Zhongbo Liu

Dalian Maritime University · Transportation Engineering College

A two-layer viscous Boussinesq-type model is developed to simulate the wave energy dissipation during wave propagation in deep water. The viscous terms are incorporated into both the dynamic and kinematic boundary conditions at the free surface, and the corresponding analytical solution of the second-order amplitude has been derived for the first t...

The two-layer Boussinesq water wave equation with the highest derivative of 2 given by Liu et al (2018）has better dispersion and nonlinearity, based on which, a 3D wave numerical model with finite difference method is established. The equations are spatially discretized on rectangular grids，and using higher order derivatives to approximate the time...

Due to the complex geometry, the research for theoretical methods on wave absorption with plunger-type wavemakers is not easy. Therefore, this paper proposes a simple approach for wave absorbing control of plunger wavemakers using machine learning. The main contribution of this approach is to control plunger wavemakers to absorb waves in the time d...

Solitary gravity waves play a key role in the field of coastal and ocean engineering. The two-layer Boussinesq model developed by the authors has excellent linear and nonlinear properties. However, the permanent solitary wave solution and the numerical performance of the model in describing the solitary waves over complex bathymetries remain unknow...

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-type model 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 sub...

This study presents an efficient Boussinesq-type wave model accelerated by a single GPU (Graphics Processing Unit). The model uses the hybrid FV (Finite Volume) and FD (Finite Difference) method to solve weakly dispersive and nonlinear Boussinesq equations in the horizontal plane, enabling the model to have the shock-capturing ability to deal with...

A series of two-dimensional flume experiments were carried out to study turbulent bore impact on the vertical wall mounted on a reef flat. Turbulent bores were generated by solitary waves propagating on typical fringing reef profiles with and without reef crest. The idealized reef model has a 1:4 face slope and a long reef flat, with an impermeable...

The increasing demand for electric power is an established trend in China, of which coal power accounts for a large proportion. This paper proposes an inventory hub location model considering multi-type coal to minimize the total transportation and inventory cost in a multi-coal-power supply chain. A second-order cone programming method is adopted...

This paper presents the experimental results for a 1:50 scaled model of an immersed tunnel element suspended from a twin-barge in tank tests. A compliant four leg catenary mooring system using steel link chains in the physical tests is investigated. The corresponding numerical model is developed to simulate the mooring characteristics and the fully...

Based on the one-layer Boussinesq model with highest spatial derivative being 3, a numerical model is established for focused wave group. The numerical model is solved with predictor-corrector scheme in finite differential method. For time integration, a third-order Adams-Bashforth scheme and a fourth-order Adams-Moulton scheme are separately used...

The coefficients embodied in a Boussinesq-type model are very important since they are determined to optimize the linear and nonlinear properties. In most conventional Boussinesq-type models, these coefficients are assigned the specific values. As for the multi-layer Boussinesq-type models with the inclusion of the vertical velocity, however, the e...

Boussinesq-type equation is one of the important tools for simulating the propagation and evolution of water waves. The theoretical derivation and numerical application of the Boussinesq-type water wave equation dating back to 1967 are reviewed with the hope of promoting its deep development and application in the fields of coastal and ocean engine...

An extended version of the XBeach model is presented in this paper to improve the accuracy of predicted wave and current fields and sandy beach evolution in scenarios where wave diffraction takes effect. Following the approach successfully implemented in the spectral wave model SWAN, a diffraction parameter is introduced into the wave action balanc...

https://authors.elsevier.com/a/1a0uRW5Z38g6G (thanks for the elsevier providing the 50's free download). In this work, a recently developed two-layer Boussinesq model with high accuracy regarding linear and nonlinear properties and interior kinematic properties from deep to shallow water is extended to include the time-varying bathymetry for model...

Thanks a lot for the permission from the Elsevier. And the publisher gives us a personalized URL providing 50 days' free access to our article: https://authors.elsevier.com/a/1Ztet6nh6sdid________________________________________The accuracy of the linear and nonlinear properties embodied in multi-layer Boussinesq-type models with the highest spatia...

A vertical two-dimensional numerical model is developed to demonstrate the application potential of the recently proposed two-layer Boussinesq-type equations, which have been theoretically shown to exhibit high accuracy in both linear and nonlinear properties, by the authors (Liu and Fang, 2016). Numerical implementation is established on a regular...

A new multi-layer irrotational Boussinesq-type model is proposed for both linear and nonlinear surface water waves over mildly sloping seabeds. The model is formulated in terms of computational horizontal and vertical velocity components within each layer and satisfies exact kinematic and dynamic free-surface conditions as well as kinematic seabed...

With the unstructured grid, the Finite Volume Coastal Ocean Model (FVCOM) is converted from its original FORTRAN code to a Compute Unified Device Architecture (CUDA) C code, and optimized on the Graphic Processor Unit (GPU). The proposed GPU-FVCOM is tested against analytical solutions for two standard cases in a rectangular basin, a tide induced f...

