Wavy properties and analytical modeling of free-surface flows in the development of the multi-domain method

The wavy (oscillatory both in space and in time) properties of free-surface flows due to presence of floating bodies are analyzed within the framework of the potential-flow theory by assuming that the fluid is perfect and flow irrotational. A so-called new multi-domain method has been developed based on the fluid domain division by an analytical control surface surrounding bodies and the application of different methods adapted in the external and internal domains. In the analytical domain external to the control surface, the fundamental solution satisfying the linear boundary condition on the free surface associated with a point singularity (often called Green function and referred here as point solution) is applied to capture all wavy features of free-surface flows extending horizontally to infinity. Unlike classical studies in which the control surface is discretized, the unknown velocity potential and its normal derivatives are expressed by expansions of orthogonal elementary functions. The velocity potential associated with each elementary distribution (elementary solutions) on the control surface can be obtained by performing multi-fold integrals in an analytical way. In the domain internal to the control surface containing the bodies, we could apply different methods like the Rankine source method based on the boundary integral equations for which the elementary solutions obtained in the external domain playing the role of Dirichlet-to-Neumann operator close the problem.