Green Naghdi Model Research Articles

Simulation of evolution of gravity wave groups with moderate steepness

The evolution of nonlinear deep-water wave groups in one-horizontal dimension is studied. The strongly nonlinear Green–Naghdi (GN) models are used to simulate this phenomenon. There are different levels of the GN models depending on the different velocity assumption used for the vertical structure of the flow field, such as GN-1, GN-2, and so forth. In this work, we use both the GN-3 and GN-4 models to do the simulations. Calculations are done for three different numbers of waves per group (or packet) as the number of envelope solitons depends on the number of waves per group (N). The numerical results show that the GN-3 model can give the converged GN results for the cases tested here. We conduct a series of physical experiments to investigate the evolution of wave groups. We also use other’s experimental data and present laboratory measurements to compare the data with the predictions of the GN models and show that good agreement is obtained overall.

Validation of Boussinesq–Green–Naghdi modeling for surf zone hydrodynamics

2D modeling for surf zone phenomenons are validated in present work using the rotational Boussinesq–Green–Naghdi model. Three benchmark test cases are simulated: tsunami wave runup a conical island; tsunami wave runup complex shelf; and rip current and wave setup over sand bars. The computed results are compared to the experimental data including the free surface deformation and depth-averaged velocities. The simulated 2D cases fundamentally validate the model׳s ability in predicting wave transformation, wave breaking, wave runup and the velocity field for complex hydrodynamic conditions and give the basis for moving on to more complex applications. The lack of irrotationality would strongly contribute to the depth-varying velocity profile of rotational modeling, which has been partially proven in 1D undertow test (Zhang et al., 2014a). Unfortunately, few 2D experiments with rotational vortex data measured could be found due to the difficulty of recording the vortex characteristics. Future work would be the model application to much more complex geophysical and engineering problems, where the lack of any irrotational constrain is expected to excel.

Effect of Magnetic Field on Reflection of Thermo-elastic Waves from the Boundary of a Half-Space Using G–N Model of Type-II for Different Nature of the Boundary Surface

The aim of this paper is to investigate the problem of reflection of plane thermoelastic waves in presence of magnetic field from the boundary of a half-space using Green–Naghdi model of type-II. Analytical solutions for the ratios of the amplitude of the reflected waves to that of incident wave have been obtained in matrix form for stress-free insulated, stress-free isothermal, rigid insulated and rigid isothermal boundaries of the half-space. Numerical results have been carried out to obtain the values the amplitude ratios only for incident P and SV wave and shown graphically.

Modelling turbulence generation in solitary waves on shear shallow water flows

We derive a dispersive model of shear shallow water flows which takes into account a non-uniform horizontal velocity. This model generalizes the Green–Naghdi model to the case of shear flows. Besides the classical dispersion term in the Green–Naghdi model related to the acceleration of the free surface, it also contains a new dispersion parameter related to the flow structure. This parameter is related to the second moment of the velocity fluctuation with respect to the vertical coordinate. The distinction between shearing and turbulence based on the scale of variation of the velocity fluctuation is proposed. In particular, an equation for the turbulence generation is derived. Solitary waves for this model are obtained in explicit form. Comparison of solitary wave profiles with experimental ones is also performed. The agreement is very good apart from the small region near the top of the wave.

Bottom pressure caused by passage of a solitary wave within the strongly nonlinear Green-Naghdi model

In practice, the bottom pressure caused by sea waves (as well as the inverse problem of reconstruction of the sea wave field according to the indications of bottom transducers) is calculated within the framework of linear theory. According to this theory, it is easy to find the one point relation between the sea level oscillations and the bottom pressure variations using the spectral approach [1–4]. As a result, the bottom pressure becomes completely determined for the known characteristics of sea waves. However, if linear theory intuitively works well in the case of small amplitude water waves, one should not rely on linear theory, when a rough sea acquires an irregular and nonlinear character as, for example, in the case of a storm at sea. For example, it is shown in [1] that the results of predictions of linear theory in solving the inverse problem differ by 15–20% from those measured under laboratory conditions and this difference is related to the nonlinearity and noise in devices.

