| Summary: | 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.
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