hence less pressure drag. The reduction in pressure drag causes the ball to travel farther.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Rectangular_object_in_a_fluid">Rectangular object in a fluid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=13" title="Edit section: Rectangular object in a fluid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The equation for a rectangular object is identical to that of a sphere, with the object being approximated as an <a href="/wiki/Ellipsoid" title="Ellipsoid">ellipsoid</a> and the axis of length being chosen as the characteristic length scale. Such considerations are important in natural streams, for example, where there are few perfectly spherical grains. For grains in which measurement of each axis is impractical, sieve diameters are used instead as the characteristic particle length-scale. Both approximations alter the values of the critical Reynolds number. </p> <div class="mw-heading mw-heading3"><h3 id="Fall_velocity">Fall velocity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=14" title="Edit section: Fall velocity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The particle Reynolds number is important in determining the fall velocity of a particle. When the particle Reynolds number indicates laminar flow, <a href="/wiki/Stokes%27_law" title="Stokes' law">Stokes' law</a> can be used to calculate its fall velocity or settling velocity. When the particle Reynolds number indicates turbulent flow, a turbulent drag law must be constructed to model the appropriate settling velocity. </p> <div class="mw-heading mw-heading3"><h3 id="Packed_bed">Packed bed</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=15" title="Edit section: Packed bed"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For fluid flow through a bed, of approximately spherical particles of diameter <span class="texhtml"><i>D</i></span> in contact, if the <i><a href="/wiki/Porosity" title="Porosity">voidage</a></i> is <span class="texhtml"><i>ε</i></span> and the <i><a href="/wiki/Superficial_velocity" title="Superficial velocity">superficial velocity</a></i> is <span class="texhtml"><i>v</i><sub>s</sub></span>, the Reynolds number can be defined as<sup id="cite_ref-FOOTNOTEDwivedi1977157–165_28-0" class="reference"><a href="#cite_note-FOOTNOTEDwivedi1977157–165-28"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></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 \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>D</mi> </mrow> <mi>μ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d3be13fb64e8c6d2c0f8de70fc68a1117ff3ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.458ex; height:5.843ex;" alt="{\displaystyle \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu }},}"></span></dd></dl> <p>or </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 \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu \varepsilon }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>D</mi> </mrow> <mrow> <mi>μ</mi> <mi>ε</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu \varepsilon }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2499fa7ef7dd6575b54fd4a5c0a0bd430a4ade7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.458ex; height:5.843ex;" alt="{\displaystyle \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu \varepsilon }},}"></span></dd></dl> <p>or </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 \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu (1-\varepsilon )}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mi>D</mi> </mrow> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu (1-\varepsilon )}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6c755e54bdbe5cb1c148dd3f6e315595a43e99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.622ex; height:6.176ex;" alt="{\displaystyle \mathrm {Re} ={\frac {\rho v_{\text{s}}D}{\mu (1-\varepsilon )}}.}"></span></dd></dl> <p>The choice of equation depends on the system involved: the first is successful in correlating the data for various types of packed and <a href="/wiki/Fluidized_bed" title="Fluidized bed">fluidized beds</a>, the second Reynolds number suits for the liquid-phase data, while the third was found successful in correlating the fluidized bed data, being first introduced for liquid fluidized bed system.<sup id="cite_ref-FOOTNOTEDwivedi1977157–165_28-1" class="reference"><a href="#cite_note-FOOTNOTEDwivedi1977157–165-28"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>Laminar conditions apply up to <span class="texhtml">Re</span> = 10, fully turbulent from <span class="texhtml">Re</span> = 2000.<sup id="cite_ref-FOOTNOTERhodes198929_25-1" class="reference"><a href="#cite_note-FOOTNOTERhodes198929-25"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Stirred_vessel">Stirred vessel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=16" title="Edit section: Stirred vessel"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a cylindrical vessel stirred by a central rotating paddle, turbine or propeller, the characteristic dimension is the diameter of the agitator <span class="texhtml"><i>D</i></span>. The velocity <span class="texhtml"><i>V</i></span> is <span class="texhtml"><i>ND</i></span> where <span class="texhtml"><i>N</i></span> is the <a href="/wiki/Angular_velocity" title="Angular velocity">rotational speed</a> in rad per second. Then the Reynolds number is: </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 \mathrm {Re} ={\frac {\rho ND^{2}}{\mu }}={\frac {\rho VD}{\mu }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mi>N</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>μ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mi>V</mi> <mi>D</mi> </mrow> <mi>μ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ={\frac {\rho ND^{2}}{\mu }}={\frac {\rho VD}{\mu }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5174cf697b1b81067ea7f3f85b67ec868bfcafc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.417ex; height:6.176ex;" alt="{\displaystyle \mathrm {Re} ={\frac {\rho ND^{2}}{\mu }}={\frac {\rho VD}{\mu }}.}"></span></dd></dl> <p>The system is fully turbulent for values of <span class="texhtml">Re</span> above <span class="nowrap"><span data-sort-value="7004100000000000000♠"></span>10<span style="margin-left:.25em;">000</span></span>.<sup id="cite_ref-FOOTNOTESinnottCoulsonRichardson200573_29-0" class="reference"><a href="#cite_note-FOOTNOTESinnottCoulsonRichardson200573-29"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Pipe_friction">Pipe friction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=17" title="Edit section: Pipe friction"><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:Moody_EN.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Moody_EN.svg/220px-Moody_EN.svg.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Moody_EN.svg/330px-Moody_EN.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Moody_EN.svg/440px-Moody_EN.svg.png 2x" data-file-width="732" data-file-height="453" /></a><figcaption>The <a href="/wiki/Moody_diagram" class="mw-redirect" title="Moody diagram">Moody diagram</a>, which describes the <a href="/wiki/Darcy%E2%80%93Weisbach_equation" title="Darcy–Weisbach equation">Darcy–Weisbach friction factor</a> <span class="texhtml"><i>f</i></span> as a function of the Reynolds number and relative pipe roughness.</figcaption></figure> <p>Pressure drops<sup id="cite_ref-nuclear-power.net_30-0" class="reference"><a href="#cite_note-nuclear-power.net-30"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> seen for fully developed flow of fluids through pipes can be predicted using the <a href="/wiki/Moody_diagram" class="mw-redirect" title="Moody diagram">Moody diagram</a> which plots the <a href="/wiki/Darcy%E2%80%93Weisbach_equation" title="Darcy–Weisbach equation">Darcy–Weisbach friction factor</a> <span class="texhtml"><i>f</i></span> against Reynolds number <span class="texhtml">Re</span> and relative roughness <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>ε</i></span><span class="sr-only">/</span><span class="den"><i>D</i></span></span>⁠</span></span>. The diagram clearly shows the laminar, transition, and turbulent flow regimes as Reynolds number increases. The nature of pipe flow is strongly dependent on whether the flow is laminar or turbulent. </p> <div class="mw-heading mw-heading2"><h2 id="Similarity_of_flows">Similarity of flows</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=18" title="Edit section: Similarity of flows"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Reynolds_behaviors.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Reynolds_behaviors.png/220px-Reynolds_behaviors.png" decoding="async" width="220" height="357" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Reynolds_behaviors.png/330px-Reynolds_behaviors.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Reynolds_behaviors.png/440px-Reynolds_behaviors.png 2x" data-file-width="543" data-file-height="882" /></a><figcaption>Qualitative behaviors of fluid flow over a cylinder depends to a large extent on Reynolds number; similar flow patterns often appear when the shape and Reynolds number is matched, although other parameters like surface roughness have a big effect.