Simulation of three‐dimensional nonlinear water waves using a pseudospectral volumetric method with an artificial boundary condition

Abstract

This article presents a pseudospectral method for the simulation of nonlinear water waves described by potential flow theory in three spatial dimensions. The method utilizes an artificial boundary condition that limits the vertical extent of the fluid domain, and it is found that the reduction in domain size offered by the boundary condition enables the solution of the Laplace problem with roughly half the degrees of freedom compared to another spectral method in the literature for (wave number times water depth) kh≥2π. Moreover, it is found that the location of the artificial boundary condition can be chosen once and for all at the beginning of simulations such that the size of the fluid domain is reduced substantially, even when the lowest point of the free surface elevation varies significantly with time. The method is tested by simulating steady nonlinear wave trains, the development of crescent waves from a steady nonlinear wave train, a nonlinear focusing event, and a Gaussian hump which is initially at rest. It is shown that in all, but the most nonlinear cases, the method is capable of obtaining accurate results.

On the statistical properties of inertia and drag forces in nonlinear multi-directional irregular water waves

“…To simulate the wave fields described above, we use the recently developed volumetric pseudospectral method of Klahn et al (2021a). This method may be divided into two parts, namely, a part concerned with the time integration of η and Φ s using (2.1c) and…”

“…The latter part is what sets it apart from other pseudospectral methods such as the methods of Dommermuth & Yue (1987), West et al (1987), Clamond & Grue (2001) and Fructus et al (2005) as it solves the Laplace equation in the fluid domain without any kind of approximation except for discretization errors. As shown by Klahn et al (2021a), this strategy enables the solution of the Laplace equation with an error 916 A59-5 that decreases exponentially with the spatial resolution for practically all values of the water depth and the wave steepness when considering steady nonlinear waves. In order to remain efficient for large problems, the method utilizes an artificial boundary condition as used by Nicholls (2011) and a preconditioning strategy inspired by the work of Fuhrman & Bingham (2004).…”

“…In this section we briefly review the method, and refer to Klahn et al (2021a) for a full description and validation. In addition, we discuss the initialization of η and Φ s , the numerical method used for the computation of I x , I y , D x and D y , as well as the computational parameters used to perform the simulations.…”

“…As such, the value of S 0 is dictated by the steepness of the wave field, ε = 2k p η 2 1/2 . We note that this definition of the wave steepness is the same as that used in the numerical studies of, for example, Toffoli et al (2010), Xiao et al (2013) and Klahn et al (2021a).…”

“…As such, if F * is affected by the steepness and directionality of the wave field then it must be because the depth-integrated quantities are affected by these parameters, and we therefore begin our investigation by studying the PDFs of the components of I and D. Having characterized the behaviour of these quantities, we then study the PDF of F * parametrically as a function of P. In order to understand the role played by nonlinear effects, we derive a number of results for the PDFs I x , I y , D x and D y as well as for F * when P = 0 and 1 based on first-order theory and compare them to the PDFs for the general nonlinear case. To calculate the latter PDFs, we perform numerical simulations of large irregular wave fields with different directional spreading and wave steepness using the numerical method of Klahn, Madsen & Fuhrman (2021a). This numerical method solves the fully nonlinear equations of potential flow theory for a single valued free surface, and has been shown to give highly accurate results on a number of challenging deep water problems.…”