Reduced topological-sector closure for periodic incompressible Navier–Stokes flow

Reduced topological-sector closure for periodic incompressible Navier–Stokes flow | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Reduced topological-sector closure for periodic incompressible Navier–Stokes flow GuoJunPan This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9688015/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Topology is central to the dynamics of linked, twisted, reconnecting, and reorganizing vortices. The value of a topology-aware fluid model is therefore not expected to be a small universal accuracy gain for routine smooth flows, where ordinary Navier–Stokes evolution and geometric curvature information may already be sufficient. Its main role is to predict, detect, and classify topological-transition states: vortex merger or reconnection, turbulent-structure regime changes, separated-flow reorganization, and larger-scale circulation pattern shifts. We formulate a reduced topological-sector state closure: a map from a periodic incompressible velocity field to a topological sector, followed by a sector-specific reduced predictor. The state coordinate is derived from a helicity-sector free energy, while local geometric curvature and ordinary Navier–Stokes dynamics remain inside each sector. A continuum force embedding is given only to show how the same sector coordinate could enter a projected dynamical equation; it is not the object claimed to be validated here. The validated claim is sharper: topological-sector and helicity-distribution variables carry predictive information that geometry-only and global-helicity descriptions miss near topological transitions. Periodic helical vortex-tube fields are constructed in pairs that either share core geometry but differ in axial twist and helicity, or have nearly the same total helicity but different spatial helicity distribution. In threshold tests, geometry-only classification fails on same-geometry twist pairs, while helicity/topology features remain accurate under up to 10% relative velocity noise. In same-global-helicity distribution pairs, topology-enriched classification improves accuracy from 50.0% for geometry-only and global-helicity baselines to 77.25%. After short classical Navier–Stokes evolution, ridge-regularized topology-enriched features outperform the best reduced baseline on both tasks: 95.35% versus 93.61% for helicitydistribution classification and 89.44% versus 76.67% for twist-level classification. In an expanded coarse-to-reference prediction test, topology-enriched coarse features predict the high-resolution final helicity-concentration observable with mean absolute error 1.82 × 10−3 , compared with 1.12 × 10−2 from the direct coarse observable and 5.94 × 10−3 from global-helicity features. Across five topology-sensitive observables, the topology-enriched predictor reduces mean normalized error from 0.111 for global-helicity features to 0.0505. An external dynamic-stall audit gives a separate phenomenon-level check: for 51 separated-flow validation cases, the sector predictor reduces lift-amplification error by 63% relative to a quasi-steady baseline and by 73% relative to an attached-flow Theodorsen baseline. Extreme-boundary and ablation tests show that the advantage is robust under held-out separation over d = 0.14–0.48, while stretching-gate features require noise-regime calibration. The result supports a bounded but physical claim: a helicity-sector topological free-energy coordinate supplies independent information for topologysensitive Navier–Stokes vortex states and can serve as the state variable for transition-aware class-specific modeling. 1 helicity topology vortex dynamics Navier-Stokes topological sector vortex reconnection twist helicity density geometry vs topology fluid modeling reduced-order modeling classification dynamic stall topological state closure incompressible flow Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9688015","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":638781806,"identity":"e611cf4c-47d7-47f0-875c-8e3473673081","order_by":0,"name":"GuoJunPan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtklEQVRIiWNgGAWjYBACNvbm4x8+GLDJ8ROthY/nWBrjjAI+Y8kGYrXISeSYMfN8kEs0OEC0w4C2PJxhYJZgfDx5A8OPim1EaAH6xeCDQVqe2ZlnBYw9Z24TZUuC5AyDY8VmN3IMmBnbiNEikWMgzWPwP3HzDBK0mAG1sCVukCBaC8+xZMMZBmzGEkC/HCTKL/LtzQcffPgDjMr25I0PflQQoQUJJBAfNQgtpOoYBaNgFIyCEQIAD7M9UHhkl8EAAAAASUVORK5CYII=","orcid":"https://orcid.org/0009-0002-5750-6523","institution":"Independent Researcher","correspondingAuthor":true,"prefix":"","firstName":"","middleName":"","lastName":"GuoJunPan","suffix":""}],"badges":[],"createdAt":"2026-05-12 07:13:31","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-9688015/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9688015/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":109162143,"identity":"447fc4a8-c757-4ec7-9e70-f0e208e4805b","added_by":"auto","created_at":"2026-05-13 07:46:32","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":803916,"visible":true,"origin":"","legend":"","description":"","filename":"pofnsv4.