Development of an Algorithm for Calculation and Application of Conformal Mapping Methods on the Calculation of Hydrodynamic Coefficients

The multipole method was first developed by Ursell [1]. His method consists of the superposition of potential functions that satisfy the Laplace equation, the free-surface boundary condition, and the condition at infinity. The potential functions represent a source and horizontal dipole at the origin, which give the radiated waves at infinity, and a series of multipoles that die off rapidly as one moves away from the origin. The strengths of the source, dipole and multipoles are all determined so that the body boundary condition is met. Ursell used this method to solve the problem of a heaving circular cylinder. For sections that are not circular in shape, conformal mapping is used.

In the multipole method, the mapping function that transforms the ship section into a semi-circle is found. The mapping function can then be used in conjunction with Ursell's known solution for a circular cylinder to find the solution for the actual ship section.

The difficulty in the technique is to determine the proper mapping function for each cross section. Various methods have been proposed to find this mapping function. The most common mapping uses the so-called Lewis-forms [2], [3] and [4].

It is recognized that this method gives smooth solutions over all frequency range (no irregular frequencies). On the other hand, sharp corners are not well represented, and sections with very low sectional area coefficient may not be well represented as well. Even though, for first rough estimates in initial stages of ship design, this method may give results that agree reasonably well with the other more computational demanding methods, in terms of order of magnitude and trend.

Considering the difficulty to find good charts of the Lewis forms data – to the best of our knowledge, the best known are those published by Bhattacharyya [5] - and even more difficult to find the data in digital form. The main purpose of this work is to present a computational method of computing the data given through the Lewis forms and apply it to naval ship sections in order to find rough estimates of the hydrodynamic coefficients in heave.