A unified steady and unsteady formulation for hydrodynamic potential flow simulations with fully nonlinear free surface boundary conditions

TL;DR

This paper presents a unified mathematical formulation and numerical approach for simulating both steady and unsteady fully nonlinear potential water flows with free surface boundary conditions, enabling consistent transient and steady-state solutions.

Contribution

It introduces a unified formulation that allows the same solver to handle both steady and unsteady potential flow problems using boundary element methods.

Findings

The solver accurately reproduces classical steady flow results.

The unified approach simplifies the modeling of water wave flows.

Numerical tests confirm the solver's effectiveness for both steady and unsteady cases.

Abstract

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.