The velocity field and Birkhoff-Rott integral for non-decaying, non-periodic vortex sheets
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917564652716032 |
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| author | Ambrose, David M. |
| author_facet | Ambrose, David M. |
| contents | The Birkhoff-Rott integral expresses the fluid velocity on a vortex sheet. This integral converges if certain quantities decay at horizontal infinity, but can also be summed over periodic images in the horizontally periodic case. However, non-decaying, non-periodic cases are also of interest, such as the interaction of periodic wavetrains with non-commensurate periods (i.e. spatially quasiperiodic solutions), or non-periodic disturbances to periodic wavetrains. We therefore develop a more general single formula for the Birkhoff-Rott integral, which unifies and extends the cases of decay and periodicity. We verify that under some reasonable conditions this new version of the Birkhoff-Rott integral is the restriction to the vortex sheet of an incompressible, irrotational velocity field, with continuous normal component but with a jump in tangential velocity across the vortex sheet. We give a number of examples of non-decaying, non-periodic sheet positions and sheet strengths for which our assumptions may be verified. While we develop this in the case of two-dimensional fluids, the methodology applies equally well to three-dimensional fluids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06055 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The velocity field and Birkhoff-Rott integral for non-decaying, non-periodic vortex sheets Ambrose, David M. Fluid Dynamics Analysis of PDEs 76B55, 76B07, 42A50 The Birkhoff-Rott integral expresses the fluid velocity on a vortex sheet. This integral converges if certain quantities decay at horizontal infinity, but can also be summed over periodic images in the horizontally periodic case. However, non-decaying, non-periodic cases are also of interest, such as the interaction of periodic wavetrains with non-commensurate periods (i.e. spatially quasiperiodic solutions), or non-periodic disturbances to periodic wavetrains. We therefore develop a more general single formula for the Birkhoff-Rott integral, which unifies and extends the cases of decay and periodicity. We verify that under some reasonable conditions this new version of the Birkhoff-Rott integral is the restriction to the vortex sheet of an incompressible, irrotational velocity field, with continuous normal component but with a jump in tangential velocity across the vortex sheet. We give a number of examples of non-decaying, non-periodic sheet positions and sheet strengths for which our assumptions may be verified. While we develop this in the case of two-dimensional fluids, the methodology applies equally well to three-dimensional fluids. |
| title | The velocity field and Birkhoff-Rott integral for non-decaying, non-periodic vortex sheets |
| topic | Fluid Dynamics Analysis of PDEs 76B55, 76B07, 42A50 |
| url | https://arxiv.org/abs/2401.06055 |