</figcaption></figure> <p>In order for two flows to be similar, they must have the same geometry and equal Reynolds and <a href="/wiki/Euler_number_(physics)" title="Euler number (physics)">Euler numbers</a>. When comparing fluid behavior at corresponding points in a model and a full-scale flow, the following holds: </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 {\begin{aligned}\mathrm {Re} _{\text{m}}&=\mathrm {Re} ,\\\mathrm {Eu} _{\text{m}}&=\mathrm {Eu} ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> <mi mathvariant="normal">u</mi> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathrm {Re} _{\text{m}}&=\mathrm {Re} ,\\\mathrm {Eu} _{\text{m}}&=\mathrm {Eu} ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/815131d80b40381c6e952f918d1468ed2eb7c13f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.849ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mathrm {Re} _{\text{m}}&=\mathrm {Re} ,\\\mathrm {Eu} _{\text{m}}&=\mathrm {Eu} ,\end{aligned}}}"></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 \mathrm {Re} _{\text{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} _{\text{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0d7572662e96847c83c8936500275e73dbbe050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.344ex; height:2.509ex;" alt="{\displaystyle \mathrm {Re} _{\text{m}}}"></span> is the Reynolds number for the model, 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 \mathrm {Re} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9a978f729af1a34028e60d45f484f0908e4be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {Re} }"></span> is full-scale Reynolds number, and similarly for the Euler numbers. </p><p>The model numbers and design numbers should be in the same proportion, hence </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 {p_{\text{m}}}{\rho _{\text{m}}v_{\text{m}}^{2}}}={\frac {p}{\rho v^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> </msub> <msubsup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>m</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mi>ρ</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p_{\text{m}}}{\rho _{\text{m}}v_{\text{m}}^{2}}}={\frac {p}{\rho v^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd993de45030e96ffb24693525b25caadde288f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.333ex; height:5.676ex;" alt="{\displaystyle {\frac {p_{\text{m}}}{\rho _{\text{m}}v_{\text{m}}^{2}}}={\frac {p}{\rho v^{2}}}.}"></span></dd></dl> <p>This allows engineers to perform experiments with reduced scale models in <a href="/wiki/Water_tunnel_(hydrodynamic)" title="Water tunnel (hydrodynamic)">water channels</a> or <a href="/wiki/Wind_tunnel" title="Wind tunnel">wind tunnels</a> and correlate the data to the actual flows, saving on costs during experimentation and on lab time. Note that true dynamic similitude may require matching other <a href="/wiki/Dimensionless_number" class="mw-redirect" title="Dimensionless number">dimensionless numbers</a> as well, such as the <a href="/wiki/Mach_number" title="Mach number">Mach number</a> used in <a href="/wiki/Compressible_flow" title="Compressible flow">compressible flows</a>, or the <a href="/wiki/Froude_number" title="Froude number">Froude number</a> that governs open-channel flows. Some flows involve more dimensionless parameters than can be practically satisfied with the available apparatus and fluids, so one is forced to decide which parameters are most important. For experimental flow modeling to be useful, it requires a fair amount of experience and judgment of the engineer. </p><p>An example where the mere Reynolds number is not sufficient for the similarity of flows (or even the flow regime – laminar or turbulent) are bounded flows, i.e. flows that are restricted by walls or other boundaries. A classical example of this is the <a href="/wiki/Taylor%E2%80%93Couette_flow" title="Taylor–Couette flow">Taylor–Couette flow</a>, where the dimensionless ratio of radii of bounding cylinders is also important, and many technical applications where these distinctions play an important role.<sup id="cite_ref-rheologic.net_31-0" class="reference"><a href="#cite_note-rheologic.net-31"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEMannevillePomeau20092072_32-0" class="reference"><a href="#cite_note-FOOTNOTEMannevillePomeau20092072-32"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> Principles of these restrictions were developed by <a href="/wiki/Maurice_Marie_Alfred_Couette" class="mw-redirect" title="Maurice Marie Alfred Couette">Maurice Marie Alfred Couette</a> and <a href="/wiki/Geoffrey_Ingram_Taylor" class="mw-redirect" title="Geoffrey Ingram Taylor">Geoffrey Ingram Taylor</a> and developed further by <a href="/wiki/Floris_Takens" title="Floris Takens">Floris Takens</a> and <a href="/wiki/David_Ruelle" title="David Ruelle">David Ruelle</a>. </p> <dl><dt>Typical values of Reynolds number<sup id="cite_ref-FOOTNOTEPatelRodiScheuerer19851308–1319_33-0" class="reference"><a href="#cite_note-FOOTNOTEPatelRodiScheuerer19851308–1319-33"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEDusenbery2009136_34-0" class="reference"><a href="#cite_note-FOOTNOTEDusenbery2009136-34"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup></dt></dl> <ul><li><a href="/wiki/Dictyostelium" title="Dictyostelium">Dictyostelium amoebae</a>: ~ 1 × 10<sup>−6</sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Bacterium" class="mw-redirect" title="Bacterium">Bacterium</a> ~ 1 × 10<sup>−4</sup></li> <li><a href="/wiki/Ciliate" title="Ciliate">Ciliate</a> ~ 1 × 10<sup>−1</sup></li> <li>Smallest <a href="/wiki/Fish" title="Fish">fish</a> ~ 1</li> <li><a href="/wiki/Blood_flow" class="mw-redirect" title="Blood flow">Blood flow</a> in <a href="/wiki/Brain" title="Brain">brain</a> ~ 1 × 10<sup>2</sup></li> <li>Blood flow in <a href="/wiki/Aorta" title="Aorta">aorta</a> ~ 1 × 10<sup>3</sup></li> <li><b>Onset of turbulent flow</b> ~ 2.3 × 10<sup>3</sup> to 5.0 × 10<sup>4</sup> for pipe flow to 10<sup>6</sup> for boundary layers</li> <li>Typical pitch in <a href="/wiki/Major_League_Baseball" title="Major League Baseball">Major League Baseball</a> ~ 2 × 10<sup>5</sup></li> <li>Person <a href="/wiki/Human_swimming" class="mw-redirect" title="Human swimming">swimming</a> ~ 4 × 10<sup>6</sup></li> <li>Fastest <a href="/wiki/Fish" title="Fish">fish</a> ~ 1 × 10<sup>8</sup></li> <li><a href="/wiki/Blue_whale" title="Blue whale">Blue whale</a> ~ 4 × 10<sup>8</sup></li> <li>A large ship (<i><a href="/wiki/Queen_Elizabeth_2" title="Queen Elizabeth 2">Queen Elizabeth 2</a></i>) ~ 5 × 10<sup>9</sup></li> <li>Atmospheric <a href="/wiki/Tropical_cyclone" title="Tropical cyclone">tropical cyclone</a> ~ 1 x 10<sup>12</sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Smallest_scales_of_turbulent_motion">Smallest scales of turbulent motion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=19" title="Edit section: Smallest scales of turbulent motion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Reynolds_number" title="Special:EditPage/Reynolds number">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">January 2019</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In a turbulent flow, there is a range of scales of the time-varying fluid motion. The size of the largest scales of fluid motion (sometimes called eddies) are set by the overall geometry of the flow. For instance, in an industrial smoke stack, the largest scales of fluid motion are as big as the diameter of the stack itself. The size of the smallest scales is set by the Reynolds number. As the Reynolds number increases, smaller and smaller scales of the flow are visible. In a smokestack, the smoke may appear to have many very small velocity perturbations or eddies, in addition to large bulky eddies. In this sense, the Reynolds number is an indicator of the range of scales in the flow. The higher the Reynolds number, the greater the range of scales. The largest eddies will always be the same size; the smallest eddies are determined by the Reynolds number. </p><p>What is the explanation for this phenomenon? A large Reynolds number indicates that viscous forces are not important at large scales of the flow. With a strong predominance of inertial forces over viscous forces, the largest scales of fluid motion are undamped—there is not enough viscosity to dissipate their motions. The kinetic energy must "cascade" from these large scales to progressively smaller scales until a level is reached for which the scale is small enough for viscosity to become important (that is, viscous forces become of the order of inertial ones). It is at these small scales where the dissipation of energy by viscous action finally takes place. The Reynolds number indicates at what scale this viscous dissipation occurs.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="In_physiology">In physiology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=20" title="Edit section: In physiology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Reynolds_number" title="Special:EditPage/Reynolds number">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">December 2014</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p><a href="/wiki/Poiseuille%27s_law" class="mw-redirect" title="Poiseuille's law">Poiseuille's law</a> on blood circulation in the body is dependent on <a href="/wiki/Laminar_flow" title="Laminar flow">laminar flow</a>.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> In turbulent flow the flow rate is proportional to the square root of the pressure gradient, as opposed to its direct proportionality to pressure gradient in laminar flow. </p><p>Using the definition of the Reynolds number we can see that a large diameter with rapid flow, where the density of the blood is high, tends towards turbulence. Rapid changes in vessel diameter may lead to turbulent flow, for instance when a narrower vessel widens to a larger one. Furthermore, a bulge of <a href="/wiki/Atheroma" title="Atheroma">atheroma</a> may be the cause of turbulent flow, where audible turbulence may be detected with a stethoscope.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Complex_systems">Complex systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=21" title="Edit section: Complex systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Reynolds number interpretation has been extended into the area of arbitrary <a href="/wiki/Complex_systems" class="mw-redirect" title="Complex systems">complex systems</a>. Such as financial flows,<sup id="cite_ref-FOOTNOTELos2006369_39-0" class="reference"><a href="#cite_note-FOOTNOTELos2006369-39"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> nonlinear networks,<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. (March 2021)">citation needed</span></a></i>]</sup> etc. In the latter case, an artificial viscosity is reduced to a nonlinear mechanism of energy distribution in <a href="/wiki/Complex_network" title="Complex network">complex network</a> media. Reynolds number then represents a basic control parameter that expresses a balance between injected and dissipated energy flows for an open boundary system. It has been shown that Reynolds critical regime separates two types of phase space motion: accelerator (attractor) and decelerator.