Practical mathematical representation of the flow due to a distribution of sources on a steadily advancing ship hull

Abstract

A practical mathematical representation of the flow velocity due to a distribution of sources on the mean wetted hull surface and the mean waterline of a ship that steadily advances along a straight path in calm water, of large depth and lateral extent, is presented. A main feature of this flow representation is a simple analytical approximation—valid within the entire flow region—to the local flow component in the expression for the gradient of the Green function associated with the classical Kelvin–Michell linearized free-surface boundary condition. Another notable feature of the flow representation is that the singularity associated with the gradient of the Green function is removed, using a straightforward regularization technique. The flow representation only involves elementary continuous functions (algebraic, exponential and trigonometric) of real arguments. These functions can then be integrated using ordinary Gaussian quadrature rules. Thus, the flow representation is particularly simple and well suited for practical flow calculations. The specific case of a low-order panel method—in which the hull geometry, the source density, and the flow velocity are consistently represented via piecewise linear approximations within flat triangular hull panels or straight waterline segments—is considered.