Discontinuous-Galerkin Discretization of a New Class of Green-Naghdi Equations

AbstractWe describe in this work a discontinuous-Galerkin Finite-Element method to approximate the solutions of a new family of 1d Green-Naghdi models. These new models are shown to be more computationally efficient, while being asymptotically equivalent to the initial formulation with regard to the shallowness parameter. Using the free surface instead of the water height as a conservative variable, the models are recasted under apre-balancedformulation and discretized using a nodal expansion basis. Independently from the polynomial degree in the approximation space, the preservation of the motionless steady-states is automatically ensured, and the water height positivity is enforced. A simple numerical procedure devoted to stabilize the computations in the vicinity of broken waves is also described. The validity of the resulting model is assessed through extensive numerical validations.

An energy-consistent depth-averaged Euler system: Derivation and properties

In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by a minimal energy constraint instead of an asymptotic expansion. The model slightly differs from the well-known Green-Naghdi model and is confronted with stationary and analytical solutions of the Euler system corresponding to rotational flows. At the end of the paper, we give time-dependent analytical solutions for the Euler system that are also analytical solutions for the proposed model but that are not solutions of the Green-Naghdi model. We also give and compare analytical solutions of the two non-hydrostatic shallow water models.

High-level Green–Naghdi wave models for nonlinear wave transformation in three dimensions

The Green–Naghdi (GN) wave models are categorized into different levels based on the assumptions made for the velocity field. The low-level GN model (Level I GN model or called the GN-1 model) is a weakly dispersive, strongly nonlinear wave model. As the level goes up, the high-level GN model becomes a strongly dispersive, strongly nonlinear wave model. This paper introduces the algorithm to solve the Green–Naghdi wave models of different levels in three dimensions. The high-level GN (GN-3 and GN-4) models are applied to three-dimensional wave problems for the first time. Three test cases are considered here. First one is on the wave evolution in a closed basin. The symmetry, in the \(x\) and \(y\) directions in this case, verifies that the algorithm introduced here works well. The GN-3 results are also compared with the linear analytical results for a small wave elevation in a closed basin, and the agreement is good. The last two cases involve wave diffraction problems caused by an uneven seabed. In both of the last two cases, the GN-3 model is proved to be the converged GN model. The agreement between the GN-3 model and the experimental data and numerical predictions of the fully nonlinear Boussinesq model of others is also very good.

A new class of fully nonlinear and weakly dispersive Green–Naghdi models for efficient 2D simulations

We introduce a new class of two-dimensional fully nonlinear and weakly dispersive Green–Naghdi equations over varying topography. These new Green–Naghdi systems share the same order of precision as the standard one but have a mathematical structure which makes them much more suitable for the numerical resolution, in particular in the demanding case of two dimensional surfaces.For these new models, we develop a high order, well balanced, and robust numerical code relying on a hybrid finite volume and finite difference splitting approach. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a finite difference approach. Higher order accuracy in space and time is achieved through WENO reconstruction methods and through an SSP-RK time stepping. Particular effort is made to ensure positivity of the water depth.Numerical validations are then performed, involving one and two dimensional cases and showing the ability of the resulting numerical model to handle waves propagation and transformation, wetting and drying; some simple treatments of wave breaking are also included. The resulting numerical code is particularly efficient from a computational point of view and very robust; it can therefore be used to handle complex two dimensional configurations.

Thermoelastic interaction in a viscoelastic functionally graded half-space under three-phase-lag model

This paper aims at studying the thermo-viscoelastic interaction in a functionally graded, infinite, Kelvin–Voigt-type viscoelastic, thermally conducting medium due to the presence of periodically varying heat sources. Three-phase-lag thermoelastic model, Green–Naghdi model II (i.e. the model which predicts thermoelasticity without energy dissipation) and Green–Naghdi model III (i.e. the model which predicts thermoelasticity with energy dissipation) are employed to study thermomechanical coupling, thermal and mechanical relaxation effects. In the absence of mechanical relaxations (viscous effect), the results for various generalised theories of thermoelasticity may be obtained as particular cases. The governing equations are expressed in Laplace–Fourier double transform domain and are solved in that domain. The inversion of the Fourier transform is carried out using residual calculus, where the poles of the integrand are obtained numerically in the complex domain by using Laguerre’s method and the inversio...

