A high-order spectral method for nonlinear wave–body interactions

A high-order spectral method originally developed to simulate nonlinear gravity wave-wave interactions is here extended to study nonlinear interaction between surface waves and a body. The method accounts for the nonlinear interactions among NF wave modes on the free surface and NHB source modes on the body surface up to an arbitrary order M in wave steepness. By using fast-transform techniques, the operational count per time step is only linearly proportional to M and NF. Significantly, for a (closed) submerged body, the exponential convergence with respect to M, NF (for moderately steep waves), and NB is obtained. To illustrate the usefulness of this method, it is applied to study the diffraction of Stokes waves by a submerged circular cylinder. Computations up to M=4 are performed to obtain the nonlinear steady and harmonic forces on the cylinder and the transmission and reflection coefficients. Comparisons to available measurements as well as existing theoretical/computational predictions are in good agreement. The most important result is the quantification of the negative horizontal drift force on the cylinder which is fourth order in the incident wave steepness. It is found that the dominant contribution of this force is due to the quadratic interaction of the first- and third-order first-harmonic waves rather than the self-interaction of the second-order second-harmonic waves, which in fact reduces the negative drift force.