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> High Reynolds number leads to a chaotic regime transition only in frame of <a href="/wiki/Strange_attractor" class="mw-redirect" title="Strange attractor">strange attractor</a> model. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_to_other_dimensionless_parameters">Relationship to other dimensionless parameters</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=22" title="Edit section: Relationship to other dimensionless parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are many <a href="/wiki/Dimensionless_numbers_in_fluid_mechanics" title="Dimensionless numbers in fluid mechanics">dimensionless numbers in fluid mechanics</a>. The Reynolds number measures the ratio of advection and diffusion effects on structures in the velocity field, and is therefore closely related to <a href="/wiki/P%C3%A9clet_number" title="Péclet number">Péclet numbers</a>, which measure the ratio of these effects on other fields carried by the flow, for example, temperature and magnetic fields. Replacement of the <a href="/wiki/Viscosity#Kinematic_viscosity" title="Viscosity">kinematic viscosity</a> <span class="texhtml"><i>ν</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><i>μ</i></span><span class="sr-only">/</span><span class="den"><i>ρ</i></span></span>⁠</span></span> in <span class="texhtml">Re</span> by the thermal or magnetic diffusivity results in respectively the thermal <a href="/wiki/P%C3%A9clet_number" title="Péclet number">Péclet number</a> and the <a href="/wiki/Magnetic_Reynolds_number" title="Magnetic Reynolds number">magnetic Reynolds number</a>. These are therefore related to <span class="texhtml">Re</span> by-products with ratios of diffusivities, namely the <a href="/wiki/Prandtl_number" title="Prandtl number">Prandtl number</a> and <a href="/wiki/Magnetic_Prandtl_number" title="Magnetic Prandtl number">magnetic Prandtl number</a>. </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=Reynolds_number&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Reynolds_transport_theorem" title="Reynolds transport theorem">Reynolds transport theorem</a> – 3D generalization of the Leibniz integral rule</li> <li><a href="/wiki/Drag_coefficient" title="Drag coefficient">Drag coefficient</a> – Dimensionless parameter to quantify fluid resistance</li> <li><a href="/wiki/Deposition_(geology)" title="Deposition (geology)">Deposition (geology)</a> – Geological process in which sediments, soil and rocks are added to a landform or landmass</li> <li><a href="/wiki/Kelvin%E2%80%93Helmholtz_instability" title="Kelvin–Helmholtz instability">Kelvin–Helmholtz instability</a></li></ul> <div style="clear:both;" class=""></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=Reynolds_number&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Footnotes">Footnotes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=25" title="Edit section: Footnotes"><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"><ol class="references"> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">The definition of the Reynolds number is not to be confused with the <a href="/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equations" title="Reynolds-averaged Navier–Stokes equations">Reynolds equation</a> or lubrication equation.</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">Full development of the flow occurs as the flow enters the pipe, the boundary layer thickens and then stabilizes after several diameters distance into the pipe.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=26" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTETansleyMarshall20013274–3283-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTETansleyMarshall20013274–3283_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFTansleyMarshall2001">Tansley & Marshall 2001</a>, pp. 3274–3283.</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"><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="CITEREFBush" class="citation web cs1">Bush, John W. M. <a rel="nofollow" class="external text" href="https://web.mit.edu/1.63/www/Lec-notes/Surfacetension/Lecture1.pdf">"Surface tension module"</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=Surface+tension+module&rft.aulast=Bush&rft.aufirst=John+W.+M.&rft_id=https%3A%2F%2Fweb.mit.edu%2F1.63%2Fwww%2FLec-notes%2FSurfacetension%2FLecture1.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEStokes18518–106-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEStokes18518–106_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFStokes1851">Stokes 1851</a>, pp. 8–106.</span> </li> <li id="cite_note-FOOTNOTESommerfeld1908116–124-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTESommerfeld1908116–124_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTESommerfeld1908116–124_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFSommerfeld1908">Sommerfeld 1908</a>, pp. 116–124.</span> </li> <li id="cite_note-FOOTNOTEReynolds1883935–982-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEReynolds1883935–982_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFReynolds1883">Reynolds 1883</a>, pp. 935–982.</span> </li> <li id="cite_note-FOOTNOTERott19901–11-6"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERott19901–11_6-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERott19901–11_6-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRott1990">Rott 1990</a>, pp. 1–11.</span> </li> <li id="cite_note-FOOTNOTEFalkovich2018-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFalkovich2018_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFalkovich2018">Falkovich 2018</a>.</span> </li> <li id="cite_note-Hall2015-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hall2015_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHall2015" class="citation web cs1">Hall, Nancy (5 May 2015). <a rel="nofollow" class="external text" href="https://www.grc.nasa.gov/www/k-12/airplane/boundlay.html">"Boundary Layer"</a>. <i>Glenn Research Center</i><span class="reference-accessdate">. Retrieved <span class="nowrap">17 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Glenn+Research+Center&rft.atitle=Boundary+Layer&rft.date=2015-05-05&rft.aulast=Hall&rft.aufirst=Nancy&rft_id=https%3A%2F%2Fwww.grc.nasa.gov%2Fwww%2Fk-12%2Fairplane%2Fboundlay.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> <li id="cite_note-Engineeringtoolbox-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Engineeringtoolbox_10-0">^</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="http://www.engineeringtoolbox.com/reynolds-number-d_237.html">"Reynolds Number"</a>. <i>Engineeringtoolbox.com</i>. 2003.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Engineeringtoolbox.com&rft.atitle=Reynolds+Number&rft.date=2003&rft_id=http%3A%2F%2Fwww.engineeringtoolbox.com%2Freynolds-number-d_237.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEHolman2002-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHolman2002_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHolman2002">Holman 2002</a>.</span> </li> <li id="cite_note-FOOTNOTEFoxMcDonaldPritchard2004348-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEFoxMcDonaldPritchard2004348_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFFoxMcDonaldPritchard2004">Fox, McDonald & Pritchard 2004</a>, p. 348.</span> </li> <li id="cite_note-FOOTNOTEIncroperaDeWitt1981-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEIncroperaDeWitt1981_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFIncroperaDeWitt1981">Incropera & DeWitt 1981</a>.</span> </li> <li id="cite_note-FOOTNOTESchlichtingGersten2017416–419-15"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTESchlichtingGersten2017416–419_15-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTESchlichtingGersten2017416–419_15-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTESchlichtingGersten2017416–419_15-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFSchlichtingGersten2017">Schlichting & Gersten 2017</a>, pp. 416–419.</span> </li> <li id="cite_note-FOOTNOTEHolman2002207-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHolman2002207_16-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHolman2002">Holman 2002</a>, p. 207.</span> </li> <li id="cite_note-FOOTNOTEPotterWiggertRamadan2012105-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPotterWiggertRamadan2012105_17-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPotterWiggertRamadan2012">Potter, Wiggert & Ramadan 2012</a>, p. 105.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSeshadri1978" class="citation journal cs1">Seshadri, K (February 1978). "Laminar flow between parallel plates with the injection of a reactant at high reynolds number". <i>International Journal of Heat and Mass Transfer</i>. <b>21</b> (2): 251–253. <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/1978IJHMT..21..251S">1978IJHMT..21..251S</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%2F0017-9310%2878%2990230-2">10.1016/0017-9310(78)90230-2</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+of+Heat+and+Mass+Transfer&rft.atitle=Laminar+flow+between+parallel+plates+with+the+injection+of+a+reactant+at+high+reynolds+number&rft.volume=21&rft.issue=2&rft.pages=251-253&rft.date=1978-02&rft_id=info%3Adoi%2F10.1016%2F0017-9310%2878%2990230-2&rft_id=info%3Abibcode%2F1978IJHMT..21..251S&rft.aulast=Seshadri&rft.aufirst=K&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEStreeter1965-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEStreeter1965_19-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFStreeter1965">Streeter 1965</a>.</span> </li> <li id="cite_note-FOOTNOTELissaman1983223–239-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELissaman1983223–239_20-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLissaman1983">Lissaman 1983</a>, pp. 223–239.