Gauss principle in incompressible flow: a unified variational perspective on pressure and projection.
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| Title: | Gauss principle in incompressible flow: a unified variational perspective on pressure and projection. |
|---|---|
| Authors: | Duraisamy, Karthik1 (AUTHOR) kdur@umich.edu |
| Source: | Journal of Fluid Mechanics. 6/25/2026, Vol. 1037, p1-12. 12p. |
| Subjects: | Incompressible flow, Variational approach (Mathematics), Fluid mechanics, Euler equations, Neumann problem, Divergence theorem |
| Abstract: | Content of image described in text. Following recent work (Gonzalez & Taha 2022 J. Fluid Mech. , vol. 941 , A58; Peters & Ormiston 2025 AIAA J. , vol. 63 , pp. 6–20), this manuscript clarifies what the Gauss–Appell principle determines in incompressible, inviscid flow and how it connects to classical projection methods. At a fixed time, freezing the velocity and varying only the material acceleration leads to minimisation of a quadratic subject to acceleration-level constraints. First-order conditions yield a Poisson–Neumann problem for a reaction pressure whose gradient removes the non-solenoidal and wall-normal content of the provisional residual, precisely the well-known Leray–Hodge projection. Thus, Gauss–Appell enforces the instantaneous kinematic constraints and recovers Euler at the instant. Once the impressed physics is specified, for instance via external body forces, the reaction pressure is uniquely determined (up to an additive constant) as the Lagrange multiplier enforcing incompressibility and wall impermeability; it does no work on divergence-free, wall-tangent motions. This is the well-established interpretation of pressure in incompressible flow. The direct, fixed-time application of this principle determines the reaction pressure for an already-specified velocity field and does not, by itself, select circulation or stagnation points, because these are properties of the velocity state, not the instantaneous acceleration correction. The formal decomposition of the pressure into impressed and reaction components admits representational freedom that does not imply physical non-uniqueness of the constraint force. Orthogonality conventions such as Dirichlet orthogonality can fix the representational freedom as an additional modelling choice. This variational viewpoint also yields a simple computational diagnostic: the minimised Appellian equals a upper L squared L 2 $L^2$ norm of the reaction-pressure gradient which vanishes for constraint-compatible updates and grows with the magnitude of divergence and wall-flux mismatch. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Fluid Mechanics is the property of Cambridge University Press and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 194751957 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Gauss principle in incompressible flow: a unified variational perspective on pressure and projection. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Duraisamy%2C+Karthik%22">Duraisamy, Karthik</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> kdur@umich.edu</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Fluid+Mechanics%22">Journal of Fluid Mechanics</searchLink>. 6/25/2026, Vol. 1037, p1-12. 12p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Incompressible+flow%22">Incompressible flow</searchLink><br /><searchLink fieldCode="DE" term="%22Variational+approach+%28Mathematics%29%22">Variational approach (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Fluid+mechanics%22">Fluid mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Euler+equations%22">Euler equations</searchLink><br /><searchLink fieldCode="DE" term="%22Neumann+problem%22">Neumann problem</searchLink><br /><searchLink fieldCode="DE" term="%22Divergence+theorem%22">Divergence theorem</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: Content of image described in text. Following recent work (Gonzalez & Taha 2022 J. Fluid Mech. , vol. 941 , A58; Peters & Ormiston 2025 AIAA J. , vol. 63 , pp. 6–20), this manuscript clarifies what the Gauss–Appell principle determines in incompressible, inviscid flow and how it connects to classical projection methods. At a fixed time, freezing the velocity and varying only the material acceleration leads to minimisation of a quadratic subject to acceleration-level constraints. First-order conditions yield a Poisson–Neumann problem for a reaction pressure whose gradient removes the non-solenoidal and wall-normal content of the provisional residual, precisely the well-known Leray–Hodge projection. Thus, Gauss–Appell enforces the instantaneous kinematic constraints and recovers Euler at the instant. Once the impressed physics is specified, for instance via external body forces, the reaction pressure is uniquely determined (up to an additive constant) as the Lagrange multiplier enforcing incompressibility and wall impermeability; it does no work on divergence-free, wall-tangent motions. This is the well-established interpretation of pressure in incompressible flow. The direct, fixed-time application of this principle determines the reaction pressure for an already-specified velocity field and does not, by itself, select circulation or stagnation points, because these are properties of the velocity state, not the instantaneous acceleration correction. The formal decomposition of the pressure into impressed and reaction components admits representational freedom that does not imply physical non-uniqueness of the constraint force. Orthogonality conventions such as Dirichlet orthogonality can fix the representational freedom as an additional modelling choice. This variational viewpoint also yields a simple computational diagnostic: the minimised Appellian equals a upper L squared L 2 $L^2$ norm of the reaction-pressure gradient which vanishes for constraint-compatible updates and grows with the magnitude of divergence and wall-flux mismatch. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Fluid Mechanics is the property of Cambridge University Press and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1017/jfm.2026.11613 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 12 StartPage: 1 Subjects: – SubjectFull: Incompressible flow Type: general – SubjectFull: Variational approach (Mathematics) Type: general – SubjectFull: Fluid mechanics Type: general – SubjectFull: Euler equations Type: general – SubjectFull: Neumann problem Type: general – SubjectFull: Divergence theorem Type: general Titles: – TitleFull: Gauss principle in incompressible flow: a unified variational perspective on pressure and projection. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Duraisamy, Karthik IsPartOfRelationships: – BibEntity: Dates: – D: 25 M: 06 Text: 6/25/2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 00221120 Numbering: – Type: volume Value: 1037 Titles: – TitleFull: Journal of Fluid Mechanics Type: main |
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