Wave transformation models with exact second-order transfer

• Published in: European journal of mechanics, B, Fluids Vol. 24; no. 6; pp. 659 - 682
• Main Authors: Bredmose, H., Agnon, Y., Madsen, P.A., Schäffer, H.A.
• Format: Journal Article
• Language: English
• Published: Paris Elsevier Masson SAS 01.11.2005; Elsevier
• Subjects: Amplitude dispersion; Bichromatic transfer; Deterministic spectral modelling; Exact sciences and technology; Fast Fourier Transform; Fluid dynamics; Fully dispersive wave theory; Fundamental areas of phenomenology (including applications); Hydrodynamic waves; Nonlinear wave transformation; Physics; Nonlinear wave transformation; Deterministic spectral modelling; Amplitude dispersion; Fast Fourier Transform; Fully dispersive wave theory; Bichromatic transfer; Dispersion (wave); Immersed body; Transfer functions; Digital simulation; Evolution equation; Shallow water; Fast Fourier Transform: Bichromatic transfer; Surface waves; Wave propagation; Fast Fourier transforms; Modelling; Non linear wave
• ISSN: 0997-7546; 1873-7390
• DOI: 10.1016/j.euromechflu.2005.05.001

Fully dispersive deterministic evolution equations for irregular water waves are derived. The equations are formulated in the complex amplitudes of an irregular, directional wave spectrum and are valid for waves propagating in directions up to ±90° from the main direction of propagation under the assumptions of weak nonlinearity, slowly varying depth and negligible reflected waves. A weak deviation from straight and parallel bottom contours is allowed for. No assumptions on the vertical structure of the velocity field is made and as a result, the equations possess exact second-order bichromatic transfer functions when comparing to the reference solution of a Stokes-type analysis. Introduction of the so-called ‘resonance assumption’ leads to the evolution equations of among others Agnon, Sheremet, Gonsalves and Stiassnie [Coastal Engrg. 20 (1993) 29–58]. For unidirectional waves, the bichromatic transfer functions of the ‘resonant’ models are found to have only small deviations in general from the reference solution. We demonstrate that the ‘resonant’ models can be solved efficiently using Fast Fourier Transforms, while this is not possible for the ‘exact’ models. Simulation results for unidirectional wave propagation over a submerged bar show that the new models provide a good improvement from linear theory with respect to wave shape. This is due to the quadratic terms, enabling a nonlinear description of shoaling and de-shoaling, including the release of higher harmonics after the bar. For these simulations, the similarity between the ‘exact’ and ‘resonant’ models is confirmed. A test case of shorter waves, however, shows that the amplitude dispersion can be quite over-predicted in the models. This behaviour is investigated and confirmed through a third-order Stokes-type perturbation analysis.