Green Naghdi Model Research Articles

English

In this paper we study the asymptotic behavior of a system composed of an integro-partial differential equation that models the longitudinal oscillation of a beam with a memory effect to which a thermal effect has been given by the Green-Naghdi model type III, being physically more accurate than the Fourier and Cattaneo models. To achieve this goal, we will use arguments from spectral theory, considering a suitable hypothesis of smoothness on the integro-partial differential equation.

Modified three-phase-lag Green–Naghdi models for thermomechanical waves in an axisymmetric annular disk

A novel modification for Green and Naghdi thermoelasticity model of type III is proposed in this article. This model is joined to other ones to treat a thermoelastic wave propagation problem in an annular disk. The closed-form solution of thermoelastic analysis of the annular disks has been presented to deduce temperature, radial displacement, dilatation and stresses. Results are illustrated and reported to compare the simple Green–Naghdi II and III and their modified single-, dual- and three-phase-lag models. Concluding remarks are added.

Surface waves in porous nonlocal thermoelastic orthotropic medium

The present article deals with the propagation of Rayleigh surface waves in a homogeneous orthotropic medium based on Eringen’s nonlocal thermoelasticity. This thermoelastic problem is studied under the purview of Green–Naghdi model type III of hyperbolic thermoelasticity in the presence of voids. The normal mode analysis is employed to obtain a vector matrix differential equation which is then solved by an eigenvalue approach. The frequency equations for different cases are derived. The path of surface particles during Rayleigh wave propagation is found to be elliptical. In order to illustrate the analytical developments, the numerical solution is carried out and the computer-simulated results with respect to phase velocity, attenuation coefficient and specific loss are presented graphically.

Generalized Thermoelastic Interactions in a Poroelastic Material Without Energy Dissipations

The purpose of this investigation is providing a method to study the effect of porosity in a porothermoelastic medium by the finite element technique. The formulations are applied under Green-Naghdi model without energy dissipations. One-dimensional application for a poroelastic half-space is considered. Due to the complex basic equations, the finite element method (FEM) has been adopted to solve this problem. The numerical outcomes for displacements, temperatures and the stress components are represented graphically for the solid and the fluid.

Hyperbolicity of the Modulation Equations for the Serre–Green–Naghdi Model

The Serre–Green–Naghdi (SGN) system is one of the most useful dispersive models for the description of long water waves having good mathematical and physical properties. First, the model is a mathematically justified approximation of the exact water-wave problem. Second, the SGN equations are the Euler–Lagrange equations derived from Hamilton’s principle of surface water waves with an approximate Lagrangian functional which appears naturally from the full Lagrangian. Finally, the equations are Galilean invariant which is necessary for physically relevant mathematical models. In this work, we derive the Whitham modulation equations for the SGN model and prove that these equations are strictly hyperbolic for any wave amplitude. This property can be interpreted by concluding that the periodic wave solutions of the SGN equations are modulationally stable. Numerical simulations of the full SGN equations are also shown, and these results confirm the theoretical stability results found using the Whitham modulation theory.

Fractional Order Two Temperature Generalized Thermoelasticity in a Symmetric Spherical Shell

This paper deals with the determination of thermoelastic stresses, strain and conductive temperature in a spherically symmetric spherical shell in the context of the fractional order two temperature generalized thermoelasticity theory (2TT). The two temperature three-phase-lag thermoelastic model (2T3P) and two temperature Green Naghdi model III (2TGN-III) 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. The physical quantities have been computed numerically and presented graphically. The effect of the fractional order parameter on the solutions has been studied and the comparisons among different thermoelastic models are made.

Thermoelastic diffusion problem for a half-space due to a refined dual-phase-lag Green-Naghdi model

The thermoelastic diffusion problem of an isotropic half-space is presented. The Green–Naghdi model with and without energy dissipation is proposed. Novel multi single/dual-phase-lag models are presented to investigate the thermoelastic diffusion behavior of the medium. The simple Green–Naghdi type II and III and their modified models are all examined here. The exact solution of thermodiffusion governing equations has been obtained considering the initial and boundary conditions. The validity of results is acceptable by comparing all variables. Benchmark results are reported to help other investigators in their future comparisons.

