Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances
The use of potential fields in fluid dynamics is retraced, ranging from classical potential theory to recent developments in this evergreen research field. The focus is centred on two major approaches and their advancements: (i) the Clebsch transformation and (ii) the classical complex variable meth...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-04-01
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| Series: | Water |
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| Online Access: | https://www.mdpi.com/2073-4441/12/5/1241 |
| _version_ | 1797569534936219648 |
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| author | Markus Scholle Florian Marner Philip H. Gaskell |
| author_facet | Markus Scholle Florian Marner Philip H. Gaskell |
| author_sort | Markus Scholle |
| collection | DOAJ |
| description | The use of potential fields in fluid dynamics is retraced, ranging from classical potential theory to recent developments in this evergreen research field. The focus is centred on two major approaches and their advancements: (i) the Clebsch transformation and (ii) the classical complex variable method utilising Airy’s stress function, which can be generalised to a first integral methodology based on the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Basic questions relating to the existence and gauge freedoms of the potential fields and the satisfaction of the boundary conditions required for closure are addressed; with respect to (i), the properties of self-adjointness and Galilean invariance are of particular interest. The application and use of both approaches is explored through the solution of four purposely selected problems; three of which are tractable analytically, the fourth requiring a numerical solution. In all cases, the results obtained are found to be in excellent agreement with corresponding solutions available in the open literature. |
| first_indexed | 2024-03-10T20:12:57Z |
| format | Article |
| id | doaj.art-ea067cf570234e919fbe87e495ebccc0 |
| institution | Directory Open Access Journal |
| issn | 2073-4441 |
| language | English |
| last_indexed | 2024-03-10T20:12:57Z |
| publishDate | 2020-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Water |
| spelling | doaj.art-ea067cf570234e919fbe87e495ebccc02023-11-19T22:51:25ZengMDPI AGWater2073-44412020-04-01125124110.3390/w12051241Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent AdvancesMarkus Scholle0Florian Marner1Philip H. Gaskell2Department of Mechatronics and Robotics, Heilbronn University, D-74081 Heilbronn, GermanyInigence GmbH, D-74626 Bretzfeld, GermanyDepartment of Engineering, Durham University, Durham DH1 3LE, UKThe use of potential fields in fluid dynamics is retraced, ranging from classical potential theory to recent developments in this evergreen research field. The focus is centred on two major approaches and their advancements: (i) the Clebsch transformation and (ii) the classical complex variable method utilising Airy’s stress function, which can be generalised to a first integral methodology based on the introduction of a tensor potential and parallels drawn with Maxwell’s theory. Basic questions relating to the existence and gauge freedoms of the potential fields and the satisfaction of the boundary conditions required for closure are addressed; with respect to (i), the properties of self-adjointness and Galilean invariance are of particular interest. The application and use of both approaches is explored through the solution of four purposely selected problems; three of which are tractable analytically, the fourth requiring a numerical solution. In all cases, the results obtained are found to be in excellent agreement with corresponding solutions available in the open literature.https://www.mdpi.com/2073-4441/12/5/1241potential fieldsClebsch variablesAiry’s stress functionGoursat functionsGalilean invariancevariational principles |
| spellingShingle | Markus Scholle Florian Marner Philip H. Gaskell Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances Water potential fields Clebsch variables Airy’s stress function Goursat functions Galilean invariance variational principles |
| title | Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances |
| title_full | Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances |
| title_fullStr | Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances |
| title_full_unstemmed | Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances |
| title_short | Potential Fields in Fluid Mechanics: A Review of Two Classical Approaches and Related Recent Advances |
| title_sort | potential fields in fluid mechanics a review of two classical approaches and related recent advances |
| topic | potential fields Clebsch variables Airy’s stress function Goursat functions Galilean invariance variational principles |
| url | https://www.mdpi.com/2073-4441/12/5/1241 |
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