Development and Validation of Quasi-Eulerian Mean Three-Dimensional Equations of Motion Using the Generalized Lagrangian Mean Method

Duoc Nguyen,Dano Roelvink,Niels Jacobsen

doi:10.3390/jmse9010076

Abstract

This study aims at developing a new set of equations of mean motion in the presence of surface waves, which is practically applicable from deep water to the coastal zone, estuaries, and outflow areas. The generalized Lagrangian mean (GLM) method is employed to derive a set of quasi-Eulerian mean three-dimensional equations of motion, where effects of the waves are included through source terms. The obtained equations are expressed to the second-order of wave amplitude. Whereas the classical Eulerian-mean equations of motion are only applicable below the wave trough, the new equations are valid until the mean water surface even in the presence of finite-amplitude surface waves. A two-dimensional numerical model (2DV model) is developed to validate the new set of equations of motion. The 2DV model passes the test of steady monochromatic waves propagating over a slope without dissipation (adiabatic condition). This is a primary test for equations of mean motion with a known analytical solution. In addition to this, experimental data for the interaction between random waves and a mean current in both non-breaking and breaking waves are employed to validate the 2DV model. As shown by this successful implementation and validation, the implementation of these equations in any 3D model code is straightforward and may be expected to provide consistent results from deep water to the surf zone, under both weak and strong ambient currents.

Highlights

The interaction between waves and currents has been the subject of much research in recent decades
The concept of “radiation stress: was first introduced by Longuet-Higgins and Stewart [1] to explain the transfer of wave energy to a uniform current
This concept was used by Longuet-Higgins and Stewart [2] to study the changes in the mean surface level and the currents caused by gravity waves

Summary

Introduction

The interaction between waves and currents has been the subject of much research in recent decades. There are two representations of wave-averaged effects on the currents called “radiation stress” and “vortex force”. The concept of “radiation stress: was first introduced by Longuet-Higgins and Stewart [1] to explain the transfer of wave energy to a uniform current. This concept was used by Longuet-Higgins and Stewart [2] to study the changes in the mean surface level and the currents caused by gravity waves. The radiation stress concept has been successful in explaining phenomena such as wave “set-up”, “surf beats”, the steepening of the surface waves on adverse currents [3], and the generation of long-shore currents by oblique incident waves [4,5,6].

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Conclusion