Depth-integrated wave–current models. Part 1. Two-dimensional formulation and applications

Depth-integrated mathematical models for simulating waves and currents from deep to shallow water are presented. These models are derived from Euler’s equations in the -coordinate system, mapping the total water depth in Cartesian coordinates onto a specified range of -coordinates. The horizontal velocity is approximated as a truncated infinite series of products of prescribed shape functions of and unknown functions of horizontal coordinates and time. Adopting the method of weighted residuals, the new models are obtained by minimizing the residuals of the horizontal momentum equations with either the Galerkin method or the subdomain method. These models’ linear and nonlinear water wave properties are investigated. The new models are implemented numerically. A hierarchy of numerical models with different degree of polynomial approximation is developed and checked against several benchmarked experiments and a new set of experiments of self-focusing wave groups. For both the Galerkin and subdomain models, excellent agreements are observed for both the free surface elevations and the velocity profiles. The new models are superior to the existing Boussinesq-type models for their applicability to a wide range of physical scenarios, including the interactions between a wave package of multiple frequency components and a linearly sheared current. The new Galerkin models have similar characteristics and accuracy as the Green–Naghdi models, but the new models are more efficient computationally. Finally, for the same degree of polynomial approximation the subdomain models perform better than the Galerkin models and require less computational time.

Magneto-thermoelastic interaction in a functionally graded medium under gravitational field

The main concern of the present contribution is the thermoelastic interactions of displacements, stress and temperature in a functionally graded unbounded medium due to the influence of gravitational field and an induced magnetic field of constant intensity while the medium is rotating with an uniform angular velocity. The governing equations have been formulated in the context of generalized thermoelasticity with energy dissipation (Green Naghdi model III (GN III)) under the influence of periodically varying heat source. Employing Laplace and Fourier transforms as tools, the problem has been solved analytically in the transformed domain. The inversion of the Fourier transform is performed analytically using Mathematica package and the numerical inversion of the Laplace transform has been executed using Zakian method. According to the graphical representations corresponding to the numerical results, conclusions about the new theory is constructed. Excellent predictive capability is demonstrated due to the effect of nonhomogeneity, effect of rotation, influence of magnetic field and gravitational field also.

A CDG-FE method for the two-dimensional Green-Naghdi model with the enhanced dispersive property

In this work, we investigate numerical solutions of the two-dimensional shallow water wave using a fully nonlinear Green-Naghdi model with an improved dispersive effect. For numerics, the Green-Naghdi model is rewritten into a formulation coupling a pseudo-conservative system and a set of pseudo-elliptic equations. Since the pseudo-conservative system is no longer hyperbolic and its Riemann problem can only be approximately solved, we consider the utilization of the central discontinuous Galerkin method, which possesses an important feature of the needlessness of Riemann solvers. Meanwhile, the stationary elliptic part will be solved using the finite element method. Both the well-balanced and the positivity-preserving features, which are highly desirable in the simulation of the shallow water wave, will be embedded into the proposed numerical scheme. The accuracy and efficiency of the numerical model and method will be illustrated through numerical tests.

Rigorous justification of the Favrie–Gavrilyuk approximation to the Serre–Green–Naghdi model

The (Serre–)Green–Naghdi system is a non-hydrostatic model for the propagation of surface gravity waves in the shallow-water regime. Recently, Favrie and Gavrilyuk proposed in Favrie and Gavrilyuk (2017 Nonlinearity 30 2718–36) an efficient way of numerically computing approximate solutions to the Green–Naghdi system. The approximate solutions are obtained through solutions of an augmented quasilinear system of balance laws, depending on a parameter. In this work, we provide quantitative estimates showing that any sufficiently regular solution to the Green–Naghdi system is the limit of solutions to the Favrie–Gavrilyuk system as the parameter goes to infinity, provided the initial data of the additional unknowns is well-chosen. The problem is therefore a singular limit related to low Mach number limits with additional difficulties stemming from the fact that both order-zero and order-one singular components are involved.

