Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives
In fluid mechanics, the bi-Laplacian operator with Neumann homogeneous boundary conditions emerges when transforming the Navier–Stokes equations to the vorticity–velocity formulation. In the case of problems with a periodic direction, the problem can be transformed into multiple, independent, two-di...
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MDPI AG
2021-10-01
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| Online Access: | https://www.mdpi.com/2227-7390/9/19/2508 |
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| author | Jesús Amo-Navarro Ricardo Vinuesa J. Alberto Conejero Sergio Hoyas |
| author_facet | Jesús Amo-Navarro Ricardo Vinuesa J. Alberto Conejero Sergio Hoyas |
| author_sort | Jesús Amo-Navarro |
| collection | DOAJ |
| description | In fluid mechanics, the bi-Laplacian operator with Neumann homogeneous boundary conditions emerges when transforming the Navier–Stokes equations to the vorticity–velocity formulation. In the case of problems with a periodic direction, the problem can be transformed into multiple, independent, two-dimensional fourth-order elliptic problems. An efficient method to solve these two-dimensional bi-Laplacian operators with Neumann homogeneus boundary conditions was designed and validated using 2D compact finite difference schemes. The solution is formulated as a linear combination of auxiliary solutions, as many as the number of points on the boundary, a method that was prohibitive some years ago due to the large memory requirements to store all these auxiliary functions. The validation has been made for different field configurations, grid sizes, and stencils of the numerical scheme, showing its potential to tackle high gradient fields as those that can be found in turbulent flows. |
| first_indexed | 2024-03-10T06:55:24Z |
| format | Article |
| id | doaj.art-c9660c68da8740d796e41c5116bf5af8 |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T06:55:24Z |
| publishDate | 2021-10-01 |
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| spelling | doaj.art-c9660c68da8740d796e41c5116bf5af82023-11-22T16:31:23ZengMDPI AGMathematics2227-73902021-10-01919250810.3390/math9192508Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal DerivativesJesús Amo-Navarro0Ricardo Vinuesa1J. Alberto Conejero2Sergio Hoyas3Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 València, SpainFLOW, Engineering Mechanics, KTH Royal Institute of Technology, SE-100 44 Stockholm, SwedenInstituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 València, SpainInstituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, 46022 València, SpainIn fluid mechanics, the bi-Laplacian operator with Neumann homogeneous boundary conditions emerges when transforming the Navier–Stokes equations to the vorticity–velocity formulation. In the case of problems with a periodic direction, the problem can be transformed into multiple, independent, two-dimensional fourth-order elliptic problems. An efficient method to solve these two-dimensional bi-Laplacian operators with Neumann homogeneus boundary conditions was designed and validated using 2D compact finite difference schemes. The solution is formulated as a linear combination of auxiliary solutions, as many as the number of points on the boundary, a method that was prohibitive some years ago due to the large memory requirements to store all these auxiliary functions. The validation has been made for different field configurations, grid sizes, and stencils of the numerical scheme, showing its potential to tackle high gradient fields as those that can be found in turbulent flows.https://www.mdpi.com/2227-7390/9/19/2508DNSCFDturbulencebi-Laplacianfourth-order elliptic |
| spellingShingle | Jesús Amo-Navarro Ricardo Vinuesa J. Alberto Conejero Sergio Hoyas Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives Mathematics DNS CFD turbulence bi-Laplacian fourth-order elliptic |
| title | Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives |
| title_full | Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives |
| title_fullStr | Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives |
| title_full_unstemmed | Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives |
| title_short | Two-Dimensional Compact-Finite-Difference Schemes for Solving the bi-Laplacian Operator with Homogeneous Wall-Normal Derivatives |
| title_sort | two dimensional compact finite difference schemes for solving the bi laplacian operator with homogeneous wall normal derivatives |
| topic | DNS CFD turbulence bi-Laplacian fourth-order elliptic |
| url | https://www.mdpi.com/2227-7390/9/19/2508 |
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