On the theory of two-temperature thermoelasticity without energy dissipation of Green–Naghdi model

This work introduces one-dimensional numerical application of an elastic and isotropic half space in which the governing equations have been taken in the context of the two-temperature thermoelasticity without energy dissipation theory to show the effects of the two-temperature parameter on all the studied fields and to stand on the contributions of this theory. The results of this work confirmed that the two-temperature thermoelasticity without energy dissipation is valid when the thermo-dynamical temperature and the temperature of the heat conduction are not coinciding.

Thermal shock problem for a fiber-reinforced anisotropic half-space placed in a magnetic field via G-N model

The generalized magneto-thermoelasticity theory, based on Green–Naghdi model, is used to study the thermal shock problem of a fiber-reinforced anisotropic half-space. The solid half-space deforms because of thermal shock, and due to the application of the magnetic field. Maxwell’s equations are formulated and the generalized coupled governing equations are derived. Finite element method with the Laplace transform technique is used to obtain the general solution. The distributions of the considered physical variables are represented graphically. The influence of the magnetic field is discussed. Some particular cases of special interest due to the application of two types of Green and Naghdi’s theory have been deduced.

On the steady solitary-wave solution of the Green–Naghdi equations of different levels

The steady-state solitary wave solution of high-level Green–Naghdi (GN) equations is obtained by use of the Newton–Raphson method. Four aspects of solitary waves are studied: the wave speed, wave profile, velocity field and particle trajectory. A convergence study is performed for each individual case. Results of the converged model are compared with the existing laboratory experiments and other theoretical solutions for an inviscid and incompressible fluid, including the solutions of the Euler equations. Particle trajectories, predicted by the GN model, show close agreement with the laboratory measurements and provide a new approach to understanding the movement of the particles under a solitary wave. It is further shown that high-level GN equations can predict the solitary wave of the highest height.

Thermoelastic response in a symmetric spherical shell

This paper is concerned with the determination of thermoelastic stresses, strain and conductive temperature in a spherically symmetric spherical shell. The two-temperature three-phase-lag thermoelastic model (2T3P) and two-temperature Green–Naghdi model III (2TGNIII) are combined into a unified formulation. There is no temperature at the outer boundary, and thermal load is applied at the inner boundary. The basic equations have been written in the form of a vector–matrix differential equation in the Laplace transform domain which is then solved by the state-space approach. The numerical inversion of the transform is carried out using Fourier series expansion techniques. Because of the short duration of the second sound effects, small time approximations of the solutions are studied. The physical quantities have been computed numerically and presented graphically in a number of figures. A complete and comprehensive analysis of the results has been presented for the 2T3P and the 2TGNIII models. These results have also been compared with those of the one-temperature three-phase-lag thermoelastic model (1T3P) and one-temperature Green–Naghdi model III (1TGNIII).

Conditions for extreme wave runup on a vertical barrier by nonlinear dispersion

AbstractThe runup of long strongly nonlinear waves impinging on a vertical wall can exceed six times the far-field amplitude of the incoming waves. This outcome stems from a precursory evolution process in which the wave height undergoes strong amplification due to the combined action of nonlinear steepening and dispersion, resulting in the formation of nonlinearly dispersive wave trains, i.e. undular bores. This part of the problem is first analysed separately, with emphasis on the wave amplitude growth rate during the development of undular bores within an evolving large-scale background. The growth of the largest wave in the group is seen to reflect the asymptotic time scaling provided by nonlinear modulation theory rather closely, even in the case of fully nonlinear evolution and moderately slow modulations. In order to address the effect of such a dynamics on the subsequent wall runup, numerical simulations of evolving long-wave groups are then carried out in a computational wave tank delimited by vertical walls. Conditions for optimal runup efficiency are sought with respect to the main physical parameters characterizing the incident waves, namely the wavelength, the length of the propagation path and the initial amplitude. Extreme runup is found to be strongly correlated to the ratio between the available propagation time and the shallow-water nonlinear time scale. The problem is studied in the twofold mathematical framework of the fully nonlinear free-surface Euler equations and the strongly nonlinear Serre–Green–Naghdi model. The performance of the reduced model in providing accurate long-time predictions can therefore be assessed.

