Pseudo-impulsive solutions of the forward-speed diffraction problem using a high-order finite-difference method

Abstract

This paper considers pseudo-impulsive numerical solutions to the forward-speed diffraction problem, as derived from classical linearized potential flow theory. Both head-and following-seas cases are treated. Fourth-order finite-difference approximations are applied on overlapping, boundary-fitted grids to obtain solutions using both the Neumann-Kelvin and the double-body flow linearizations of the problem. A method for computing the pseudo-impulsive incident wave forcing in finite water depth using the Fast Fourier Transform (FFT) is presented. The pseudo-impulsive scattering solution is then Fourier transformed into the frequency domain to obtain the wave excitation forces and the body motion response. The calculations are validated against reference solutions for a submerged circular cylinder and a submerged sphere. Calculations are also made for a modern bulk carrier, showing good agreement with experimental measurements.

Salvesen’s Method for Added Resistance Revisited

“…[37] is shown for F r = 0.2. For F r = 0.3, the added resistance calculated using the 3D finite-difference model of Amini-Afshar & Bingham (OWD3D-Seakeeping) [38,39,40] is also presented for the comparison.…”

A spectral element solution of the two‐dimensional linearized potential flow radiation problem

“…7,13,14 Approaches for solving the PF problem have also been developed using the Finite Difference Method (FDM), both for pure nonlinear wave propagation, [15][16][17] nonlinear wave-structure interaction, 18 linear wave-structure interaction using pseudo-impulsive time domain formulations at zero speed, [19][20][21] and with forward speed. 22,23 Using an FDM-based model, the entire fluid domain is discretized to solve the Laplace problem, hereby resulting in a larger linear system of equations compared to the BEM discretization; however, the system is sparse due to the locally applied stencils. This can-via iterative solvers-lead to an optimal (N) scaling in work effort for higher-order accurate schemes.…”

“…On the body boundary, $$$ {\Gamma}^{\mathrm{body}} $$$, a pseudo‐impulsive velocity, $$$ {\dot{x}}_k $$$, is to be imposed. This velocity will be based on a Gaussian displacement signal, $$$ {x}_k $$$, 22,23 and designed such that: (1) the Gaussian is tailored to contain a specific range of frequencies, (2) the signal is practically zero at $$$ t=0 $$$, and (3) the resultant radiation force signal, $$$ {F}_{jk} $$$, should reach a zero steady‐state solution at $$$ t=T $$$. With these requirements, a unit height Gaussian displacement signal is defined as $$$$ {x}_k(t)={e}^{-2{\pi}^2{s}^2{\left(t-{t}_0\right)}^2},\kern1em \mathrm{with}\kern1em {t}_0=\sqrt{\frac{\log \left(\epsilon \right)}{-2{\pi}^2{s}^2}},\kern1em \mathrm{and},\kern1em s=\sqrt{\frac{-{f}_r^2}{2\ln (r)}}, $$$$ where $$$ {t}_0 $$$ is the time location of the Gauss...…”

“…On the body boundary, Γ body , a pseudo-impulsive velocity, ̇xk , is to be imposed. This velocity will be based on a Gaussian displacement signal, x k , 22,23 and designed such that: (1) the Gaussian is tailored to contain a specific range of frequencies, (2) the signal is practically zero at t = 0, and (3) the resultant radiation force signal, F jk , should reach a zero steady-state solution at t = T. With these requirements, a unit height Gaussian displacement signal is defined as…”

Stability analysis of high-order finite-difference discretizations of the linearized forward-speed seakeeping problem

“…The application of this strategy to solving the forward-speed seakeeping problem was presented in [1,2]. Here the details of the basic solution strategy were described and demonstrations of both convergence and the scaling of the solution effort were provided.…”