This work discusses the correct modeling of the fully nonlinear free surface boundary conditions to be prescribed in water waves flow simulations based on potential flow theory. The main goal of such a discussion is that of identifying a mathematical formulation and a numerical treatment that can be used both to carry out transient simulations, and to compute steady solutions -- for any flow admitting them. In the literature on numerical towing tank in fact, steady and unsteady fully nonlinear potential flow solvers are characterized by different mathematical formulations. The kinematic and dynamic fully nonlinear free surface boundary conditions are discussed, and in particular it is proven that the kinematic free surface boundary condition, written in semi-Lagrangian form, can be manipulated to derive an alternative non penetration boundary condition by all means identical to the one used on the surface of floating bodies or on the basin bottom. The simplified mathematical problem obtained is discretized over space and time via Boundary Element Method (BEM) and Implicit Backward Difference Formula (BDF) scheme, respectively. The results confirm that the solver implemented is able to solve steady potential flow problems just by eliminating null time derivatives in the unsteady formulation. Numerical results obtained confirm that the solver implemented is able to accurately reproduce results of classical steady flow solvers available in the literature.

翻译：本文讨论了基于势流理论的水波流动模拟中完全非线性自由表面边界条件的正确建模。讨论的主要目标是确定一种数学公式和数值处理方法，既能用于瞬态模拟，也能用于计算稳态解——适用于任何允许稳态解的流动。事实上，在关于数值拖曳水槽的文献中，稳态和非稳态完全非线性势流求解器具有不同的数学公式。本文讨论了运动学和动力学完全非线性自由表面边界条件，并特别证明了以半拉格朗日形式表示的运动学自由表面边界条件可以经过推导，得到一个替代的非穿透边界条件，该条件与浮体表面或水槽底部使用的边界条件完全相同。通过边界元法和隐式后向差分公式，分别对简化后的数学问题在空间和时间上进行离散。结果表明，所实现的求解器能够通过消除非稳态公式中的零时间导数来解决稳态势流问题。数值结果证实，所实现的求解器能够精确复现文献中经典稳态流动求解器的结果。