Analytical solutions for hydrodynamic responses of arrays of floating truncated cylinders using multi-term Galerkin method and its application to a new wave energy converter device

Abstract

In the context of linear water wave theory, the analytical solutions for the diffraction and radiation of a truncated cylinder array are developed in the presence of ambient incident waves. Each cylinder in the array can oscillate with five degrees of freedom (DOFs), i.e., surge, sway, heave, roll, and pitch. This paper adopts the multi-term Galerkin method to expand the fluid velocity at the interface of different regions into a set of basis functions containing Gegenbauer polynomials, which accurately and efficiently characterizes the cube root singularity of the fluid velocity near the edges of the truncated cylinders. Using the dynamic equilibrium equations, the amplitudes of each DOF of the cylinders in the array are solved. The analytical solution presented in this paper converges rapidly, and high-precision hydrodynamic response results can be obtained using just a few truncated terms (e.g., the upper bounds of m0 = 5 and p0 = 22 can yield results of five-figure accuracy). For the 4-cylinder array, under the same accuracy conditions (the error less than 1%), the computation time of the conventional method developed by Zeng et al. [“Hydrodynamic interactions between waves and cylinder arrays of relative motions composed of truncated floating cylinders with five degrees of freedom,” J. Fluids Struct. 115, 103785 (2022d)] based on the exact algebraic method [Kagemoto and Yue, “Interactions among multiple three-dimensional bodies in water waves: An exact algebraic method,” J. Fluid Mech. 166, 189–209 (1986)] is 3.9 times longer than that of the present method. As the number of cylinders increases, the advantage of the present method in terms of convergence speed becomes more apparent, e.g., for the 16-cylinder array, the conventional solution takes 6.3 times longer than the present solution. To extract wave energy more efficiently, a new 5DOF wave energy converter (WEC) device that can extract energy in 5DOFs is proposed. The present method is adopted to investigate the hydrodynamic performance of the 5DOF WEC arrays. Compared with the traditional 1DOF (heave) WEC, the 5DOF WEC can significantly improve the energy capture performance of arrays, especially in the high-frequency wave region.