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Very high-order accurate wall vorticity treatment on curved boundaries with polygonal meshes for incompressible vorticity equations.

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Title: Very high-order accurate wall vorticity treatment on curved boundaries with polygonal meshes for incompressible vorticity equations.
Authors: Costa, Ricardo1,2 (AUTHOR) rcosta@dep.uminho.pt, Clain, Stéphane3 (AUTHOR), Machado, Gaspar J.4 (AUTHOR), Nóbrega, João M.1,2 (AUTHOR)
Source: Computers & Mathematics with Applications. Jul2026, Vol. 214, p94-134. 41p.
Subjects: Finite volume method, Incompressible flow, Discretization methods, Computational fluid dynamics, Navier-Stokes equations, Boundary value problems
Abstract: • Boundary conditions for incompressible vorticity equations on arbitrary curved domains are derived. • A finite volume discretisation on unstructured meshes ensures efficiency and high-order accuracy. • A novel boundary discretisation technique preserves high-order accuracy without curved meshes. • Benchmark tests on curved geometries confirm very high-orders of convergence. Vorticity formulations of the incompressible Navier-Stokes equations circumvent the challenges associated with the pressure-velocity numerical coupling. Furthermore, vorticity dynamics in challenging fluid flow problems can be analysed more effectively within a vorticity-based numerical framework. However, deriving suitable vorticity boundary conditions for numerical simulations with high-order accuracy on curved domains remains a significant challenge for these formulations, particularly in three dimensions. Consequently, most existing work considers only rectangular domains with regular grids, limiting its application to geometries of practical interest. This study presents suitable wall vorticity boundary conditions for incompressible vorticity equations on arbitrary curved domains, introducing a simple, effective, and highly accurate numerical treatment based on a finite volume discretisation on unstructured meshes. Moreover, a novel technique is employed to discretise the boundary conditions on a linear piecewise approximation of the curved boundary, circumventing the challenges associated with curved meshes while preserving very high-order accuracy. The two-dimensional streamfunction-vorticity formulation is analysed both theoretically and numerically, but these advancements establish a foundation for other vorticity-based formulations and the three-dimensional case, in which the treatment of vorticity boundary conditions remains fundamentally unchanged. To demonstrate the accuracy and effectiveness of the proposed methodology, several benchmark incompressible fluid flow problems in non-trivial two-dimensional curved domains are presented, including the semi-elliptical lid-driven cavity problem, for which very high orders of convergence are achieved without any regularisation. [ABSTRACT FROM AUTHOR]
Copyright of Computers & Mathematics with Applications is the property of Pergamon Press - An Imprint of Elsevier Science 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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DbLabel: Engineering Source
An: 193805854
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  Data: Very high-order accurate wall vorticity treatment on curved boundaries with polygonal meshes for incompressible vorticity equations.
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  Data: <searchLink fieldCode="AR" term="%22Costa%2C+Ricardo%22">Costa, Ricardo</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)<i> rcosta@dep.uminho.pt</i><br /><searchLink fieldCode="AR" term="%22Clain%2C+Stéphane%22">Clain, Stéphane</searchLink><relatesTo>3</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Machado%2C+Gaspar+J%2E%22">Machado, Gaspar J.</searchLink><relatesTo>4</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Nóbrega%2C+João+M%2E%22">Nóbrega, João M.</searchLink><relatesTo>1,2</relatesTo> (AUTHOR)
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  Data: <searchLink fieldCode="JN" term="%22Computers+%26+Mathematics+with+Applications%22">Computers & Mathematics with Applications</searchLink>. Jul2026, Vol. 214, p94-134. 41p.
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  Data: <searchLink fieldCode="DE" term="%22Finite+volume+method%22">Finite volume method</searchLink><br /><searchLink fieldCode="DE" term="%22Incompressible+flow%22">Incompressible flow</searchLink><br /><searchLink fieldCode="DE" term="%22Discretization+methods%22">Discretization methods</searchLink><br /><searchLink fieldCode="DE" term="%22Computational+fluid+dynamics%22">Computational fluid dynamics</searchLink><br /><searchLink fieldCode="DE" term="%22Navier-Stokes+equations%22">Navier-Stokes equations</searchLink><br /><searchLink fieldCode="DE" term="%22Boundary+value+problems%22">Boundary value problems</searchLink>
– Name: Abstract
  Label: Abstract
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  Data: • Boundary conditions for incompressible vorticity equations on arbitrary curved domains are derived. • A finite volume discretisation on unstructured meshes ensures efficiency and high-order accuracy. • A novel boundary discretisation technique preserves high-order accuracy without curved meshes. • Benchmark tests on curved geometries confirm very high-orders of convergence. Vorticity formulations of the incompressible Navier-Stokes equations circumvent the challenges associated with the pressure-velocity numerical coupling. Furthermore, vorticity dynamics in challenging fluid flow problems can be analysed more effectively within a vorticity-based numerical framework. However, deriving suitable vorticity boundary conditions for numerical simulations with high-order accuracy on curved domains remains a significant challenge for these formulations, particularly in three dimensions. Consequently, most existing work considers only rectangular domains with regular grids, limiting its application to geometries of practical interest. This study presents suitable wall vorticity boundary conditions for incompressible vorticity equations on arbitrary curved domains, introducing a simple, effective, and highly accurate numerical treatment based on a finite volume discretisation on unstructured meshes. Moreover, a novel technique is employed to discretise the boundary conditions on a linear piecewise approximation of the curved boundary, circumventing the challenges associated with curved meshes while preserving very high-order accuracy. The two-dimensional streamfunction-vorticity formulation is analysed both theoretically and numerically, but these advancements establish a foundation for other vorticity-based formulations and the three-dimensional case, in which the treatment of vorticity boundary conditions remains fundamentally unchanged. To demonstrate the accuracy and effectiveness of the proposed methodology, several benchmark incompressible fluid flow problems in non-trivial two-dimensional curved domains are presented, including the semi-elliptical lid-driven cavity problem, for which very high orders of convergence are achieved without any regularisation. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Computers & Mathematics with Applications is the property of Pergamon Press - An Imprint of Elsevier Science 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:
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        Value: 10.1016/j.camwa.2026.04.004
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      – Code: eng
        Text: English
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        PageCount: 41
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        Type: general
      – SubjectFull: Incompressible flow
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      – SubjectFull: Discretization methods
        Type: general
      – SubjectFull: Computational fluid dynamics
        Type: general
      – SubjectFull: Navier-Stokes equations
        Type: general
      – SubjectFull: Boundary value problems
        Type: general
    Titles:
      – TitleFull: Very high-order accurate wall vorticity treatment on curved boundaries with polygonal meshes for incompressible vorticity equations.
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            NameFull: Costa, Ricardo
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            NameFull: Clain, Stéphane
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            NameFull: Machado, Gaspar J.
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            NameFull: Nóbrega, João M.
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              Text: Jul2026
              Type: published
              Y: 2026
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