Proceedings of the Thirty-second (2022) International Ocean and Polar Engineering Conference
Shanghai, China, June 5–10, 2022
Copyright © 2022 by the International Society of Offshore and Polar Engineers (ISOPE)
ISBN 978-1-880653-81-4; ISSN 1098-6189

www.isope.org

A two-layer Boussinesq modeling of regular wave evolution over a submerged
trapezoid breakwater with two steeper slopes
Zhongbo Liu 1, Peixiu Han 1, Yan Wang1, Kezhao Fang 2
1. Transportation Engineering College, Dalian Maritime University, Dalian, Liaoning, China
2. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, Liaoning, China

wave information is very important for building structure design,
sediment movement and seabed evolution. A submerged breakwater is
a common nearshore hydraulic structure which is usually built in frontof a protected coastal structure to reduce the wave height. To save thecost of engineering materials and other practical conditions, thesubmerged breakwater usually has a steep slope in both sides. The
wave propagation and transformation over the submerged breakwater is
a complex hydrodynamic phenomenon, involving wave reflection,
nonlinear shoaling process and even wave breaking. The study of theinteraction between waves and submerged dikes is a topic of interest toboth the coastal engineering and academic researches.

ABSTRACT
Accurate simulation of wave evolution over a submerged trapezoid
breakwater requires a high accuracy in both linear and nonlinear
properties of the numerical model. The two-layer Boussinesq-typemodel with highest spatial derivatives being 2 (Liu et al, 2018) is
derived with a mild-slope assumption. The model is applied to wave
evolution over a submerged trapezoid breakwater with a 1:20 afore
slope and 1:10 back slope, good agreement is found (Liu et al., 2019).
The applicability of this model in simulating regular wave evolution
over a submerged breakwater with slope higher than 1:5 is not known
yet. Recently, Sun et al. (2021) reveal that the different values for α(0.13≤α≤0.25) in this two-layer model do not have great effects on thehigh accuracy of the linear shoaling, linear phase celerity and eventhird-order nonlinearity for the range of 0<kh ≤ 10 (where k is wave
number and h is water depth). The present study investigates thecapability and flexibility of the two-layer Boussinesq-type model inmodeling of strongly nonlinear nonbreaking wave propagation andtransformation over a submerged trapezoid breakwater when the valueof dispersion coefficient α is chosen at the interval [0.13, 0.25]. Twoavailable published experimental data (Ohyama et al, 1995; Rao et al,2021) of wave evolution over a breakwater are used to validate themodel. The slope of the submerged trapezoid breakwater in theseexperiments is 1:2 (Ohyama et al., 1995) and 1:4 (Rao et al., 2021). The
agreement index (Wilmott, 1981) is applied to testify the agreement ofsurface elevation or velocity between the numerical model and theexperimental data. The good agreement demonstrates that the two-layer
Boussinesq-type model with different values for α (0.13 ≤ α ≤ 0.25) ofdispersion coefficient can be applied to wave evolution over asubmerged breakwater with a steeper slope as high as 1:2.

To accurate predict wave conditions in nearshore zone and capture
wave evolution over a submerged breakwater, numerous models havebeen developed and the Boussinesq-type formula (Madsen andSørensen, 1992; Nwogu, 1993; Wei et al., 1995; Gobbi and Kirby, 1999;
Madsen et al., 2002; Lynett and Liu, 2004; Zou and Fang, 2008; Liu andFang, 2015; Liu and Fang, 2016; Liu et al., 2018; Liu et al., 2019) isamong them. As shown in some experimental works (Luth et al. 1994,and Ohyama et al. 1995), regular waves will decompose into higher
frequency free waves as they propagate past a submerged breakwater.
When waves propagate onto the front slope of the breakwater,
nonlinear interactions transfer energy from the leading wave
component to higher harmonics, causing the wave to become steeper.
The waves are negatively asymmetric and have pitched-forward shape.
When the bottom slope becomes negative, the nonlinear coupling of thehigher harmonics with the fundamental wave becomes progressively
weaker, each wave components are released as free waves and travelwith their speeds. And the process of wave evolution is fairlycomplicated. It is a quite demanding test for any Boussinesq-typemodel. As reported in the literatures (Ohyama et al. 1995; Gobbi et al,1999), the Boussinesq-type models (Nwogu, 1993; Wei et al., 1995) withPadé[2,2] approximant in dispersion cannot accurately predict surface
elevations after waves propagates over lee side of the breakwater. As
far as we have known, this holds true for any Boussinesq-type modelwith Padé[2,2] approximant in dispersion. With 1:20 afore slope and1:10 back slope in the experiment of Luth et al (1994), those
Boussinesq-type models (Gobbi and Kirby, 1999; Lynett and Liu, 2004;
Liu and Fang, 2015; Liu et al., 2018; Liu and Fang, 2019; Liu et al., 2019)

KEY WORDS: Boussinesq-type model; numerical simulation;
surface elevation; velocity; dispersion; submerged breakwater.
INTRODUCTION
Waves are the main hydrodynamic force in nearshore water, and2665