A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)⁴

Abstract

A Boussinesq-type model is derived which is accurate to O(kh)4 and which retains the full representation of the fluid kinematics in nonlinear surface boundary condition terms, by not assuming weak nonlinearity. The model is derived for a horizontal bottom, and is based explicitly on a fourth-order polynomial representation of the vertical dependence of the velocity potential. In order to achieve a (4,4) Padé representation of the dispersion relationship, a new dependent variable is defined as a weighted average of the velocity potential at two distinct water depths. The representation of internal kinematics is greatly improved over existing O(kh)2 approximations, especially in the intermediate to deep water range. The model equations are first examined for their ability to represent weakly nonlinear wave evolution in intermediate depth. Using a Stokes-like expansion in powers of wave amplitude over water depth, we examine the bound second harmonics in a random sea as well as nonlinear dispersion and stability effects in the nonlinear Schrödinger equation for a narrow-banded sea state. We then examine numerical properties of solitary wave solutions in shallow water, and compare model performance to the full solution of Tanaka (1986) as well as the level 1, 2 and 3 solutions of Shields & Webster (1988).

Coupling of Boussinesq and sediment transport model in a wave flume

“…Two terms of F 21 and F 22 are variables of the second order accuracy O(µ 2 ) of the model. Detail explanation of F 21 and F 22 can be found in Gobbi 4) .…”

“…The extended Boussinesq equations 4) , solved in this model are recovered by continuity and momentum equations. The fully nonlinear Boussinesq equations solve the surface elevation η and the velocity field U evaluated at some reference elevations.…”

“…In this study, further investigation on choosing the appropriate value of the important parameter will be discussed. A coupling of the Boussinesq model 4) and a sediment transport model is developed in order to predict cross-shore sediment transport in a wave flume. The total sediment transport is obtained as the sum of the net bed load and suspended load.…”

Importance of second-order wave generation for focused wave group run-up and overtopping

“…Numerous improved versions of Boussinesq-type equations have also been derived extending their validity both offshore and inshore (see e.g. Wei et al (1995), Agnon et al (1999), Gobbi et al (2000), Madsen et al (2003)). The validation tests, described later, show that the Madsen and Sørensen (1992) Boussinesq equations are sufficient to assess the importance of second-order wave generation for the focused wave groups considered herein.…”

Otimizações Computacionais Aplicadas a um Simulador de Ondas Aquáticas

“…Dessa forma, o usuário pode escolher o modelo mais apropriado para a simulação de um problema específico e pode também comparar a eficiência e acurácia de cada modelo. Dentre os modelos disponíveis, estão os modelos linear e não-linear deáguas rasas, o modelo de Boussinesq padrão e os modelos de Nwogu [10], Wei [13], Kennedy [7], Chen [4] e Gobbi [6].…”