According to the paper by Wei et al. (2014), the assumption of the search and rescue (SAR) bases of Chinese patrol vessels does not conform to the current reality. This renders the probability distribution of Chinese patrol vessels in the South China Sea to be inaccurate; The selection of the count and speed of Chinese patrol vessels does not confo...

To accurately describe strongly nonlinear wave motion in deep water, a new two-layer Boussinesq model for water waves is derived in this paper with excellent dispersive and nonlinear properties in extremely deep water. First, we separated the fluid into two parts: the upper layer and lower layer. Then, using Taylor expansion, we expanded the veloci...

We derive a new two-layer Boussinesq model with high accuracy in linear and nonlinear properties and in interior kinematic property from deep to shallow water. This model is formulated in terms of computational horizontal and vertical velocities defined in each layer. The highest derivative in the equations is limited to three, which is convenient...

A double-layer depth-averaged Boussinesq-type model for wave propagation over an uneven bottom is derived. The governing equations are formulated in two depth-averaged velocities within each water layer and of the second-order fully nonlinearity. To improve the model properties, higher-order terms are introduced to momentum equations and theoretica...

A numerical model, which solves the horizontal two-dimensional fully nonlinear Boussinesq equations using a well-balanced shock-capturing scheme, is developed and used to investigate wave transformation in fringing reef environment. The governing equations are firstly reformulated into a conservative form, and a Godunov-type finite volume method is...

This paper presents a depth-integrated, non-hydrostatic model for coastal water waves. The shock-capturing ability of this model is its most attractive aspect and is essential for computation of energetic breaking waves and wet–dry fronts. The model is solved in a fraction step manner, where the total pressure is decomposed into hydrostatic and non...

Seven sets of higher order Boussinesq equations are inter-compared in order to verify the effects of different approximation levels of nonlinearity and the mild slope assumption on numerical results. The models have linear dispersion accurate to a Padé [2,2] or Padé [4,4] expansion, respectively and different nonlinearities with ε=Ο(µ 2), ε=Ο(µ 2/3...

The procedure proposed by Chondros and Memos [Chondros and Memos, 2014, A 2DH nonlinear Boussinesq-type wave model of improved dispersion, shoaling, and wave generation characteristics, Coast. Eng., 91: 99–122] for improving dispersion, shoaling and wave generation characteristics of a 2DH Boussinesq-type wave model is discussed. We realize that th...

Based on uniform grid, a depth integrated non-hydrostatic model for coastal water flow is presented. The numerical implementation of the model is split into hydrostatic step and non-hydrostatic step. In the hydrostatic step, an efficient finite volume scheme is used to solve the fully nonlinear shallow water equations, where the hydrostatic reconst...

A Boussinesq-type model is developed for simulating wave and runup generated by sliding masses in the coastal region. A shock-capturing Boussinesq type model, which is originally designed for simulation of coastal water waves over fixed sea bottom, is extended to include the effect of time-varying bed. Landslide is then assumed as a rigid body with...

A hybrid finite-difference/finite-volume scheme is developed to solve the 2D fully nonlinear Boussinesq equations. Finite volume method, in conjunction with the MUSTA scheme, is used to evaluate the flux terms while finite difference method is used to approximate the rest terms in the conservative governing equations. The third order Runge-Kutta me...

To better understand the complex process of wave transformation and associated hydrodynamics over various fringing reef profiles, numerical experiments were conducted with a one-dimensional (1D) Boussinesq wave model. The model is based on higher-order Boussinesq equations and a higher-accuracy finite difference method. The dominant energy dissipat...

When Boussinesq model is applied to compute wave transformation from deep water to shallow water, the proper shoaling effect is essentially important for accurately predicting wave evolution. Improper shoaling parameters value results in bad shoaling property, and decreases the applicability of the Boussinesq model. Based on a fourth order dispersi...

In this paper, a hybrid finite-difference and finite-volume numerical scheme is developed to solve the 2-D Boussinesq equations. The governing equations are the extended version of Madsen and Sorensen's formulations. The governing equations are firstly rearranged into a conservative form. The finite volume method with the HLLC Riemann solver is use...

The pioneering work of Haller [8] on physically investigating bathymetry-controlled rip currents in the laboratory is a standard benchmark test for verifying numerical nearshore circulation models. In this paper, a numerical model based on higher-order Boussinesq equations was developed to reproduce the number of experiments involved in such an inv...

A Boussinesq-type wave model is developed to numerically investigate the breaking waves and wave-induced currents. All the nonlinear terms are retained in the governing equations to keep fully nonlinearity characteristics and it hence more suitable to describe breaking waves with strong nonlinearity in the nearshore region. The Boussinesq equations...

Unstructured grid has been widely used to establish hydrodynamic models. To rapidly get the calculation results without a cluster when the number of computational grids is too large, a high-performance computing technology which is based on GPU (graphic processing unit), is adopted to design a parallel algorithm and establish a 2D unstructured grid...

A Boussinesq-type wave model is presented in this paper to simulate the tsunami solitary wave transformation over a reef profile. The governing equation is solved by combining the finite difference method and the finite volume method. The convective flux terms are discretized using the finite volume method, in conjunction with the HLL Riemann Solve...

