An arbitrary Lagrangian Eulerian method for moving-boundary problems and its application to jumping over water

We develop an ALE (Arbitrary Lagrangian Eulerian) moving mesh method suitable for solving two-dimensional and axisymmetric moving-boundary problems, including the interaction between a free-surface and a solid structure. This method employs a body-fitted grid system where the gas-liquid interface and solid-liquid interface are lines of the grid system, and complicated dynamic boundary conditions are incorporated naturally and accurately in a Finite-Volume formulation. The resulting nonlinear system of mass and momentum conservation is then solved by a fractional step (projection) method. The method is validated on the uniform flow passing a cylinder (a two-dimensional flow with a solid structure) and several problems of bubble dynamics (axi-symmetrical flows with a free surface) for both steady and unsteady flows. Good agreement with other theoretical, numerical and experimental results is obtained. A further application is the investigation of a two-dimensional mechanical strider (a mass-spring system) interacting with a water surface, demonstrating the ability of the method in handling the interaction between a solid structure and a free surface. We find that the critical compression required to jump off the water surface varies linearly with spring constant for stiff springs and algebraically with exponent 0.7 for weak springs.