From anomalous dissipation through Euler singularities to stabilized finite element methods for turbulent flows

From anomalous dissipation through Euler singularities to stabilized finite element methods for turbulent flows | 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 From anomalous dissipation through Euler singularities to stabilized finite element methods for turbulent flows Niklas Fehn, Martin Kronbichler, Gert Lube This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4187657/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 22 Apr, 2025 Read the published version in Flow, Turbulence and Combustion → Version 1 posted 9 You are reading this latest preprint version Abstract It is well-known that kinetic energy produced artificially by an inadequate numerical discretization of nonlinear transport terms may lead to a blow-up of the numerical solution in simulations of fluid dynamical problems such as incompressible turbulent flows. However, the community seems to be divided whether this problem should be resolved by the use of discretely energy-preserving or dissipative discretization schemes. The rationale for discretely energy-preserving schemes is often based on the expectation of exact conservation of kinetic energy in the inviscid limit, which mathematically relies on the assumption of sufficient regularity of the solution. There is the (contradictory) phenomenological observation in turbulence that flows dissipate energy in the limit of vanishing molecular viscosity, an "anomalous" phenomenon termed dissipation anomaly or the zeroth law of turbulence. As already conjectured by Onsager, the Euler equations may dissipate kinetic energy through the formation of singularities of the velocity field. With the proof of Onsager's conjecture in recent years, a consequence for designing numerical methods for turbulent flows is that the smoothness assumption behind conservation of energy in the inviscid limit becomes indeed critical for turbulent flows. The velocity field rather has to be expected to show singular behavior towards the inviscid limit, supporting the dissipation of kinetic energy. Our main argument is that designing numerical methods against the background of this physical behavior is a strong rationale for the construction of dissipative (or dissipation-aware) numerical schemes for convective terms. From that perspective, numerical dissipation does not appear artificial, but as an important ingredient to overcome problems introduced by energy-conserving numerical methods such as the inability to represent anomalous dissipation as well as the accumulation of energy in small scales, which is known as thermalization. This work discusses stabilized H1, L2, and H(div)-conforming finite element methods with a focus on the energy-stability of the numerical method and its dissipation mechanisms to predict inertial dissipation. Finally, we discuss the achievable convergence rate for the kinetic energy in under-resolved turbulent flow simulations. anomalous energy dissipation finite-time Euler singularities large-eddy simulation Navier–Stokes equations Onsager’s conjecture turbulence Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 22 Apr, 2025 Read the published version in Flow, Turbulence and Combustion → Version 1 posted Editorial decision: Revision requested 12 Sep, 2024 Reviews received at journal 12 Sep, 2024 Reviews received at journal 26 May, 2024 Reviewers agreed at journal 24 Apr, 2024 Reviewers agreed at journal 15 Apr, 2024 Reviewers invited by journal 09 Apr, 2024 Editor assigned by journal 02 Apr, 2024 Submission checks completed at journal 01 Apr, 2024 First submitted to journal 29 Mar, 2024 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. 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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-4187657","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":287231494,"identity":"c2890fb2-b289-47bb-8598-5d48f63495b6","order_by":0,"name":"Niklas Fehn","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5klEQVRIie3OsQrCMBCA4ZNAurRkvSLqK6QIVUHwVTo5desoaKDQzd1BH8LNMVLQRXQtdNFdoWOHDqYo6BQ7OuQfEgj5cgEwmf4xQltSbQwQWldQq4r/IlATVyAQ3ozAh1B8H+nJwKKBLCtA1l4eZs5uCEyQ7VVHRjGV+2UC6K7P09w5IaCkkXYMTy0hHQFznoV+7iS4ENL28RfZV+pjE0UiRaAn7UGpJ1SmNgXkGPqkJlxN0YmaBGknQcQs7LsbRbyURvqPXQ7e7V6Nka1Cr3gkc+ge422hHfPq+1nS4L7JZDKZ9D0Bx9tB7qipG0gAAAAASUVORK5CYII=","orcid":"","institution":"University of Augsburg","correspondingAuthor":true,"prefix":"","firstName":"Niklas","middleName":"","lastName":"Fehn","suffix":""},{"id":287231495,"identity":"df97e419-99c0-46a8-9ddd-aa076bf651ca","order_by":1,"name":"Martin Kronbichler","email":"","orcid":"","institution":"Ruhr University Bochum","correspondingAuthor":false,"prefix":"","firstName":"Martin","middleName":"","lastName":"Kronbichler","suffix":""},{"id":287231496,"identity":"0b393807-17b0-48e2-a8b8-32658a04c082","order_by":2,"name":"Gert Lube","email":"","orcid":"","institution":"University of Göttingen","correspondingAuthor":false,"prefix":"","firstName":"Gert","middleName":"","lastName":"Lube","suffix":""}],"badges":[],"createdAt":"2024-03-29 11:36:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4187657/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4187657/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s10494-025-00639-6","type":"published","date":"2025-04-22T15:58:12+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":81569774,"identity":"6e8ed7c4-fc9e-4bf1-bc7c-ce620313e48e","added_by":"auto","created_at":"2025-04-28 16:11:03","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1252800,"visible":true,"origin":"","legend":"","description":"","filename":"latex.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4187657/v1_covered_ca4e585b-b269-47b3-9a7f-af6b8f33ac93.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"From anomalous dissipation through Euler singularities to stabilized finite element methods for turbulent flows","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"flow-turbulence-and-combustion","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"appl","sideBox":"Learn more about [Flow, Turbulence and Combustion](http://link.springer.com/journal/10494)","snPcode":"10494","submissionUrl":"https://submission.nature.com/new-submission/10494/3","title":"Flow, Turbulence and Combustion","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"anomalous energy dissipation, finite-time Euler singularities, large-eddy simulation, Navier–Stokes equations, Onsager’s conjecture, turbulence","lastPublishedDoi":"10.21203/rs.3.rs-4187657/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4187657/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"It is well-known that kinetic energy produced artificially by an inadequate numerical discretization of nonlinear transport terms may lead to a blow-up of the numerical solution in simulations of fluid dynamical problems such as incompressible turbulent flows. 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