Proof of the equivalence of Tanizawa–Berkvens’ and Cointe–van Daalen's formulations for the time derivative of the velocity potential for non-linear potential flow solvers

Abstract

Dealing with freely-floating bodies in the framework of non-linear potential flow theory may require solving Laplace's equation for the time derivative of the velocity potential. At present, there are two competing formulations for the body boundary condition. The first one was derived by Cointe [Cointe(1989)] in 2D. It was later extended to 3D by van Daalen [van Daalen(1993)]. The second formulation was derived by Tanizawa [Tanizawa(1995)] in 2D. It was extended to 3D by Berkvens [Berkvens(1998)]. In this paper, a proof is given that the Cointe-Van Daalen's and the Tanizawa-Berkvens' formulations are equivalent. It leads to a simplified version of Cointe-Van Daalen's formulation. The formulation is validated against the analytical solution for a moving sphere in an unbounded water domain.

Boundary element and integral methods in potential flow theory: a review with a focus on wave energy applications

“…Some codes compute the nonlinear radiation-diffraction forces by linearising, around z = 0, the free-surface equations of the weak-scatterer approach (discussed in Sect. 8.3), such as LAMP-4 (Beck and Reed 2001), or the code WS_Cn, developed by the LHEEA in Nantes (Letournel 2015). Using the body-exact nonlinear radiationdiffraction option of WS_Cn is around three times faster than the weak-scatterer approach.…”

“…s 1 and s 2 are the local coordinates vectors, and c 1 and c 2 are the local curvature along the respective local vectors. More details on the method to compute the time derivative of the potential are given in Letournel et al (2017) and Letournel et al (2018).…”

Development of a Three‐Dimensional Fully Nonlinear Potential Numerical Wave Tank for a Heaving Buoy Wave Energy Converter

“…e body boundary condition is difficult to describe in the acceleration potential field because of the nonlinear term ϕ t . e expression reported by Letournel et al [32] was adopted to express the body boundary condition, which was derived from the approach of Tanizawa [44] and Cointe [45]. e following equations show the body boundary condition in the Mathematical Problems in Engineering acceleration field.…”

“…e acceleration potential approach using the indirect method was used to calculate the total force and displacement of the body. To describe the body boundary condition in an acceleration field, the simplified formulation from Letournal et al [32] was adopted. To develop the PNWT based on the CPM, the least-square gradient reconstruction method and the inverse distance weighting method were adopted.…”