Proceedings of the Thirty-second (2022) International Ocean and Polar Engineering ConferenceShanghai, China, June 5–10, 2022
Copyright © 2022 by the International Society of Offshore and Polar Engineers (ISOPE)
ISBN 978-1-880653-81-4; ISSN 1098-6189

www.isope.org

Analytical Representation of Flows due to Arbitrary Distributions of Singularities
for Steady Motion of Ships in Finite Water Depth
Huiyu Wu, Ren-Chuan Zhu, Jiayi He
State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean & Civil Engineering, Shanghai Jiao Tong University, Shanghai, China

ABSTRACTtimization, as is largely demonstrated in e.g. (Yang et al., 2014; Huang etal., 2016; Huang & Yang, 2016; Yang & Huang, 2016; Zha et al., 2021).
The realistic and commonly-used framework of potential-flow theory and the related Green-function and boundary-integral method require
accurate and efficient numerical evaluation of flows created by general(typically polynomial) distributions of singularities (sources and dipoles)
over (flat or curved) panels of various shapes (notably quadrilateral ortriangular) that approximate the hull surface of a ship. Specifically, theflow due to a singularity distribution over a panel is commonly evaluatedin a two-step manner, which is called ‘direct Green function method’
hereafter: the Green function (given by a Fourier integration) and its gradient are first evaluated and are subsequently integrated over the panel (aspace integration). Accordingly, the corresponding Green function is abasic core element in the theory of steady motion of ships.
In the simpler case of deep water, the Green function has been studied in a broad literature, e.g. (Peters, 1949; Lunde, 1951; Kostyukov,1968; Noblesse, 1981) Alternative approximate methods for computing the Green function include complementary near-field and far-field
asymptotic approximations (Bessho, 1964; Noblesse, 1978; Newman,1987), Chebyshev polynomial approximations within complementary
contiguous flow regions (Newman, 1987), and table interpolation (Telste& Noblesse, 1989; Ponizy et al., 1994). In addition, analytical approximations valid within the entire flow region, i.e. global approximations,
are given in (Noblesse et al., 2011; Wu et al. 2016). These global approximations only involve elementary continuous functions of real arguments, and provide a particularly simple and efficient method for evaluating the Green function for steady motion of ships in deep water.
However, in the case of finite water depth, the corresponding Greenfunction, which can be found in (Lunde, 1951; Kostyukov, 1968; Chen& Zhao, 2001; Yang et al., 2006), is much more complicated than theGreen function for deep water and is considerably less studied. Indeed,
the Green function and its gradient are functions of three nondimensionalvariables for deep water, but are functions of five variables for finite water
depth. The approximate methods for computing the deep-water Greenfunction are then difficult to be applied to the case of finite water depth.
Moreover, the Green function G is usually decomposed into a wave
component GW and a local-flow component G L that represent the waves
found aft of the translating source or a non-oscillatory local flow. Although the sum GW + G L is a smooth function, the wave and the localflow components GW and G L are not smooth, and their derivatives can bediscontinuous, at the source point.
Direct evaluation of the wave component GW is a difficult task.
Specifically, the wave component GW of the Green function G consists

Flow around a ship that advances at a constant speed in calm waterof finite depth is considered within the usual framework of the Greenfunction and boundary-integral method associated with potential-flow
theory. This method requires accurate and efficient evaluation of flows
created by distributions of singularities over panels used to approximate
the ship hull surface. This basic element of the Green-function method is
considered for steady motion of ships in the subcritical and supercriticalflow regimes. An analytical representation of the flow created by a general distribution of singularities over a hull-surface panel is given. This
flow-representation is based on the Fourier-Kochin approach, in whichspace-integration over the panel is performed first and Fourier-integrationis performed subsequently, unlike the common approach in which theGreen function (defined via a Fourier integration) is evaluated first andsubsequently integrated over the panel. The analytical and numericalcomplexities associated with the numerical evaluation and subsequent
panel integration of the Green function are then avoided in the FourierKochin approach. The analytical flow-representation provides a smoothdecomposition of free-surface effects into waves and a non-oscillatory
local flow. Illustrative numerical applications to the flow potentials andvelocities associated with a typical distribution of sources over a panel
show that the flow-representation given in the study is well suited for accurate and efficient numerical evaluation.

KEY WORDS: Ship waves; finite water depth; singularity distributions; Green function; Fourier-Kochin.
INTRODUCTION
Flow around a ship that advances at a constant speed in calm water of
uniform finite depth is considered within the classical framework of linear potential-flow theory and the related Green-function and boundaryintegral method in which the Green function satisfies the linearized
Kelvin-Michell free-surface boundary condition. This theoretical framework is realistic and suited for routine practical applications.
Indeed, in deep water, this theory has been shown to yield predictions of the wave drag, the hydrodynamic lift and pitch-moment (and theresulting sinkage and trim), as well as the wave profile along a ship hull,that are sufficiently accurate for practical purposes within a broad rangeof Froude numbers (Noblesse et al., 2013; Huang et al., 2013; Zhang etal., 2014; Ma et al., 2017, 2018). In addition, numerical methods basedon potential-flow theory are well suited for ship design and hull-form op-3769