</span> </li> <li id="cite_note-eng.cam.ac.uk-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-eng.cam.ac.uk_21-0">^</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="http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/atmos/atmos.html">"International Standard Atmosphere"</a>. <i>eng.cam.ac.uk</i><span class="reference-accessdate">. Retrieved <span class="nowrap">17 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=eng.cam.ac.uk&rft.atitle=International+Standard+Atmosphere&rft_id=http%3A%2F%2Fwww-mdp.eng.cam.ac.uk%2Fweb%2Flibrary%2Fenginfo%2Faerothermal_dvd_only%2Faero%2Fatmos%2Fatmos.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEEhrensteinEloy2013321–346-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEEhrensteinEloy2013321–346_22-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFEhrensteinEloy2013">Ehrenstein & Eloy 2013</a>, pp. 321–346.</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"><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.ese.iitb.ac.in/sites/default/files/Falling%20Ball%20Viscometer.pdf">"Falling Ball Viscometer"</a> <span class="cs1-format">(PDF)</span>. <i>Falling Ball Viscometer</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Falling+Ball+Viscometer&rft.atitle=Falling+Ball+Viscometer&rft_id=https%3A%2F%2Fwww.ese.iitb.ac.in%2Fsites%2Fdefault%2Ffiles%2FFalling%2520Ball%2520Viscometer.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></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"><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.americanscientist.org/article/not-just-going-with-the-flow">"Not Just Going with the Flow"</a>. <i>American Scientist</i>. 2017-02-06<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-05-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=American+Scientist&rft.atitle=Not+Just+Going+with+the+Flow&rft.date=2017-02-06&rft_id=https%3A%2F%2Fwww.americanscientist.org%2Farticle%2Fnot-just-going-with-the-flow&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTERhodes198929-25"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTERhodes198929_25-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTERhodes198929_25-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFRhodes1989">Rhodes 1989</a>, p. 29.</span> </li> <li id="cite_note-FOOTNOTEDusenbery200949-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDusenbery200949_26-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDusenbery2009">Dusenbery 2009</a>, p. 49.</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 web cs1"><a rel="nofollow" class="external text" href="http://www.aerospaceweb.org/question/aerodynamics/q0215.shtml">"Golf Ball Dimples & Drag"</a>. <i>Aerospaceweb.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">11 August</span> 2011</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Aerospaceweb.org&rft.atitle=Golf+Ball+Dimples+%26+Drag&rft_id=http%3A%2F%2Fwww.aerospaceweb.org%2Fquestion%2Faerodynamics%2Fq0215.shtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEDwivedi1977157–165-28"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEDwivedi1977157–165_28-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEDwivedi1977157–165_28-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDwivedi1977">Dwivedi 1977</a>, pp. 157–165.</span> </li> <li id="cite_note-FOOTNOTESinnottCoulsonRichardson200573-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTESinnottCoulsonRichardson200573_29-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFSinnottCoulsonRichardson2005">Sinnott, Coulson & Richardson 2005</a>, p. 73.</span> </li> <li id="cite_note-nuclear-power.net-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-nuclear-power.net_30-0">^</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.nuclear-power.net/nuclear-engineering/fluid-dynamics/major-head-loss-friction-loss/">"Major Head Loss - Friction Loss"</a>. <i>Nuclear Power</i><span class="reference-accessdate">. Retrieved <span class="nowrap">17 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Nuclear+Power&rft.atitle=Major+Head+Loss+-+Friction+Loss&rft_id=https%3A%2F%2Fwww.nuclear-power.net%2Fnuclear-engineering%2Ffluid-dynamics%2Fmajor-head-loss-friction-loss%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> <li id="cite_note-rheologic.net-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-rheologic.net_31-0">^</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://web.archive.org/web/20200217120012/https://rheologic.net/en/cfd-simulation-laminar-and-turbulent-flow#disk_diag">"Laminar, transitional and turbulent flow"</a>. <i>rheologic.net</i>. Archived from <a rel="nofollow" class="external text" href="https://rheologic.net/en/cfd-simulation-laminar-and-turbulent-flow#disk_diag">the original</a> on 17 February 2020<span class="reference-accessdate">. Retrieved <span class="nowrap">17 September</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=rheologic.net&rft.atitle=Laminar%2C+transitional+and+turbulent+flow&rft_id=https%3A%2F%2Frheologic.net%2Fen%2Fcfd-simulation-laminar-and-turbulent-flow%23disk_diag&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEMannevillePomeau20092072-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMannevillePomeau20092072_32-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMannevillePomeau2009">Manneville & Pomeau 2009</a>, p. 2072.</span> </li> <li id="cite_note-FOOTNOTEPatelRodiScheuerer19851308–1319-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEPatelRodiScheuerer19851308–1319_33-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFPatelRodiScheuerer1985">Patel, Rodi & Scheuerer 1985</a>, pp. 1308–1319.</span> </li> <li id="cite_note-FOOTNOTEDusenbery2009136-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDusenbery2009136_34-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDusenbery2009">Dusenbery 2009</a>, p. 136.</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="CITEREFWangOthmer2016" class="citation journal cs1">Wang, Qixuan; Othmer, Hans G. (2016-06-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1007/s00285-015-0925-9">"Computational analysis of amoeboid swimming at low Reynolds number"</a>. <i>Journal of Mathematical Biology</i>. <b>72</b> (7): 1893–1926. <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/1509.03504">1509.03504</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/10.1007%2Fs00285-015-0925-9">10.1007/s00285-015-0925-9</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/1432-1416">1432-1416</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/26362281">26362281</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:253808496">253808496</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+Mathematical+Biology&rft.atitle=Computational+analysis+of+amoeboid+swimming+at+low+Reynolds+number&rft.volume=72&rft.issue=7&rft.pages=1893-1926&rft.date=2016-06-01&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A253808496%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs00285-015-0925-9&rft_id=info%3Aarxiv%2F1509.03504&rft.issn=1432-1416&rft_id=info%3Apmid%2F26362281&rft.aulast=Wang&rft.aufirst=Qixuan&rft.au=Othmer%2C+Hans+G.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%2Fs00285-015-0925-9&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.cfd-online.com/Wiki/Reynolds_number">"Reynolds number -- CFD-Wiki, the free CFD reference"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Reynolds+number+--+CFD-Wiki%2C+the+free+CFD+reference&rft_id=https%3A%2F%2Fwww.cfd-online.com%2FWiki%2FReynolds_number&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" 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="CITEREFHelpsMcDonald1954" class="citation journal cs1">Helps, E. P. W.; McDonald, D. A. (1954-06-28). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1366298">"Observations on laminar flow in veins"</a>. <i>The Journal of Physiology</i>. <b>124</b> (3): 631–639. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1113%2Fjphysiol.1954.sp005135">10.1113/jphysiol.1954.sp005135</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1366298">1366298</a></span>. <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/13175205">13175205</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Journal+of+Physiology&rft.atitle=Observations+on+laminar+flow+in+veins&rft.volume=124&rft.issue=3&rft.pages=631-639&rft.date=1954-06-28&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1366298%23id-name%3DPMC&rft_id=info%3Apmid%2F13175205&rft_id=info%3Adoi%2F10.1113%2Fjphysiol.1954.sp005135&rft.aulast=Helps&rft.aufirst=E.+P.+W.&rft.au=McDonald%2C+D.+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1366298&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" 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="CITEREFRafieian-KopaeiSetorkiDoudiBaradaran2014" class="citation journal cs1">Rafieian-Kopaei, M.; Setorki, M.; Doudi, M.; Baradaran, A.; Nasri, H. (2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4258672">"Atherosclerosis: Process, Indicators, Risk Factors and New Hopes - PMC"</a>. <i>International Journal of Preventive Medicine</i>. <b>5</b> (8): 927–946. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4258672">4258672</a></span>. <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/25489440">25489440</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+of+Preventive+Medicine&rft.atitle=Atherosclerosis%3A+Process%2C+Indicators%2C+Risk+Factors+and+New+Hopes+-+PMC&rft.volume=5&rft.issue=8&rft.pages=927-946&rft.date=2014&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4258672%23id-name%3DPMC&rft_id=info%3Apmid%2F25489440&rft.aulast=Rafieian-Kopaei&rft.aufirst=M.&rft.au=Setorki%2C+M.&rft.au=Doudi%2C+M.&rft.au=Baradaran%2C+A.&rft.au=Nasri%2C+H.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4258672&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTELos2006369-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELos2006369_39-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLos2006">Los 2006</a>, p. 369.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGómez_Blázquez2016" class="citation thesis cs1">Gómez Blázquez, Alberto (2016-06-23). <a rel="nofollow" class="external text" href="https://upcommons.upc.edu/handle/2117/97079"><i>Aerodynamic analysis of the flat floor</i></a> (Bachelor thesis). Universitat Politècnica de Catalunya.