Driven Nonlinear Potential Flow With Wave Breaking at Shallow-Water Beaches

2017 ◽  
Author(s):  
Floriane Gidel ◽  
Onno Bokhove ◽  
Mark Kelmanson
Keyword(s):  
Shallow Water ◽  
Potential Flow ◽  
Water Waves ◽  
Wave Breaking ◽  
Equations Of Motion ◽  
Water Model ◽  
Dispersive Waves ◽  
Variational Structure ◽  
Nonlinear Potential ◽  
Nonlinear Equations Of Motion

We introduce a new model of nonlinear dispersive waves generated by wavemakers in deep water, coupled to a shallow-water model with wave breaking, modelled as hydraulics bores, in the shore zone. Coupling of deep- and shallow-water models requires the formulation of an advanced space-time technology able to stably capture the free-surface dynamics. Our approach comprises the direct discretisation of the variational principle for the continuum modelling of potential-flow water-wave dynamics. Preservation of the variational structure in the discretisation ensures that important conservation properties of the original continuum system are inherited to a high degree by the discrete system. The nonlinear equations of motion resulting from the coupling of the potential-flow water-waves and the (breaking) shallow-water waves are solved in unison. By construction, this process results in a stable and robust numerical scheme that is well suited to demanding maritime applications.

Wave-driven currents and vortex dynamics on barred beaches

2001 ◽  
Vol 449 ◽  
pp. 313-339 ◽  
Author(s):  
OLIVER BÜHLER ◽  
TIVON E. JACOBSON
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Wave Breaking ◽  
Bottom Topography ◽  
Breaking Waves ◽  
Vortex Structures ◽  
Water Model ◽  
Mean Flow ◽  
The Mean

We present a theoretical and numerical investigation of longshore currents driven by breaking waves on beaches, especially barred beaches. The novel feature considered here is that the wave envelope is allowed to vary in the alongshore direction, which leads to the generation of strong dipolar vortex structures where the waves are breaking. The nonlinear evolution of these vortex structures is studied in detail using a simple analytical theory to model the effect of a sloping beach. One of our findings is that the vortex evolution provides a robust mechanism through which the preferred location of the longshore current can move shorewards from the location of wave breaking. Such current dislocation is an often-observed (but ill-understood) phenomenon on real barred beaches.To underpin our results, we present a comprehensive theoretical description of the relevant wave–mean interaction theory in the context of a shallow-water model for the beach. Therein we link the radiation-stress theory of Longuet-Higgins & Stewart to recently established results concerning the mean vorticity generation due to breaking waves. This leads to detailed results for the entire life-cycle of the mean-flow vortex evolution, from its initial generation by wave breaking until its eventual dissipative decay due to bottom friction.In order to test and illustrate our theory we also present idealized nonlinear numerical simulations of both waves and vortices using the full shallow-water equations with bottom topography. In these simulations wave breaking occurs through shock formation of the shallow-water waves. We note that because the shallow-water equations also describe the two-dimensional flow of a homentropic perfect gas, our theoretical and numerical results can also be applied to nonlinear acoustics and sound–vortex interactions.

scholarly journals Shallow water waves in a rotating rectangular basin

2000 ◽  
Vol 24 (10) ◽  
pp. 649-661 ◽  
Author(s):  
Mohamed Atef Helal
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Equations Of Motion ◽  
System Of Nonlinear Equations ◽  
System Of Equations ◽  
Order Of Approximation ◽  
Shallow Water Waves ◽  
Nonlinear System Of Equations ◽  
Rectangular Basin ◽  
Shallow Water Approximation

This paper is mainly concerned with the motion of an incompressible fluid in a slowly rotating rectangular basin. The equations of motion of such a problem with its boundary conditions are reduced to a system of nonlinear equations, which is to be solved by applying the shallow water approximation theory. Each unknown of the problem is expanded asymptotically in terms of the small parameterϵwhich generally depends on some intrinsic quantities of the problem of study. For each order of approximation, the nonlinear system of equations is presented successively. It is worthy to note that such a study has useful applications in the oceanography.

