Asymptotic Solutions and Numerical Methods for some Free-boundary Problems in Fluid Mechanics
Free-boundary problems are encountered in a wide range of applications in fluid mechanics, such as the interaction of a ship with the surface of the ocean, and the failure of a dam. Since the free boundary is unknown and part of the solution, such problems are nonlinear and rarely have analytical so...
| Main Author: | |
|---|---|
| Format: | Thesis (University of Nottingham only) |
| Language: | English |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://eprints.nottingham.ac.uk/76824/ |
| _version_ | 1848800942151958528 |
|---|---|
| author | Fan, Yiyun |
| author_facet | Fan, Yiyun |
| author_sort | Fan, Yiyun |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | Free-boundary problems are encountered in a wide range of applications in fluid mechanics, such as the interaction of a ship with the surface of the ocean, and the failure of a dam. Since the free boundary is unknown and part of the solution, such problems are nonlinear and rarely have analytical solutions. In this thesis, we formulate and solve some free-boundary problems in inviscid fluid mechanics using asymptotic and numerical methods. We construct new asymptotic solutions for the two-fluid dam-break problem and a solid/two-fluid interaction problem with an inclined accelerating plate, and develop the numerical methods based on the finite element method for generic free-boundary problems.
The main outcomes of this research are as follows. The small-time outer asymptotic solutions have a singularity at the intersection point between the interface and the solid boundary for both problems, which can be resolved by rescaling into an inner region. A numerical approach based on the finite element method and Newton’s method is developed to resolve the inner problem of the solid/single fluid inner region problem, which agrees with the results obtained by the boundary integral method in earlier work. Furthermore, we derive a Shape-Newton method as a fast nonlinear numerical solver to solve the generic free-boundary problem with Bernoulli-type boundary conditions on the free surface, which is tested on the problem of flow over a triangular obstacle. The application of this method can be extended to a range of more complicated free-boundary problems in fluid mechanics. |
| first_indexed | 2025-11-14T20:59:34Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-76824 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:59:34Z |
| publishDate | 2023 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-768242024-01-10T14:55:33Z https://eprints.nottingham.ac.uk/76824/ Asymptotic Solutions and Numerical Methods for some Free-boundary Problems in Fluid Mechanics Fan, Yiyun Free-boundary problems are encountered in a wide range of applications in fluid mechanics, such as the interaction of a ship with the surface of the ocean, and the failure of a dam. Since the free boundary is unknown and part of the solution, such problems are nonlinear and rarely have analytical solutions. In this thesis, we formulate and solve some free-boundary problems in inviscid fluid mechanics using asymptotic and numerical methods. We construct new asymptotic solutions for the two-fluid dam-break problem and a solid/two-fluid interaction problem with an inclined accelerating plate, and develop the numerical methods based on the finite element method for generic free-boundary problems. The main outcomes of this research are as follows. The small-time outer asymptotic solutions have a singularity at the intersection point between the interface and the solid boundary for both problems, which can be resolved by rescaling into an inner region. A numerical approach based on the finite element method and Newton’s method is developed to resolve the inner problem of the solid/single fluid inner region problem, which agrees with the results obtained by the boundary integral method in earlier work. Furthermore, we derive a Shape-Newton method as a fast nonlinear numerical solver to solve the generic free-boundary problem with Bernoulli-type boundary conditions on the free surface, which is tested on the problem of flow over a triangular obstacle. The application of this method can be extended to a range of more complicated free-boundary problems in fluid mechanics. 2023-12-12 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en cc_by_nc https://eprints.nottingham.ac.uk/76824/1/thesis_final.pdf Fan, Yiyun (2023) Asymptotic Solutions and Numerical Methods for some Free-boundary Problems in Fluid Mechanics. PhD thesis, University of Nottingham. Free-boundary problems partial differential operators fluid mechanics |
| spellingShingle | Free-boundary problems partial differential operators fluid mechanics Fan, Yiyun Asymptotic Solutions and Numerical Methods for some Free-boundary Problems in Fluid Mechanics |
| title | Asymptotic Solutions and Numerical Methods for some Free-boundary Problems in Fluid Mechanics |
| title_full | Asymptotic Solutions and Numerical Methods for some Free-boundary Problems in Fluid Mechanics |
| title_fullStr | Asymptotic Solutions and Numerical Methods for some Free-boundary Problems in Fluid Mechanics |
| title_full_unstemmed | Asymptotic Solutions and Numerical Methods for some Free-boundary Problems in Fluid Mechanics |
| title_short | Asymptotic Solutions and Numerical Methods for some Free-boundary Problems in Fluid Mechanics |
| title_sort | asymptotic solutions and numerical methods for some free-boundary problems in fluid mechanics |
| topic | Free-boundary problems partial differential operators fluid mechanics |
| url | https://eprints.nottingham.ac.uk/76824/ |