Augmented Multiscale Formulation (AMS) for Incompressible Turbulence

Augmented Multiscale Formulation (AMS) for Incompressible Turbulence | 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 Augmented Multiscale Formulation (AMS) for Incompressible Turbulence Rakesh Ranjan, Guillermo Araya This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5338976/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 We revise the multiscale formulation for incompressible flow computations. In addition to the classical multiscale formulation, we incorporate an iterative penalty approach to ensure the incompressibility condition during the solution of the Navier-Stokes equations. The constraint of incompressibility, enforced by the continuity equation, is approached via an iterative penalty method. The multiscale formulation (MS) of Hughes et al. [1] is reconsidered, introducing a new formulation named Augmented Multiscale Formulation (AMS). We maintain consistency of the formulation by carrying the velocity and pressure variables rather than an L2 projection of the same variables, as previously investigated by researchers in the past. Inexact Newton solvers are utilized to solve discrete problems. In this aspect, the formulation departs from the Multiscale Formulation for incompressible turbulence. In the current formulation, there is no requirement to resort to least squares stabilization as a separate stabilization mechanism, and this departure is intuitive since there are no shock discontinuities when solving incompressible flow. We validated the AMS formulation with the Kovasznay flow problem and standard solutions available. We benchmark the temporal evolution of AMS formulation with the driven cavity solutions provided by the ASUPG/AGLS formulation. Furthermore, the revised formulation is tested on a series of problems such as flow past a series of cylinders, cylinders in rotation, to serve as incompressible flow benchmarks and excellent agreement is reported for all problems, including experimental data from Zdravkovich [2] and Schlichting et al. [3] for the problem of flow past a single cylinder. We report non-linear convergence histories of the formulation, as well which are difficult to provide within the MS formulation. The predictor-corrector scheme that has been utilized for the solutions of incompressible flow computations has been the reason the method has not been able to provide non linear convergence histories for the problems solved. There is departure from the procedure in the present framework: present AMS and MS formulation both recover lower L2 error with analytical solutions of Navier-Stokes equations and the error is found to be lower than the ASUPG formulation of Ranjan [4] for low plevels of 3 and 4. For higher polynomial orders, we report suboptimal convergence of the pressures. This paper provides an insightful look at the non-linear convergence of the incompressible flow Navier-Stokes equations within a stabilized finite element setting. AMS formulation Spectral Element Method Stabilized Finite Element Incompressible Turbulence Inexact Newton Krylov Method. Full Text Additional Declarations No competing interests reported. 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-5338976","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":372434002,"identity":"9cbb2c49-b71a-4d05-91d3-404e9bd6ccaa","order_by":0,"name":"Rakesh Ranjan","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3UlEQVRIiWNgGAWjYHACZiA+AGIwPgASPHzEapEAMQxAWthI0cIGIhgIajGfffixwYc/d+r4p50xq/yaYyfDxsD88NENPFpkzqUZJ85seyYhcTvH7LbstmSgw9iMjXPwaJHgYTA+zNtwWIIBpEVyGzNQCw+bNH4t7J8P//lzWEIeqKVYcls9MVp4jJMZ2A5LGAC1MH7cdpgoLcWGvW2HJTfeTiuWZtx2nIeNmaBf2DdL/PhzmF/udvLGjz+3Vdvzszc/fIxPCwpg5gGTxCoHAcYfpKgeBaNgFIyCEQMAGcRC/hZL3pQAAAAASUVORK5CYII=","orcid":"","institution":"General Dynamics","correspondingAuthor":true,"prefix":"","firstName":"Rakesh","middleName":"","lastName":"Ranjan","suffix":""},{"id":372434003,"identity":"78926f37-c0da-40b2-8015-e8616dd48fb9","order_by":1,"name":"Guillermo Araya","email":"","orcid":"","institution":"University of Texas","correspondingAuthor":false,"prefix":"","firstName":"Guillermo","middleName":"","lastName":"Araya","suffix":""}],"badges":[],"createdAt":"2024-10-26 20:23:08","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5338976/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5338976/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":77075092,"identity":"60087bed-d968-46bf-9bb5-955262fc0fa9","added_by":"auto","created_at":"2025-02-25 00:01:30","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1941388,"visible":true,"origin":"","legend":"","description":"","filename":"AMSEComp.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5338976/v1_covered_384161f1-f546-41ed-9131-ae38acbc839c.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Augmented Multiscale Formulation (AMS) for Incompressible Turbulence","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"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":"AMS formulation, Spectral Element Method, Stabilized Finite Element, Incompressible Turbulence, Inexact Newton Krylov Method.","lastPublishedDoi":"10.21203/rs.3.rs-5338976/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5338976/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"We revise the multiscale formulation for incompressible flow computations. In addition to the classical multiscale formulation, we incorporate an iterative penalty approach to ensure the incompressibility condition during the solution of the Navier-Stokes equations. The constraint of incompressibility, enforced by the continuity equation, is approached via an iterative penalty method. The multiscale formulation (MS) of Hughes et al. [1] is reconsidered, introducing a new formulation named Augmented Multiscale Formulation (AMS). We maintain consistency of the formulation by carrying the velocity and pressure variables rather than an L2 projection of the same variables, as previously investigated by researchers in the past. Inexact Newton solvers are utilized to solve discrete problems. In this aspect, the formulation departs from the Multiscale Formulation for incompressible turbulence. In the current formulation, there is no requirement to resort to least squares stabilization as a separate stabilization mechanism, and this departure is intuitive since there are no shock discontinuities when solving incompressible flow. We validated the AMS formulation with the Kovasznay flow problem and standard solutions available. We benchmark the temporal evolution of AMS formulation with the driven cavity solutions provided by the ASUPG/AGLS formulation. Furthermore, the revised formulation is tested on a series of problems such as flow past a series of cylinders, cylinders in rotation, to serve as incompressible flow benchmarks and excellent agreement is reported for all problems, including experimental data from Zdravkovich [2] and Schlichting et al. [3] for the problem of flow past a single cylinder. We report non-linear convergence histories of the formulation, as well which are difficult to provide within the MS formulation. The predictor-corrector scheme that has been utilized for the solutions of incompressible flow computations has been the reason the method has not been able to provide non linear convergence histories for the problems solved. There is departure from the procedure in the present framework: present AMS and MS formulation both recover lower L2 error with analytical solutions of Navier-Stokes equations and the error is found to be lower than the ASUPG formulation of Ranjan [4] for low plevels of 3 and 4. For higher polynomial orders, we report suboptimal convergence of the pressures. This paper provides an insightful look at the non-linear convergence of the incompressible flow Navier-Stokes equations within a stabilized finite element setting.","manuscriptTitle":"Augmented Multiscale Formulation (AMS) for Incompressible Turbulence","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-11-06 10:59:23","doi":"10.21203/rs.3.rs-5338976/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":"d7b22e77-d926-4c2b-a519-ff4fdc000aa9","owner":[],"postedDate":"November 6th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-02-24T23:53:14+00:00","versionOfRecord":[],"versionCreatedAt":"2024-11-06 10:59:23","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5338976","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5338976","identity":"rs-5338976","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}