A new σ‐transform based Fourier‐Legendre‐Galerkin model for nonlinear water waves. (7th August 2020)

• Record Type: Journal Article
• Authors: Klahn, Mathias; Madsen, Per A.; Fuhrman, David R.
• Is Part Of: International journal for numerical methods in fluids. Volume 93:Number 1(2021)
• Page Start: 220
• Page End: 248
• Publication Date: 2020-08-07
• Subjects: σ‐transform -- accuracy -- nonlinear water waves -- potential flow -- spectral methods -- stability; Fluid dynamics -- Mathematics -- Periodicals; 532
• DOI: 10.1002/fld.4881
• Languages: English
• ISSNs: 0271-2091

Abstract

Abstract: This paper presents a new spectral model for solving the fully nonlinear potential flow problem for water waves in a single horizontal dimension. At the heart of the numerical method is the solution to the Laplace equation which is solved using a variant of the σ ‐transform. The method discretizes the spatial part of the governing equations using the Galerkin method and the temporal part using the classical fourth‐order Runge‐Kutta method. A careful investigation of the numerical method's stability properties is carried out, and it is shown that the method is stable up to a certain threshold steepness when applied to nonlinear monochromatic waves in deep water. Above this threshold artificial damping may be employed to obtain stable solutions. The accuracy of the model is tested for: (i) highly nonlinear progressive wave trains, (ii) solitary wave reflection, and (iii) deep water wave focusing events. In all cases it is demonstrated that the model is capable of obtaining excellent results, essentially up to very near breaking. Abstract : This paper presents a new spectral model for solving the fully nonlinear potential flow problem for water waves in a single horizontal dimension. The accuracy of the model is tested for: (i) highly nonlinear progressive wave trains, (ii) solitary wave reflection, and (iii) deep water wave focusing events. In all cases it is demonstrated that the model is capable of obtaining excellent results, essentially up to very near breaking.