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Aerodynamic+analysis+of+the+flat+floor&rft.inst=Universitat+Polit%C3%A8cnica+de+Catalunya&rft.date=2016-06-23&rft.aulast=G%C3%B3mez+Bl%C3%A1zquez&rft.aufirst=Alberto&rft_id=https%3A%2F%2Fupcommons.upc.edu%2Fhandle%2F2117%2F97079&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=27" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-hanging-indents refbegin-columns references-column-width" style="column-width: 30em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirdStewartLightfoot2006" class="citation book cs1">Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=L5FnNlIaGfcC"><i>Transport Phenomena</i></a>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-11539-8" title="Special:BookSources/978-0-470-11539-8"><bdi>978-0-470-11539-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transport+Phenomena&rft.pub=John+Wiley+%26+Sons&rft.date=2006&rft.isbn=978-0-470-11539-8&rft.aulast=Bird&rft.aufirst=R.+Byron&rft.au=Stewart%2C+Warren+E.&rft.au=Lightfoot%2C+Edwin+N.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DL5FnNlIaGfcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDusenbery2009" class="citation book cs1">Dusenbery, David B. (2009). <i>Living at Micro Scale</i>. Cambridge, Massachusetts: Harvard University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780674031166" title="Special:BookSources/9780674031166"><bdi>9780674031166</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Living+at+Micro+Scale&rft.place=Cambridge%2C+Massachusetts&rft.pub=Harvard+University+Press&rft.date=2009&rft.isbn=9780674031166&rft.aulast=Dusenbery&rft.aufirst=David+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDwivedi1977" class="citation journal cs1">Dwivedi, P. N. (1977). "Particle-fluid mass transfer in fixed and fluidized beds". <i>Industrial & Engineering Chemistry Process Design and Development</i>. <b>16</b> (2): 157–165. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Fi260062a001">10.1021/i260062a001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Industrial+%26+Engineering+Chemistry+Process+Design+and+Development&rft.atitle=Particle-fluid+mass+transfer+in+fixed+and+fluidized+beds&rft.volume=16&rft.issue=2&rft.pages=157-165&rft.date=1977&rft_id=info%3Adoi%2F10.1021%2Fi260062a001&rft.aulast=Dwivedi&rft.aufirst=P.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEhrensteinEloy2013" class="citation journal cs1">Ehrenstein, Uwe; Eloy, Christophe (2013). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190302134918/http://pdfs.semanticscholar.org/b698/40d639dd2902b2447273d9968d379c022f3d.pdf">"Skin friction on a moving wall and its implications for swimming animals"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Fluid Mechanics</i>. <b>718</b>: 321–346. <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/2013JFM...718..321E">2013JFM...718..321E</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%2Fjfm.2012.613">10.1017/jfm.2012.613</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/0022-1120">0022-1120</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:56331294">56331294</a>. Archived from <a rel="nofollow" class="external text" href="http://pdfs.semanticscholar.org/b698/40d639dd2902b2447273d9968d379c022f3d.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2019-03-02.</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=Skin+friction+on+a+moving+wall+and+its+implications+for+swimming+animals&rft.volume=718&rft.pages=321-346&rft.date=2013&rft_id=info%3Adoi%2F10.1017%2Fjfm.2012.613&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A56331294%23id-name%3DS2CID&rft.issn=0022-1120&rft_id=info%3Abibcode%2F2013JFM...718..321E&rft.aulast=Ehrenstein&rft.aufirst=Uwe&rft.au=Eloy%2C+Christophe&rft_id=http%3A%2F%2Fpdfs.semanticscholar.org%2Fb698%2F40d639dd2902b2447273d9968d379c022f3d.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFalkovich2018" class="citation book cs1">Falkovich, Gregory (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=OEFPDwAAQBAJ&pg=PA193"><i>Fluid Mechanics</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-12956-6" title="Special:BookSources/978-1-107-12956-6"><bdi>978-1-107-12956-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics&rft.pub=Cambridge+University+Press&rft.date=2018&rft.isbn=978-1-107-12956-6&rft.aulast=Falkovich&rft.aufirst=Gregory&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOEFPDwAAQBAJ%26pg%3DPA193&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFoxMcDonaldPritchard2004" class="citation book cs1">Fox, R. W.; McDonald, A. T.; Pritchard, Phillip J. (2004). <i>Introduction to Fluid Mechanics</i> (6th ed.). Hoboken: John Wiley and Sons. p. 348. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-20231-8" title="Special:BookSources/978-0-471-20231-8"><bdi>978-0-471-20231-8</bdi></a>.</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.place=Hoboken&rft.pages=348&rft.edition=6th&rft.pub=John+Wiley+and+Sons&rft.date=2004&rft.isbn=978-0-471-20231-8&rft.aulast=Fox&rft.aufirst=R.+W.&rft.au=McDonald%2C+A.+T.&rft.au=Pritchard%2C+Phillip+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHolman2002" class="citation book cs1">Holman, J. P. (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1wyjAgAAQBAJ"><i>Heat Transfer</i></a> (Si Units ed.). McGraw-Hill Education (India) Pvt Limited. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-106967-0" title="Special:BookSources/978-0-07-106967-0"><bdi>978-0-07-106967-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Heat+Transfer&rft.edition=Si+Units&rft.pub=McGraw-Hill+Education+%28India%29+Pvt+Limited&rft.date=2002&rft.isbn=978-0-07-106967-0&rft.aulast=Holman&rft.aufirst=J.+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1wyjAgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIncroperaDeWitt1981" class="citation book cs1">Incropera, Frank P.; DeWitt, David P. (1981). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xAdRAAAAMAAJ"><i>Fundamentals of heat transfer</i></a>. New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-42711-7" title="Special:BookSources/978-0-471-42711-7"><bdi>978-0-471-42711-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+heat+transfer&rft.place=New+York&rft.pub=Wiley&rft.date=1981&rft.isbn=978-0-471-42711-7&rft.aulast=Incropera&rft.aufirst=Frank+P.&rft.au=DeWitt%2C+David+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxAdRAAAAMAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLissaman1983" class="citation journal cs1">Lissaman, P. B. S. (1983). "Low-Reynolds-Number Airfoils". <i>Annu. Rev. Fluid Mech</i>. <b>15</b> (15): 223–39. <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/1983AnRFM..15..223L">1983AnRFM..15..223L</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=10.1.1.506.1131">10.1.1.506.1131</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/10.1146%2Fannurev.fl.15.010183.001255">10.1146/annurev.fl.15.010183.001255</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:123639541">123639541</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annu.+Rev.+Fluid+Mech.&rft.atitle=Low-Reynolds-Number+Airfoils&rft.volume=15&rft.issue=15&rft.pages=223-39&rft.date=1983&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.506.1131%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123639541%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1146%2Fannurev.fl.15.010183.001255&rft_id=info%3Abibcode%2F1983AnRFM..15..223L&rft.aulast=Lissaman&rft.aufirst=P.+B.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLos2006" class="citation book cs1">Los, Cornelis (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Bx3Z9y5mg0MC&pg=PA"><i>Financial Market Risk: Measurement and Analysis</i></a>. Routledge. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-134-46932-1" title="Special:BookSources/978-1-134-46932-1"><bdi>978-1-134-46932-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Financial+Market+Risk%3A+Measurement+and+Analysis&rft.pub=Routledge&rft.date=2006&rft.isbn=978-1-134-46932-1&rft.aulast=Los&rft.aufirst=Cornelis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DBx3Z9y5mg0MC%26pg%3DPA&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMannevillePomeau2009" class="citation journal cs1">Manneville, Paul; Pomeau, Yves (25 March 2009). <a rel="nofollow" class="external text" href="https://doi.org/10.4249%2Fscholarpedia.2072">"Transition to turbulence"</a>. <i>Scholarpedia</i>. <b>4</b> (3): 2072. <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/2009SchpJ...4.2072M">2009SchpJ...4.2072M</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/10.4249%2Fscholarpedia.2072">10.4249/scholarpedia.2072</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scholarpedia&rft.atitle=Transition+to+turbulence&rft.volume=4&rft.issue=3&rft.pages=2072&rft.date=2009-03-25&rft_id=info%3Adoi%2F10.4249%2Fscholarpedia.2072&rft_id=info%3Abibcode%2F2009SchpJ...4.2072M&rft.aulast=Manneville&rft.aufirst=Paul&rft.au=Pomeau%2C+Yves&rft_id=https%3A%2F%2Fdoi.org%2F10.4249%252Fscholarpedia.2072&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPatelRodiScheuerer1985" class="citation journal cs1">Patel, V. C.; Rodi, W.; Scheuerer, G. (1985). "Turbulence Models for Near-Wall and Low Reynolds Number Flows—A Review". <i>AIAA Journal</i>. <b>23</b> (9): 1308–1319. <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/1985AIAAJ..23.1308P">1985AIAAJ..23.1308P</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%2F3.9086">10.2514/3.9086</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=Turbulence+Models+for+Near-Wall+and+Low+Reynolds+Number+Flows%E2%80%94A+Review&rft.volume=23&rft.issue=9&rft.pages=1308-1319&rft.date=1985&rft_id=info%3Adoi%2F10.2514%2F3.9086&rft_id=info%3Abibcode%2F1985AIAAJ..23.1308P&rft.aulast=Patel&rft.aufirst=V.+C.&rft.au=Rodi%2C+W.&rft.au=Scheuerer%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPotterWiggertRamadan2012" class="citation book cs1">Potter, Merle C.; Wiggert, David C.; Ramadan, Bassem H. