The boundary-integral flow-representation associated with the boundary-value problem, commonly called Neumann–Kelvin (NK) problem, that corresponds to linear potential flow around a ship steadily advancing in calm water involves an integral around the mean waterline of the ship. This ‘waterline integral’ is a notorious source of numerical difficulties and has been extensively studied. The waterline integral in the NK theory is largely – but not fully – eliminated in the modification, called Neumann–Michell (NM) theory, of the NK theory. Specifically, the NM theory includes a residual waterline-distribution of weak Rankine singularities, ignored in practical applications. A crucial element of the NM theory is a mathematical transformation that is based on a vector Green function, which is associated with the common scalar Green function used in the NK theory. This transformation is revisited in the present study. A rigorous analysis yields an answer to a fifty-year old puzzle: an exact boundary-integral flow representation that does not include a waterline integral. A remarkable feature of this new flow representation, which is a modification of the NM flow representation given previously, is that it explicitly determines the flow potential and the flow velocity at a ship hull surface in terms of the flow velocity at the hull surface, rather than in terms of the hull-surface potential as in usual boundary-integral flow representations obtained via Green’s classical identity.

与边界值问题相关的边界积分流表示，通常称为 Neumann-Kelvin (NK) 问题，对应于在平静水中稳步前进的船舶周围的线性势流，涉及船舶平均水线周围的积分。这种“水线积分”是一个臭名昭著的数值困难来源，并已被广泛研究。NK 理论中的水线积分在 NK 理论的修改中被大部分——但不是完全——消除，称为 Neumann-Michell (NM) 理论。具体来说，NM 理论包括弱朗肯奇点的残余水线分布，在实际应用中被忽略。NM 理论的一个关键要素是基于向量格林函数的数学变换，这与 NK 理论中使用的常见标量格林函数有关。在本研究中重新审视了这种转变。严格的分析为一个 50 年前的谜题提供了答案：不包括水线积分的精确边界积分流表示。这个新的流表示，这是前面给出的NM流表示的变形例的一个显着特征是，它明确地确定在船体表面上的流动电位和流速在流动方面速度在船体表面，而与通过格林经典恒等式获得的通常边界积分流表示中的船体表面势相比。