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.

The Neumann–Michell theory of ship waves

“…The dipole distribution φ n ·∇L in (3) is ignored. The Green function given in [5] is used. The Wigley hull is approximated by 16,000 flat triangular panels.…”

“…This Green function is commonly expressed as the sum of a wave component W that is given by a Fourier superposition of elementary waves and satisfies the radiation condition, and a local-flow component L that is given by a somewhat complicated integral representation [4]. However, the local component L (and its gradient) can be well approximated by the simple analytical expression, valid in the entire flow field, given in [5] . Furthermore, calculations show that numerical predictions obtained using the simple analytical approximation given in [5] or the even simpler approximation, given by a combination of three elementary free-space Rankine sources, proposed in [6,7] do not differ significantly.…”

“…However, the local component L (and its gradient) can be well approximated by the simple analytical expression, valid in the entire flow field, given in [5] . Furthermore, calculations show that numerical predictions obtained using the simple analytical approximation given in [5] or the even simpler approximation, given by a combination of three elementary free-space Rankine sources, proposed in [6,7] do not differ significantly. The Green function G( x ; x), where x ≡ ( x , y , z) and x ≡ ( x , y , z ) stand for a source point and a flow field point, can therefore be greatly simplified as…”

“…The Green function (1) and the straightforward regularization approach given in [5] lead to a practical, indeed particularly simple, mathematical flow representation. This flow representation only involves ordinary continuous functions, which can readily be integrated using straightforward Gaussian integration rules.…”

“…The waves that stem from the waterline integral around Γ and the hull-surface integral over Σ H are well known to largely cancel out, e.g. [5], which unavoidably causes a loss of numerical accuracy. This loss of accuracy is rendered more severe by the fact that the waterline integral around Γ involves flow evaluation at the mean free-surface plane z = 0, which cannot be done accurately as noted below.…”