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mRYNKlFcPCQC"><i>Mechanics of Fluids</i></a> (4th, SI units ed.). Cengage Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-66773-5" title="Special:BookSources/978-0-495-66773-5"><bdi>978-0-495-66773-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Fluids&rft.edition=4th%2C+SI+units&rft.pub=Cengage+Learning&rft.date=2012&rft.isbn=978-0-495-66773-5&rft.aulast=Potter&rft.aufirst=Merle+C.&rft.au=Wiggert%2C+David+C.&rft.au=Ramadan%2C+Bassem+H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DmRYNKlFcPCQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReynolds1883" class="citation journal cs1"><a href="/wiki/Osborne_Reynolds" title="Osborne Reynolds">Reynolds, Osborne</a> (1883). <a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1883.0029">"An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels"</a>. <i><a href="/wiki/Philosophical_Transactions_of_the_Royal_Society" title="Philosophical Transactions of the Royal Society">Philosophical Transactions of the Royal Society</a></i>. <b>174</b>: 935–982. <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/1883RSPT..174..935R">1883RSPT..174..935R</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/10.1098%2Frstl.1883.0029">10.1098/rstl.1883.0029</a></span>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/109431">109431</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society&rft.atitle=An+experimental+investigation+of+the+circumstances+which+determine+whether+the+motion+of+water+shall+be+direct+or+sinuous%2C+and+of+the+law+of+resistance+in+parallel+channels&rft.volume=174&rft.pages=935-982&rft.date=1883&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F109431%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.1098%2Frstl.1883.0029&rft_id=info%3Abibcode%2F1883RSPT..174..935R&rft.aulast=Reynolds&rft.aufirst=Osborne&rft_id=https%3A%2F%2Fdoi.org%2F10.1098%252Frstl.1883.0029&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRhodes1989" class="citation book cs1">Rhodes, M. (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P9Qgvh7kMP8C&pg=PA29"><i>Introduction to Particle Technology</i></a>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-98482-5" title="Special:BookSources/978-0-471-98482-5"><bdi>978-0-471-98482-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Particle+Technology&rft.pub=Wiley&rft.date=1989&rft.isbn=978-0-471-98482-5&rft.aulast=Rhodes&rft.aufirst=M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DP9Qgvh7kMP8C%26pg%3DPA29&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRott1990" class="citation journal cs1">Rott, N. (1990). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190225114301/http://pdfs.semanticscholar.org/6d8d/cdbad09e1028c641856e550cb11e1a1997e8.pdf">"Note on the history of the Reynolds number"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Annual_Review_of_Fluid_Mechanics" title="Annual Review of Fluid Mechanics">Annual Review of Fluid Mechanics</a></i>. <b>22</b> (1): 1–11. <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/1990AnRFM..22....1R">1990AnRFM..22....1R</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.fl.22.010190.000245">10.1146/annurev.fl.22.010190.000245</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:54583669">54583669</a>. Archived from <a rel="nofollow" class="external text" href="http://pdfs.semanticscholar.org/6d8d/cdbad09e1028c641856e550cb11e1a1997e8.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2019-02-25.</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=Note+on+the+history+of+the+Reynolds+number&rft.volume=22&rft.issue=1&rft.pages=1-11&rft.date=1990&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A54583669%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1146%2Fannurev.fl.22.010190.000245&rft_id=info%3Abibcode%2F1990AnRFM..22....1R&rft.aulast=Rott&rft.aufirst=N.&rft_id=http%3A%2F%2Fpdfs.semanticscholar.org%2F6d8d%2Fcdbad09e1028c641856e550cb11e1a1997e8.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchlichtingGersten2017" class="citation book cs1">Schlichting, Hermann; Gersten, Klaus (2017). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=bOUyDQAAQBAJ"><i>Boundary-Layer Theory</i></a>. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-52919-5" title="Special:BookSources/978-3-662-52919-5"><bdi>978-3-662-52919-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Boundary-Layer+Theory&rft.pub=Springer&rft.date=2017&rft.isbn=978-3-662-52919-5&rft.aulast=Schlichting&rft.aufirst=Hermann&rft.au=Gersten%2C+Klaus&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DbOUyDQAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSinnottCoulsonRichardson2005" class="citation book cs1">Sinnott, R. K.; Coulson, John Metcalfe; Richardson, John Francis (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=XLzOngEACAAJ"><i>Chemical Engineering Design</i></a>. Vol. 6 (4th ed.). Elsevier Butterworth-Heinemann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7506-6538-4" title="Special:BookSources/978-0-7506-6538-4"><bdi>978-0-7506-6538-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemical+Engineering+Design&rft.edition=4th&rft.pub=Elsevier+Butterworth-Heinemann&rft.date=2005&rft.isbn=978-0-7506-6538-4&rft.aulast=Sinnott&rft.aufirst=R.+K.&rft.au=Coulson%2C+John+Metcalfe&rft.au=Richardson%2C+John+Francis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXLzOngEACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSommerfeld1908" class="citation journal cs1"><a href="/wiki/Arnold_Sommerfeld" title="Arnold Sommerfeld">Sommerfeld, Arnold</a> (1908). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20161115133937/http://www.mathunion.org/ICM/ICM1908.3/Main/icm1908.3.0116.0124.ocr.pdf">"Ein Beitrag zur hydrodynamischen Erkläerung der turbulenten Flüssigkeitsbewegüngen (A Contribution to Hydrodynamic Explanation of Turbulent Fluid Motions)"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/International_Congress_of_Mathematicians" title="International Congress of Mathematicians">International Congress of Mathematicians </a></i>. <b>3</b>: 116–124. Archived from <a rel="nofollow" class="external text" href="http://www.mathunion.org/ICM/ICM1908.3/Main/icm1908.3.0116.0124.ocr.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2016-11-15.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Congress+of+Mathematicians&rft.atitle=Ein+Beitrag+zur+hydrodynamischen+Erkl%C3%A4erung+der+turbulenten+Fl%C3%BCssigkeitsbeweg%C3%BCngen+%28A+Contribution+to+Hydrodynamic+Explanation+of+Turbulent+Fluid+Motions%29&rft.volume=3&rft.pages=116-124&rft.date=1908&rft.aulast=Sommerfeld&rft.aufirst=Arnold&rft_id=http%3A%2F%2Fwww.mathunion.org%2FICM%2FICM1908.3%2FMain%2Ficm1908.3.0116.0124.ocr.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStokes1851" class="citation journal cs1"><a href="/wiki/George_Gabriel_Stokes" class="mw-redirect" title="George Gabriel Stokes">Stokes, George</a> (1851). "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums". <i><a href="/wiki/Transactions_of_the_Cambridge_Philosophical_Society" class="mw-redirect" title="Transactions of the Cambridge Philosophical Society">Transactions of the Cambridge Philosophical Society</a></i>. <b>9</b>: 8–106. <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/1851TCaPS...9....8S">1851TCaPS...9....8S</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Cambridge+Philosophical+Society&rft.atitle=On+the+Effect+of+the+Internal+Friction+of+Fluids+on+the+Motion+of+Pendulums&rft.volume=9&rft.pages=8-106&rft.date=1851&rft_id=info%3Abibcode%2F1851TCaPS...9....8S&rft.aulast=Stokes&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStreeter1965" class="citation book cs1">Streeter, Victor Lyle (1965). <i>Fluid mechanics</i> (3rd ed.). New York: McGraw-Hill. <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/878734937">878734937</a>.</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.place=New+York&rft.edition=3rd&rft.pub=McGraw-Hill&rft.date=1965&rft_id=info%3Aoclcnum%2F878734937&rft.aulast=Streeter&rft.aufirst=Victor+Lyle&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTansleyMarshall2001" class="citation journal cs1">Tansley, Claire E.; Marshall, David P. (2001). <a rel="nofollow" class="external text" href="https://wayback.archive-it.org/all/20110401142411/http://www.met.reading.ac.uk/~ocean/Dynamics/pub/tm01b.pdf">"Flow past a Cylinder on a Plane, with Application to Gulf Stream Separation and the Antarctic Circumpolar Current"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Physical Oceanography</i>. <b>31</b> (11): 3274–3283. <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/2001JPO....31.3274T">2001JPO....31.3274T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1175%2F1520-0485%282001%29031%3C3274%3AFPACOA%3E2.0.CO%3B2">10.1175/1520-0485(2001)031<3274:FPACOA>2.0.CO;2</a>. Archived from <a rel="nofollow" class="external text" href="http://www.met.reading.ac.uk/~ocean/Dynamics/pub/tm01b.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-04-01.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physical+Oceanography&rft.atitle=Flow+past+a+Cylinder+on+a+Plane%2C+with+Application+to+Gulf+Stream+Separation+and+the+Antarctic+Circumpolar+Current&rft.volume=31&rft.issue=11&rft.pages=3274-3283&rft.date=2001&rft_id=info%3Adoi%2F10.1175%2F1520-0485%282001%29031%3C3274%3AFPACOA%3E2.0.CO%3B2&rft_id=info%3Abibcode%2F2001JPO....31.3274T&rft.aulast=Tansley&rft.aufirst=Claire+E.&rft.au=Marshall%2C+David+P.&rft_id=http%3A%2F%2Fwww.met.reading.ac.uk%2F~ocean%2FDynamics%2Fpub%2Ftm01b.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Further_reading">Further reading</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Reynolds_number&action=edit&section=28" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ol><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBatchelor1967" class="citation book cs1">Batchelor, G. K. (1967). <i>An Introduction to Fluid Dynamics</i>. Cambridge University Press. pp. 211–215.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Fluid+Dynamics&rft.pages=211-215&rft.pub=Cambridge+University+Press&rft.date=1967&rft.aulast=Batchelor&rft.aufirst=G.+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AReynolds+number" class="Z3988"></span></li> <li>Brezina, Jiri, 1979, Particle size and settling rate distributions of sand-sized materials: 2nd European Symposium on Particle Characterisation (<a rel="nofollow" class="external text" href="http://www.grano.de/PARTEC1s.pdf">PARTEC</a>), Nürnberg, West Germany.