To efficiently solve the extended Boussinesq equations, a hybrid finite-difference and finite-volume scheme is developed. The one-dimensional governing equations are kept in conservation form. The flux term is discretized using the finite volume method, while the remaining terms are discretized using the finite difference method. A Godunov-type hig...

To consider the effect of porous media on wave propagation in mathematical model, the drag resistance force and inertial force of the porous media were included in the fluid motion, and the corresponding Laplace equation and boundary conditions were given. First cancelling out the dimensions of control equations, and then starting from the velocity...

It is very important to adopt reasonable discrete way for Boussinesq-type equation which is often used to simulate offshore wave field. In order to obtain the accurate numerical results, we present the finite different numerical model for the modified Boussinesq equations which are introduced by Beji and Nadaoka. In the numerical model, the differe...

A resistance equation is introduced for considering the porous effect, and a set of high-order Boussinesq-type equations for water waves is extended to be applicable for porous seabed. The new equations are analyzed theoretically in constant water depth with different porous layer depths, and the phase velocity and damping rate are discussed compar...

Large-scale marine reclamation is expected to have some effects on the marine environment, but these effects vary depending on the reclamation layout schemes. For the offshore airport programming of Dalian, China, an insular artificial island scheme and a peninsula artificial island scheme are considered. To contrast the effects of these two scheme...

To better understand the wave-induced longshore currents in nearshore region, a numerical model is developed based on the second order fully nonlinear Boussinesq equations. The governing equations possess optimum linear properties and second order fully nonlinearity characteristics in medium water depths. The relaxation zone method is used to gener...

We propose alternative forms of the Boussinesq equations which extend the equations of Madsen and Schäffer by introducing extra nonlinear terms during enhancement. Theoretical analysis shows that nonlinear characteristics are considerably improved. A numerical implementation of one-dimensional equations is described. Three tests involving strongly...

To obtain the mathematical model of wave propagation in a better performance, a new set of Boussinesq equations with full nonlinearity approximating to the order O(μ 2) was developed by enhancing an existing set of Boussinesq equations. Different from the traditional enhancement of Boussinesq-type equations, all the nonlinear terms were retained, i...

A set of nonlinear Boussinesq equations with fully nonlinearity property is solved numerically in generalized coordinates, to develop a Boussinesq-type wave model in dealing with irregular computation boundaries in complex nearshore regions and to facilitate the grid refinements in simulations. The governing equations expressed in contravariant com...

Rip current is quite common in coastal zone and plays a key role in coastal engineering related issues, for example, nearshore pollutants transport and coastal morphology. Among variety of documented research results on rip current, the experiment of Haller(1999) is widely used for validating the nearshore circulation models, such as Boussinesq-typ...

A 2D wave breaking model based on second order fully nonlinear Boussinesq-type equations is developed to simulate the rip current generated on a barred beach. Firstly, a set of equations with fully nonlinearity characteristics is extended, i.e., eddy viscosity method and slot method are adopted to mimic energy dissipation of breaking waves and movi...

With the rapid development of port, the negative influences such as the resources occupation, environmental pollution have become immensely serious. Therefore evaluating the green contribution of port to city systematically from the economic, resource and environmental perspective has become an important problem to seek the overall balance between...

Enhancements for the Bragg reflection are introduced for three sets of 2D higher order Boussinesq equations to improve the prediction of the Bragg reflection. The extension of the approach to other sets of Boussinesq equations is discussed. The analytical solutions for the Bragg reflection over an infinite number of sinusoidal bars are derived for...

Based on the hyperbolic mild-slope equations derived by Copeland (1985), a numerical model is established in unstaggered grids. A composite 4 th-order Adam-Bashforth-Moulton (ABM) scheme is used to solve the model in the time domain. Terms involving the first order spatial derivatives are differenced to O(Δx)4 accuracy utilizing a five-point formul...

The dispersion characteristic of the modified Boussinesq equations with two parameters is only equal to the second order Pade expansion of the linear dispersion relation. Further improvement is done by introducing a new velocity vector to replace the depth-averaged velocity in the modified Boussinesq equations. The dispersion of the further modifie...

Based on the classical Boussinesq model by Peregrine [Peregrine, D.H., 1967. Long waves on a beach. J. Fluid Mech. 27 (4), 815–827], two parameters are introduced to improve dispersion and linear shoaling characteristics. The higher order non-linear terms are added to the modified Boussinesq equations. The non-linearity of the Boussinesq model is a...

There are many ways to derive Boussinesq equations, one of the simple methods was proposed by Beji and Nadaoka. Based on the classical Boussinesq equations, the simple method is used with two parameters employed to improve the linear dispersion of the Boussinesq equations. Also by the comparison with the model derived by Schaffer and Madsen, the tw...

When an oil tanker under the combined action of wind, waves and tidal current and is berthed or moored to a platform, the impact forces on the fenders and the tensile force in the mooring lines are important factors in the studies of berthing and mooring conditions. Based on the experiment of a berthing and mooring tanker model under the action of...