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9688015/v1_covered_e118f2cc-36ad-419e-885d-780f59141221.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eReduced topological-sector closure for periodic incompressible Navier–Stokes flow\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"helicity, topology, vortex dynamics, Navier-Stokes, topological sector, vortex reconnection, twist, helicity density, geometry vs topology, fluid modeling, reduced-order modeling, classification, dynamic stall, topological state closure, incompressible flow","lastPublishedDoi":"10.21203/rs.3.rs-9688015/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9688015/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eTopology is central to the dynamics of linked, twisted, reconnecting, and reorganizing vortices. The value of a topology-aware fluid model is therefore not expected to be a small universal accuracy gain for routine smooth flows, where ordinary Navier–Stokes evolution and geometric curvature information may already be sufficient. Its main role is to predict, detect, and classify topological-transition states: vortex merger or reconnection, turbulent-structure regime changes, separated-flow reorganization, and larger-scale circulation pattern shifts. We formulate a reduced topological-sector state closure: a map from a periodic incompressible velocity field to a topological sector, followed by a sector-specific reduced predictor. The state coordinate is derived from a helicity-sector free energy, while local geometric curvature and ordinary Navier–Stokes dynamics remain inside each sector. A continuum force embedding is given only to show how the same sector coordinate could enter a projected dynamical equation; it is not the object claimed to be validated here. The validated claim is sharper: topological-sector and helicity-distribution variables carry predictive information that geometry-only and global-helicity descriptions miss near topological transitions. Periodic helical vortex-tube fields are constructed in pairs that either share core geometry but differ in axial twist and helicity, or have nearly the same total helicity but different spatial helicity distribution. In threshold tests, geometry-only classification fails on same-geometry twist pairs, while helicity/topology features remain accurate under up to 10% relative velocity noise. In same-global-helicity distribution pairs, topology-enriched classification improves accuracy from 50.0% for geometry-only and global-helicity baselines to 77.25%. After short classical Navier–Stokes evolution, ridge-regularized topology-enriched features outperform the best reduced baseline on both tasks: 95.35% versus 93.61% for helicitydistribution classification and 89.44% versus 76.67% for twist-level classification. In an expanded coarse-to-reference prediction test, topology-enriched coarse features predict the high-resolution final helicity-concentration observable with mean absolute error 1.82 × 10−3 , compared with 1.12 × 10−2 from the direct coarse observable and 5.94 × 10−3 from global-helicity features. Across five topology-sensitive observables, the topology-enriched predictor reduces mean normalized error from 0.111 for global-helicity features to 0.0505. An external dynamic-stall audit gives a separate phenomenon-level check: for 51 separated-flow validation cases, the sector predictor reduces lift-amplification error by 63% relative to a quasi-steady baseline and by 73% relative to an attached-flow Theodorsen baseline. Extreme-boundary and ablation tests show that the advantage is robust under held-out separation over d = 0.14–0.48, while stretching-gate features require noise-regime calibration. The result supports a bounded but physical claim: a helicity-sector topological free-energy coordinate supplies independent information for topologysensitive Navier–Stokes vortex states and can serve as the state variable for transition-aware class-specific modeling. 1\u003c/p\u003e","manuscriptTitle":"Reduced topological-sector closure for periodic incompressible Navier–Stokes flow","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-05-13 07:46:12","doi":"10.21203/rs.3.rs-9688015/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"ebee59c4-10f3-432b-a2e9-42d63663fe71","owner":[],"postedDate":"May 13th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-05-13T07:46:12+00:00","versionOfRecord":[],"versionCreatedAt":"2026-05-13 07:46:12","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9688015","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9688015","identity":"rs-9688015","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}