Boundary integral representations of steady flow around a ship

Abstract

The classical Neumann–Kelvin (NK) linear potential flow theory of ship waves in calm water and the related Neumann–Michell (NM) theory are considered. Five alternative boundary integral representations are given: (1) the classical NK integro-differential representation, called “classical NK formulation”, which corresponds to an inconsistent linear flow model, (2) a modification of the classical NK flow formulation that corresponds to a consistent linear flow model and is called “consistent NK formulation”, (3) a flow representation, called “NM potential and velocity formulation”, that involves the flow potential ϕ and the velocity components ϕd and ϕt along two unit vectors d and t tangent to the ship hull surface, and yields an integro-differential equation for determining ϕ, (4) a flow representation, called “NM velocity formulation”, that only involves ϕd and ϕt and yields a pair of coupled integral equations for determining (ϕd,ϕt), and (5) a flow representation called “NM potential formulation” that only involves the flow potential ϕ and yields an integral equation for determining ϕ. The two NK formulations involve both a surface integral over the ship hull surface and a line integral around the ship waterline, whereas the three NM formulations do not involve a waterline integral. All flow representations other than the classical NK representation are based on a consistent linear flow model.