On modeling of undular bores based on the second approximation of the shallow water theory

It is studied the possibility of modeling of undular bores on the basis of the second approximation of the shallow water theory. The classical differential Green-Naghdi model cannot be used for correct numerical simulation of wave flows with undular bores. The reason for this is that this model is derived within the framework of the long-wave approximation, by virtue of which the characteristic depth of the stream is much less than the characteristic length of the surface waves, which is not performed in the undular bore front. An integro-differential Green-Naghdi model is proposed for numerical simulation of undular bores. In this model we used the divergent differential form of the continuity equation and the integral conservation law of horizontal momentum. This model is derived from two-dimensional integral conservation laws of mass and momentum, describing a plane-parallel flow of an ideal incompressible fluid over a horizontal bottom. The basis of this conclusion is the concept of a local hydrostatic approximation, which generalizes the concept of the long-wave approximation and is used to justify the applicability of shallow water models to describe wave flows with the hydraulic bores.

Influence of Rotation and Magnetic Fields on a Functionally Graded Thermoelastic Solid Subjected to a Mechanical Load

The current manuscript is presented to study two-dimensional deformations in a nonhomogeneous, isotropic, rotating, magneto-thermoelastic medium in the context of Green-Naghdi model III. It is assumed that the functionally graded material has nonhomogeneous mechanical and thermal properties in the x-direction. The exact expressions for the displacement components, temperature field, and stresses are obtained in the physical domain by using normal mode technique. These are also computed numerically for a copper-like material and presented graphically to observe the variations of the considered physical variables. Comparisons of the physical quantities are shown in figures to depict the effects of angular velocity, nonhomogeneity parameter, and magnetic field.

Dispersive and dispersive-like bores in channels with sloping banks

In this paper a detailed analysis of undular bore dynamics in channels of variable cross-section is presented. Two undular bore regimes, low Froude number (LFN) and high Froude number (HFN), are simulated with a Serre–Green–Naghdi model, and the results are compared with the experiments by Treske (1994). We show that contrary to Favre waves and HFN bores, which are controlled by dispersive non-hydrostatic mechanisms, LFN bores correspond to a hydrostatic phenomenon. The dispersive-like properties of the LFN bores is related to wave refraction on the banks in a way similar to that of edge waves in the near shore. A fully hydrostatic asymptotic model for these dispersive-like bores is derived and compared to the observations, confirming our claim.

Analytical aspects in strain gradient theory for chiral Cosserat thermoelastic materials within three Green-Naghdi models

This work is motivated by the recent interest in using strain gradient theory to model the chiral behavior of elastic materials. In this paper, we derive a linear strain gradient theory for Cosserat thermoelastic materials according to the three models (types I, II and III) of Green-Naghdi theory. Models II and III permit propagation of thermal waves at finite speeds, while model I coincides with the classical Fourier’s law. The thermal field is influenced by the displacement and the microrotation fields and by some additional parameters that describe the chiral behavior. We prove the well-posedness for the three models and the asymptotic behavior for models I and III by the semigroup theory of linear operators.

One-dimensional nonlinear vibration analysis and coupled thermoelasticity based on Green-Naghdi model

The present article reports on a study on nonlinear coupled thermoelasticity based on Green-Naghdi type III model in one-dimensional form. Unlike other research in which temperature change is low and the strain relations are linear, in the present study the nonlinear finite strain assumption is governed. Two coupled nonlinear partial differential equations, namely; energy and linear momentum are established. The basic equations are formulated in material coordinates and then transformed into dimensionless form. The homotopy analysis method is used to analyze the one-dimensional structure. Two different cases are studied. The thermo-mechanical wave propagation with reflection from the boundary, and the influence of the damping parameter–presented in Green-Naghdi type III model–on temperature. The linear and nonlinear equations are compared. It should be noted that the nonlinear finite strain theory governs better when the loading temperature is high.

