Boundary integral representations of steady flow around a ship

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.