Reflection of a shallow-water soliton. Part 1. Edge layer for shallow-water waves

1984 ◽  
Vol 146 ◽  
pp. 369-382 ◽  
Author(s):  
N. Sugimoto ◽  
T. Kakutani
Keyword(s):  
Boundary Condition ◽  
Shallow Water ◽  
Water Waves ◽  
Shallow Water Equation ◽  
Expansion Method ◽  
Matching Condition ◽  
Finite Amplitude ◽  
Matched Asymptotic Expansion ◽  
Dispersive Waves ◽  
Edge Layer

To investigate reflection of a shallow-water soliton at a sloping beach, the edge-layer theory is developed to obtain a ‘reduced’ boundary condition relevant to the simplified shallow-water equation describing the weakly dispersive waves of small but finite amplitude. An edge layer is introduced to take account of the essentially two-dimensional motion that appears in the narrow region adjacent to the beach. By using the matched-asymptotic-expansion method, the edge-layer theory is formulated to cope with the shallow-water theory in the offshore region and the boundary condition at the beach. The ‘reduced’ boundary condition is derived as a result of the matching condition between the two regions. An explicit edge-layer solution is obtained on assuming a plane beach.

scholarly journals Shallow water equations for equatorial tsunami waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170100 ◽  
Author(s):  
Anna Geyer ◽  
Ronald Quirchmayr
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Waves ◽  
Water Model ◽  
Shallow Water Model ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Tsunami Waves ◽  
De Vries ◽  
Model Equations

We present derivations of shallow water model equations of Korteweg–de Vries and Boussinesq type for equatorial tsunami waves in the f -plane approximation and discuss their applicability. This article is part of the theme issue ‘Nonlinear water waves’.

scholarly journals Wave Breaking in a Shallow Water Model

2018 ◽  
Vol 50 (1) ◽  
pp. 354-380 ◽  
Author(s):  
Vera Mikyoung Hur ◽  
Lizheng Tao
Keyword(s):  
Shallow Water ◽  
Wave Breaking ◽  
Water Model ◽  
Shallow Water Model

Over-reflection and shear instability in a shallow-water model

1992 ◽  
Vol 236 ◽  
pp. 259-279 ◽  
Author(s):  
Shin-Ichi Takehiro ◽  
Yoshi-Yuki Hayashi
Keyword(s):  
Shear Flow ◽  
Shallow Water ◽  
Water Waves ◽  
Shear Instability ◽  
Water Model ◽  
Shallow Water Waves ◽  
Wave Amplification ◽  
Linear Shear ◽  
Conservation Of Momentum ◽  
The Relationship

The characteristics of shallow-water waves in a linear shear flow are studied, and the relationship between waves and unstable modes is examined. Numerical integration of the linear shallow-water equations shows that over-reflection occurs when a wave packet is incident at the turning surface. This phenomenon can be explained by the conservation of momentum as discussed by Acheson (1976). The unstable modes of linear shear flow in a shallow water found by Satomura (1981) are described in terms of the properties of wave propagation as proposed by Lindzen and others. Ripas's (1983) theorem, which is the sufficient condition for stability of flows in shallow water, is also related to the wave geometry. The Orr mechanism, which is proposed by Lindzen (1988) as the primary mechanism of wave amplification, cannot explain the over-reflection of shallow-water waves. The amplification of these waves occurs in the opposite sense to that of Orr's solution.

A coupled variational principle for 2D interactions between water waves and a rigid body containing fluid

2017 ◽  
Vol 827 ◽  
Author(s):  
Hamid Alemi Ardakani
Keyword(s):  
Variational Principle ◽  
Potential Flow ◽  
Water Waves ◽  
Variational Principles ◽  
Equations Of Motion ◽  
The Body ◽  
Two Dimensional ◽  
Hydrodynamic Equations ◽  
Gravity Driven ◽  
Complete Set

New variational principles are given for the two-dimensional interactions between gravity-driven water waves and a rotating and translating rectangular vessel dynamically coupled to its interior potential flow with uniform vorticity. The complete set of equations of motion for the exterior water waves, the exact nonlinear hydrodynamic equations of motion for the vessel in the roll/pitch, sway/surge and heave directions, and also the full set of equations of motion for the interior fluid of the vessel, relative to the body coordinate system attached to the rotating–translating vessel, are derived from two Lagrangian functionals.

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