Bottom pressure distribution under a solitonic wave reflecting on a vertical wall

The bottom pressure distribution under solitonic waves, travelling or fully reflected at a wall is analysed here. Results given by two kind of numerical models are compared. One of the models is based on the Green–Naghdi equations, while the other one is based on the fully nonlinear potential equations. The two models differ through the way in which wave dispersion is taken into account. This approach allows us to emphasize the influence of dispersion, in the case of travelling or fully reflected waves. The Green–Naghdi model is found to predict well the bottom pressure distribution, even when the quantitative representation of the runup height is not satisfactorily described.

Fractional order generalised thermoelasticity to an infinite body with a cylindrical cavity and variable material properties

This paper is concerned with the determination of thermoelastic displacement, stress and temperature produced in an infinite isotropic elastic body having a cylindrical cavity in the context of the new consideration of heat conduction with fractional order generalised thermoelasticity with Lord–Shulman model (LS model) and Green–Naghdi model with energy dissipation (GN-III model). Here, the elastic parameters and the thermal conductivity are temperature dependent. The boundary of the cavity is subjected to time-dependent thermal and mechanical shocks. The fractional order generalised coupled thermoelasticity theories for the problem are combined into a unified formulation introducing the unified parameters. The governing equations of generalised thermoelasticity theory are obtained in the Laplace transform domain and are solved in that domain by finding out the roots by using the Laguerre’s method. The inversion of the transform solution is carried out numerically by applying a method based on the Fourier...

High order well-balanced CDG-FE methods for shallow water waves by a Green-Naghdi model

In this paper, we consider a one-dimensional fully nonlinear weakly dispersive Green-Naghdi model for shallow water waves over variable bottom topographies. Such model describes a large spectrum of...

Shallow water asymptotic models for the propagation of internal waves

We are interested in asymptotic models for the propagation of internal waves at the interface between two shallow layers of immiscible fluid, under the rigid-lid assumption. We review and complete existing works in the literature, in order to offer a unified and comprehensive exposition. Anterior models such as the shallow water and Boussinesq systems, as well as unidirectional models of Camassa-Holm type, are shown to descend from a broad Green-Naghdi model, that we introduce and justify in the sense of consistency. Contrarily to earlier works, our Green-Naghdi model allows a non-flat topography, and horizontal dimension $d=2$. Its derivation follows directly from classical results concerning the one-layer case, and we believe such strategy may be used to construct interesting models in different regimes than the shallow-water/shallow-water studied in the present work.

High order well-balanced CDG–FE methods for shallow water waves by a Green–Naghdi model

In this paper, we consider a one-dimensional fully nonlinear weakly dispersive Green–Naghdi model for shallow water waves over variable bottom topographies. Such model describes a large spectrum of shallow water waves, and it is thus of great importance to design accurate and robust numerical methods for solving it. The governing equations contain mixed spatial and temporal derivatives of the unknowns. They also have still-water stationary solutions which should be preserved in stable numerical simulations. In our numerical approach, we first reformulate the Green–Naghdi equations into balance laws coupled with an elliptic equation. We then propose a family of high order numerical methods which discretize the balance laws with well-balanced central discontinuous Galerkin methods and the elliptic part with continuous finite element methods. Linear dispersion analysis for both the (reformulated) Green–Naghdi system and versions of the proposed numerical scheme is performed when the bottom topography is flat. Numerical tests are presented to illustrate the accuracy and stability of the proposed schemes as well as the capability of the Green–Naghdi model to describe a wide range of shallow water wave phenomena.