</li> <li>Brezina, Jiri, 1980, Sedimentological interpretation of errors in size analysis of sands; 1st European Meeting of the International Association of Sedimentologists, Ruhr University at Bochum, Federal Republic of Germany, March 1980.</li> <li>Brezina, Jiri, 1980, Size distribution of sand - sedimentological interpretation; 26th International Geological Congress, <a href="/wiki/Paris" title="Paris">Paris</a>, July 1980, Abstracts, vol. 2.</li> <li>Fouz, Infaz "Fluid Mechanics," Mechanical Engineering Dept., University of Oxford, 2001, p. 96</li> <li>Hughes, Roger "Civil Engineering Hydraulics," Civil and Environmental Dept., University of Melbourne 1997, pp. 107–152</li> <li>Jermy M., "Fluid Mechanics A Course Reader," Mechanical Engineering Dept., University of Canterbury, 2005, pp. d5.10.</li> <li>Purcell, E. M. "Life at Low Reynolds Number", <i>American Journal of Physics</i> vol 45, pp. 3–11 (1977)<a rel="nofollow" class="external autonumber" href="https://www2.gwu.edu/~phy21bio/Reading/Purcell_life_at_low_reynolds_number.pdf">[1]</a></li> <li>Truskey, G. A., Yuan, F, Katz, D. F. (2004). <i>Transport Phenomena in Biological Systems</i> Prentice Hall, pp. 7. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-042204-5" title="Special:BookSources/0-13-042204-5">0-13-042204-5</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-042204-0" title="Special:BookSources/978-0-13-042204-0">978-0-13-042204-0</a>.</li> <li>Zagarola, M. V. and Smits, A. J., "Experiments in High Reynolds Number Turbulent Pipe Flow." AIAA paper #96-0654, 34th AIAA Aerospace Sciences Meeting, Reno, Nevada, January 15–18, 1996.</li> <li><a href="/w/index.php?title=Isobel_Clark&action=edit&redlink=1" class="new" title="Isobel Clark (page does not exist)">Isobel Clark</a>, 1977, ROKE, a Computer Program for Non-Linear Least Squares Decomposition of Mixtures of Distributions; Computer & Geosciences (Pergamon Press), vol. 3, p. 245 - 256.</li> <li><a href="/w/index.php?title=B._C._Colby&action=edit&redlink=1" class="new" title="B. C. Colby (page does not exist)">B. C. Colby</a> and R. P. Christensen, 1957, Some Fundamentals of Particle Size Analysis; St. Anthony Falls Hydraulic Laboratory, Minneapolis, Minnesota, USA, Report Nr. 12/December, 55 pages.</li> <li><a href="/w/index.php?title=Arthur_T._Corey&action=edit&redlink=1" class="new" title="Arthur T. Corey (page does not exist)">Arthur T. Corey</a>, 1949, Influence of Shape on the Fall Velocity of Sand Grains; M. S. Thesis, Colorado Agricultural and Mechanical College, Fort Collins, Colorado, USA, December 102 pages.</li> <li><a href="/w/index.php?title=Joseph_R._Curray&action=edit&redlink=1" class="new" title="Joseph R. Curray (page does not exist)">Joseph R. Curray</a>, 1961, <a rel="nofollow" class="external text" href="http://www.grano.de/CURR1960.pdf">Tracing</a> sediment masses by grain size modes; Proc. Internat. Association of Sedimentology, Report of the 21st Session Norden, Internat. Geol. Congress, p. 119 - 129.</li> <li><a href="/w/index.php?title=Burghard_Walter_Flemming&action=edit&redlink=1" class="new" title="Burghard Walter Flemming (page does not exist)">Burghard Walter Flemming</a> & Karen Ziegler, 1995, High-resolution grain size distribution patterns and textural trends in the back-barrier environment of Spiekeroog Island (Southern North Sea); Senckenbergiana Maritima, vol. 26, No. 1+2, p. 1 - 24.</li> <li><a href="/w/index.php?title=Robert_Louis_Folk&action=edit&redlink=1" class="new" title="Robert Louis Folk (page does not exist)">Robert Louis Folk</a>, 1962, Of skewnesses and sands; Jour. Sediment. Petrol., vol. 8, No. 3/September, p. 105 - 111</li> <li>FOLK, Robert Louis & William C. WARD, 1957: Brazos River bar: a study in the significance of grain size parameters; Jour. Sediment. Petrol., vol. 27, No. 1/March, p. 3 - 26</li> <li><a href="/w/index.php?title=George_Herdan&action=edit&redlink=1" class="new" title="George Herdan (page does not exist)">George Herdan</a>, M. L. Smith & W. H. Hardwick (1960): Small Particle Statistics. 2nd revised edition, Butterworths (London, Toronto, etc.), 418 pp.</li> <li><a href="/wiki/Douglas_Inman" title="Douglas Inman">Douglas Inman</a>, 1952: Measures for describing the size distribution of sediments. Jour. Sediment. Petrology, vol. 22, No. 3/September, p. 125 - 145</li> <li><a href="/w/index.php?title=Miroslaw_Jonasz&action=edit&redlink=1" class="new" title="Miroslaw Jonasz (page does not exist)">Miroslaw Jonasz</a>, 1991: Size, shape, composition, and structure of microparticles from light scattering; in SYVITSKI, James P. M., 1991, Principles, Methods, and Application of Particle Size Analysis; Cambridge Univ. Press, <a href="/wiki/Cambridge" title="Cambridge">Cambridge</a>, 368 pp., p. 147.</li> <li><a href="/wiki/William_C._Krumbein" title="William C. Krumbein">William C. Krumbein</a>, 1934: Size frequency distribution of sediments; Jour. Sediment. Petrol., vol. 4, No. 2/August, p. 65 - 77.</li> <li>Krumbein, William Christian & Francis J. Pettijohn, 1938: Manual of Sedimentary Petrography; Appleton-Century-Crofts, Inc., New York; 549 pp.</li> <li><a href="/w/index.php?title=John_S._McNown&action=edit&redlink=1" class="new" title="John S. McNown (page does not exist)">John S. McNown</a> & Pin-Nam Lin, 1952, Sediment concentration and fall velocity; Proc. of the 2nd Midwestern Conf. on Fluid Mechanics, <a href="/wiki/Ohio_State_University" title="Ohio State University">Ohio State University</a>, <a href="/wiki/Columbus,_Ohio" title="Columbus, Ohio">Columbus, Ohio</a>; State Univ. of Iowa Reprints in Engineering, Reprint No. 109/1952, p. 401 - 411.</li> <li>McNownn, John S. & J. Malaika, 1950, Effects of Particle Shape of Settling Velocity at Low Reynolds' Numbers; American Geophysical Union Transactions, vol. 31, No. 1/February, p. 74 - 82.</li> <li><a href="/wiki/Gerard_V._Middleton" title="Gerard V. Middleton">Gerard V. Middleton</a> 1967, Experiments on density and turbidity currents, III; Deposition; Canadian Jour. of Earth Science, vol. 4, p. 475 - 505 (PSI definition: p. 483 - 485).</li> <li><a href="/wiki/Osborne_Reynolds" title="Osborne Reynolds">Osborne Reynolds</a>, 1883: An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. Roy. Soc., 174, Papers, vol. 2, p. 935 - 982</li> <li><a href="/w/index.php?title=E._F._Schultz&action=edit&redlink=1" class="new" title="E. F. Schultz (page does not exist)">E. F. Schultz</a>, R. H. Wilde & M. L. Albertson, 1954, Influence of Shape on the Fall Velocity of Sedimentary Particles; Colorado Agricultural & Mechanical College, Fort Collins, Colorado, MRD Sediment Series, No. 5/July (CER 54EFS6), 161 pages.</li> <li><a href="/w/index.php?title=H._J._Skidmore&action=edit&redlink=1" class="new" title="H. J. Skidmore (page does not exist)">H. J. Skidmore</a>, 1948, Development of a stratified-suspension technique for size-frequency analysis; Thesis, Department of Mechanics and Hydraulics, State Univ. of Iowa, p. 2 (? pages).</li> <li><a href="/w/index.php?title=James_P._M._Syvitski&action=edit&redlink=1" class="new" title="James P. M. Syvitski (page does not exist)">James P. M. Syvitski</a>, 1991, Principles, Methods, and Application of Particle Size Analysis; Cambridge Univ. Press, Cambridge, 368 pp.</li></ol> </div> <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=Reynolds_number&action=edit&section=29" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Reynolds_number" class="extiw" title="commons:Category:Reynolds number">Reynolds number</a></span>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/II_41.html#Ch41-S3">The Reynolds number - The Feynman Lectures on Physics</a></li> <li><a rel="nofollow" class="external text" href="https://sixtysymbols.com/videos/reynolds.htm">The Reynolds Number</a> at Sixty Symbols</li> <li><a rel="nofollow" class="external text" href="https://uh.edu/engines/epi1529.htm">Reynolds mini-biography and picture of original apparatus at Manchester University.</a></li> <li><a rel="nofollow" class="external text" href="https://www.calculatoratoz.com/en/reynolds-number-of-fluid-flowing-in-pipe-calculator/Calc-24037">Reynolds Number Calculation</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Dimensionless_numbers_in_fluid_mechanics" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><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:Dimensionless_numbers_in_fluid_mechanics" title="Template:Dimensionless numbers in fluid mechanics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Dimensionless_numbers_in_fluid_mechanics" title="Template talk:Dimensionless numbers in fluid mechanics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Dimensionless_numbers_in_fluid_mechanics" title="Special:EditPage/Template:Dimensionless numbers in fluid mechanics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Dimensionless_numbers_in_fluid_mechanics" style="font-size:114%;margin:0 4em"><a href="/wiki/Dimensionless_numbers_in_fluid_mechanics" title="Dimensionless numbers in fluid mechanics">Dimensionless numbers in fluid mechanics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Archimedes_number" title="Archimedes number">Archimedes</a></li> <li><a href="/wiki/Atwood_number" title="Atwood number">Atwood</a></li> <li><a href="/wiki/Bagnold_number" title="Bagnold number">Bagnold</a></li> <li><a href="/wiki/Bejan_number" title="Bejan number">Bejan</a></li> <li><a href="/wiki/Biot_number" title="Biot number">Biot</a></li> <li><a href="/wiki/E%C3%B6tv%C3%B6s_number" title="Eötvös number">Bond</a></li> <li><a href="/wiki/Brinkman_number" title="Brinkman number">Brinkman</a></li> <li><a href="/wiki/Capillary_number" title="Capillary number">Capillary</a></li> <li><a href="/wiki/Cauchy_number" title="Cauchy number">Cauchy</a></li> <li><a href="/wiki/Chandrasekhar_number" title="Chandrasekhar number">Chandrasekhar</a></li> <li><a href="/wiki/Damk%C3%B6hler_numbers" title="Damköhler numbers">Damköhler</a></li> <li><a href="/wiki/Darcy_number" title="Darcy