Application of Green−Naghdi Equations for Modeling Wave Flows with Undular Bores

The basic conservation laws in the Green–Naghdi model of shallow-water theory are derived from the two-dimensional integral conservation laws of mass and the total momentum describing the plane-parallel flow in an ideal incompressible fluid above a horizontal bottom. This conclusion is based on the concept of a local hydrostatic approximation, which generalizes the concept of the long-wavelength approximation and is used for analyzing the applicability of the Green–Naghdi equations in modeling the wave flows of a fluid with undular bores.

Effects of fractional and two-temperature parameters on stress distributions for an unbounded generalized thermoelastic medium with spherical cavity

Effects of fractional and two-temperature parameters on the distribution of stresses of an unbounded isotropic thermoelastic medium with spherical cavity are studied in the context of the theory of two-temperature generalized thermoelasticity based on the Green-Naghdi model III using fractional order heat conduction equation. The surface of the cavity is considered to be free from traction and is subjected to a smooth and time-dependent-heating effect. A spherical polar coordinate system has been used to describe the problem and the resulting governing equations are solved in Laplace transform domain. Numerical Laplace transform inversion method has been then applied to get the stresses in time domain. The numerical estimates of the distributions of stresses are obtained and are presented graphically to study the effects of fractional and two-temperature parameters.

The development of a high-accuracy, broadband, Green–Naghdi model for steep, deep-water ocean waves

The focus of the research presented here is the development of an efficient analytical model for the time-domain simulation of the evolution of a train of a three-dimensional steep random waves and its associated flow. Of particular interest here is the development of a tool that not only accurately predicts the surface elevation history of this wave train, but also the kinematics within the waves, particularly near the free surface. Fenton (Advances in coastal and ocean engineering, vol 5, World Scientific, Singapore, pp 241–324, 1999) reviews the rich literature of various computational models for water waves and is divided between analytical approaches that develop systems of equations to describe the evolution of waves in space and time, and computational approaches such as CFD (which will not be the focus here). What is required to analyze properly the practical ocean engineering problem described in the motivation below is a theory that predicts the three-dimensional evolution of waves that simultaneously has four essential characteristics: deep-water, random broadband seaways, steep waves (close to breaking), and demonstrated accuracy in both wave shape and near-surface wave kinematics. The number and variety of theories that satisfy some but not all of these characteristics is too voluminous to reference here; dynamic theories that exhibit the confluence of all four characteristics are apparently nonexistent. The intent of this paper is to make a step in the direction of filling this void by extending the Green–Naghdi theory of deep-water waves. It is shown that higher level GN models using distributed directors do have a bandwidth that is significantly larger than former GN models and have the same computational effort as using the traditional directors. The bandwidths achieved with the new approach are large enough to be useful in the context of many ocean engineering problems. Applications of these models to random wave situations will be reported in a subsequent article.

Model Skill and Sensitivity for Simulating Wave Processes on Coral Reefs Using a Shock-Capturing Green-Naghdi Solver

ABSTRACT Beetham, E.; Kench, P.S., and Popinet, S., 2018. Model skill and sensitivity for simulating wave processes on coral reefs using a shock-capturing Green-Naghdi solver. Wave-flume data from published benchmark experiments were used to extensively evaluate numerical model skill and sensitivity for applying a shock-capturing Green-Naghdi (GN) model to simulate nonlinear wave-transformation processes on complex coral reefs. Boussinesq-type models that utilise nonlinear shallow-water equations (NSWEs) to represent wave breaking and dissipation hold significant potential for understanding coastal hazards associated with global environmental change and sea-level rise. These fully nonlinear phase-resolving models typically require a threshold condition to switch from dispersive equations to shock-capturing NSWEs in areas of active wave breaking. However, limited information exists regarding how this splitting approach influences the behaviour of different surf-zone processes that contribute to wave runup ...