number">Darcy</a></li> <li><a href="/wiki/Dean_number" title="Dean number">Dean</a></li> <li><a href="/wiki/Deborah_number" title="Deborah number">Deborah</a></li> <li><a href="/wiki/Dukhin_number" title="Dukhin number">Dukhin</a></li> <li><a href="/wiki/Eckert_number" title="Eckert number">Eckert</a></li> <li><a href="/wiki/Ekman_number" title="Ekman number">Ekman</a></li> <li><a href="/wiki/E%C3%B6tv%C3%B6s_number" title="Eötvös number">Eötvös</a></li> <li><a href="/wiki/Euler_number_(physics)" title="Euler number (physics)">Euler</a></li> <li><a href="/wiki/Froude_number" title="Froude number">Froude</a></li> <li><a href="/wiki/Galilei_number" title="Galilei number">Galilei</a></li> <li><a href="/wiki/Graetz_number" title="Graetz number">Graetz</a></li> <li><a href="/wiki/Grashof_number" title="Grashof number">Grashof</a></li> <li><a href="/wiki/G%C3%B6rtler_vortices" title="Görtler vortices">Görtler</a></li> <li><a href="/wiki/Hagen_number" title="Hagen number">Hagen</a></li> <li><a href="/wiki/Iribarren_number" title="Iribarren number">Iribarren</a></li> <li><a href="/wiki/Kapitza_number" title="Kapitza number">Kapitza</a></li> <li><a href="/wiki/Keulegan%E2%80%93Carpenter_number" title="Keulegan–Carpenter number">Keulegan–Carpenter</a></li> <li><a href="/wiki/Knudsen_number" title="Knudsen number">Knudsen</a></li> <li><a href="/wiki/Laplace_number" title="Laplace number">Laplace</a></li> <li><a href="/wiki/Lewis_number" title="Lewis number">Lewis</a></li> <li><a href="/wiki/Mach_number" title="Mach number">Mach</a></li> <li><a href="/wiki/Marangoni_number" title="Marangoni number"> Marangoni</a></li> <li><a href="/wiki/Morton_number" title="Morton number">Morton</a></li> <li><a href="/wiki/Nusselt_number" title="Nusselt number">Nusselt</a></li> <li><a href="/wiki/Ohnesorge_number" title="Ohnesorge number">Ohnesorge</a></li> <li><a href="/wiki/P%C3%A9clet_number" title="Péclet number">Péclet</a></li> <li><a href="/wiki/Prandtl_number" title="Prandtl number">Prandtl</a> <ul><li><a href="/wiki/Magnetic_Prandtl_number" title="Magnetic Prandtl number">magnetic</a></li> <li><a href="/wiki/Turbulent_Prandtl_number" title="Turbulent Prandtl number">turbulent</a></li></ul></li> <li><a href="/wiki/Rayleigh_number" title="Rayleigh number">Rayleigh</a></li> <li><a class="mw-selflink selflink">Reynolds</a> <ul><li><a href="/wiki/Magnetic_Reynolds_number" title="Magnetic Reynolds number">magnetic</a></li></ul></li> <li><a href="/wiki/Richardson_number" title="Richardson number">Richardson</a></li> <li><a href="/wiki/Roshko_number" title="Roshko number">Roshko</a></li> <li><a href="/wiki/Rossby_number" title="Rossby number">Rossby</a></li> <li><a href="/wiki/Rouse_number" title="Rouse number">Rouse</a></li> <li><a href="/wiki/Schmidt_number" title="Schmidt number">Schmidt</a></li> <li><a href="/wiki/Scruton_number" title="Scruton number">Scruton</a></li> <li><a href="/wiki/Sherwood_number" title="Sherwood number">Sherwood</a></li> <li><a href="/wiki/Shields_parameter" title="Shields parameter">Shields</a></li> <li><a href="/wiki/Stanton_number" title="Stanton number">Stanton</a></li> <li><a href="/wiki/Stokes_number" title="Stokes number">Stokes</a></li> <li><a href="/wiki/Strouhal_number" title="Strouhal number">Strouhal</a></li> <li><a href="/wiki/Stuart_number" title="Stuart number">Stuart</a></li> <li><a href="/wiki/Laplace_number" title="Laplace number">Suratman</a></li> <li><a href="/wiki/Taylor_number" title="Taylor number">Taylor</a></li> <li><a href="/wiki/Ursell_number" title="Ursell number">Ursell</a></li> <li><a href="/wiki/Weber_number" title="Weber number">Weber</a></li> <li><a href="/wiki/Weissenberg_number" title="Weissenberg number">Weissenberg</a></li> <li><a href="/wiki/Womersley_number" title="Womersley number">Womersley</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q178932#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Reynolds-Zahl"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4177959-9">Germany</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Reynolds number"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85113557">United States</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Reynoldsovo číslo"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph1002002&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007536463305171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Reynolds_number&oldid=1260072694">https://en.wikipedia.org/w/index.php?title=Reynolds_number&oldid=1260072694</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Aerodynamics" title="Category:Aerodynamics">Aerodynamics</a></li><li><a href="/wiki/Category:Convection" title="Category:Convection">Convection</a></li><li><a href="/wiki/Category:Dimensionless_numbers" title="Category:Dimensionless numbers">Dimensionless numbers</a></li><li><a href="/wiki/Category:Dimensionless_numbers_of_fluid_mechanics" title="Category:Dimensionless numbers of fluid mechanics">Dimensionless numbers of fluid mechanics</a></li><li><a href="/wiki/Category:Dimensionless_numbers_of_thermodynamics" title="Category:Dimensionless numbers of thermodynamics">Dimensionless numbers of thermodynamics</a></li><li><a href="/wiki/Category:Fluid_dynamics" title="Category:Fluid dynamics">Fluid dynamics</a></li><li><a href="/wiki/Category:Piping" title="Category:Piping">Piping</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Pages_using_multiple_image_with_auto_scaled_images" title="Category:Pages using multiple image with auto scaled images">Pages using multiple image with auto scaled images</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_February_2014" title="Category:Articles needing additional references from February 2014">Articles needing additional references from February 2014</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_January_2019" title="Category:Articles needing additional references from January 2019">Articles needing additional references from January 2019</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_December_2014" title="Category:Articles needing additional references from December 2014">Articles needing additional references from December 2014</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_March_2021" title="Category:Articles with unsourced statements from March 2021">Articles with unsourced statements from March 2021</a></li><li><a href="/wiki/Category:Commons_category_link_is_on_Wikidata" title="Category:Commons category link is on Wikidata">Commons category link is on Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 28 November 2024, at 18:05<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Reynolds_number&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5c59558b9d-qr7ld","wgBackendResponseTime":177,"wgPageParseReport":{"limitreport":{"cputime":"1.179","walltime":"1.491","ppvisitednodes":{"value":9195,"limit":1000000},"postexpandincludesize":{"value":139047,"limit":2097152},"templateargumentsize":{"value":10884,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":8,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":152029,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1206.655 1 -total"," 17.28% 208.532 2 Template:Reflist"," 12.75% 153.830 3 Template:Annotated_link"," 10.49% 126.594 10 Template:Cite_web"," 10.48% 126.425 29 Template:Sfn"," 7.55% 91.070 62 Template:Math"," 7.53% 90.858 14 Template:Cite_journal"," 7.03% 84.806 1 Template:Dimensionless_numbers_in_fluid_mechanics"," 6.75% 81.427 1 Template:Navbox"," 6.57% 79.318 1 Template:Short_description"]},"scribunto":{"limitreport-timeusage":{"value":"0.761","limit":"10.000"},"limitreport-memusage":{"value":25020249,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFBatchelor1967\"] = 1,\n [\"CITEREFBirdStewartLightfoot2006\"] = 1,\n [\"CITEREFBush\"] = 1,\n [\"CITEREFDusenbery2009\"] = 1,\n [\"CITEREFDwivedi1977\"] = 1,\n [\"CITEREFEhrensteinEloy2013\"] = 1,\n [\"CITEREFFalkovich2018\"] = 1,\n [\"CITEREFFoxMcDonaldPritchard2004\"] = 1,\n [\"CITEREFGómez_Blázquez2016\"] = 1,\n [\"CITEREFHall2015\"] = 1,\n [\"CITEREFHelpsMcDonald1954\"] = 1,\n [\"CITEREFHolman2002\"] = 1,\n [\"CITEREFIncroperaDeWitt1981\"] = 1,\n [\"CITEREFLissaman1983\"] = 1,\n [\"CITEREFLos2006\"] = 1,\n [\"CITEREFMannevillePomeau2009\"] = 1,\n [\"CITEREFPatelRodiScheuerer1985\"] = 1,\n [\"CITEREFPotterWiggertRamadan2012\"] = 1,\n [\"CITEREFRafieian-KopaeiSetorkiDoudiBaradaran2014\"] = 1,\n [\"CITEREFReynolds1883\"] = 1,\n [\"CITEREFRhodes1989\"] = 1,\n [\"CITEREFRott1990\"] = 1,\n [\"CITEREFSchlichtingGersten2017\"] = 1,\n [\"CITEREFSeshadri1978\"] = 1,\n [\"CITEREFSinnottCoulsonRichardson2005\"] = 1,\n [\"CITEREFSommerfeld1908\"] = 1,\n [\"CITEREFStokes1851\"] = 1,\n [\"CITEREFStreeter1965\"] = 1,\n [\"CITEREFTansleyMarshall2001\"] = 1,\n [\"CITEREFWangOthmer2016\"] = 1,\n}\ntemplate_list = table#1 {\n [\"=\"] = 3,\n [\"Annotated link\"] = 3,\n [\"Authority control\"] = 1,\n [\"Citation needed\"] = 1,\n [\"Cite book\"] = 13,\n [\"Cite journal\"] = 14,\n [\"Cite thesis\"] = 1,\n [\"Cite web\"] = 10,\n [\"Clear\"] = 1,\n [\"Commons category\"] = 1,\n [\"Dimensionless numbers in fluid mechanics\"] = 1,\n [\"Equation box 1\"] = 1,\n [\"ISBN\"] = 2,\n [\"Main\"] = 1,\n [\"Math\"] = 62,\n [\"Multiple images\"] = 1,\n [\"Ordered list\"] = 1,\n [\"Overline\"] = 1,\n [\"Refbegin\"] = 2,\n [\"Refend\"] = 2,\n [\"Reflist\"] = 2,\n [\"Sfn\"] = 29,\n [\"Sfrac\"] = 7,\n [\"Short description\"] = 1,\n [\"Sub\"] = 9,\n [\"Unreferenced section\"] = 3,\n [\"Val\"] = 3,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-5c59558b9d-48l5j","timestamp":"20241130014353","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Reynolds number","url":"https:\/\/en.wikipedia.org\/wiki\/Reynolds_number","sameAs":"http:\/\/www.wikidata.org\/entity\/Q178932","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q178932","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-06-16T05:20:29Z","dateModified":"2024-11-28T18:05:07Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/0\/03\/Laminar-turbulent_transition.jpg","headline":"dimensionless quantity that is used to help predict similar flow patterns in different fluid flow situations"}</script> </body> </html>