Energy equation for certain approximate models of long-wave hydrodynamics

Fedotova, Z. I.; Khakimzyanov, G. S.; Dutykh, Denys

doi:10.1515/rnam-2014-0013

Cited by 16 publications

(20 citation statements)

References 10 publications

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“…Here we continue the recent investigations in [3][4][5][6]. It is necessary to acknowledge [7][8][9][10] and others as the first studies on this theme.…”

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confidence: 59%

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Hierarchy of nonlinear models of the hydrodynamics of long surface waves

2015

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224For numerical solution of problems of hydrody namics of long surface waves propagating in real reser voirs, it is advantageous to use models in which the region of applicability is associated with the character istic scales of the wave process [1,2]. An expansion of the circle of actual problems related to the develop ment of coastal territories stimulates the creation of new models, and this fact results in variety of the sys tems of differential equations corresponding to these models due to a variety of methods and simplifying assumptions. Among these differential equations, there are also some that formally approximate the ini tial problem without providing an adequate descrip tion of the physical process or are inconvenient for numerical implementation.The purpose of this study is formation of a uniform approach for constructing long wave approximations in order to provide a hierarchical chain of shallow water equations of the first and second approximations having a succession of physically substantial proper ties. Here we continue the recent investigations in [3][4][5][6]. It is necessary to acknowledge [7-10] and others as the first studies on this theme.The derivation of shallow water models taking into account the dispersion is based on the Euler equations for an ideal incompressible fluid on a rotating sphere, the mobility of the bottom surface being taken into account, while further passage along the hierarchy from completely nonlinear equations with dispersion towards simplifications proceeds with inheritance of the most important properties, in particular, the laws of conservation. We succeeded in writing the obtained nonlinear dispersive (NLD) equations both on the plane and on the sphere in a universal compact form, which structurally coincides with the set of gas dynamic equations. EULER'S EQUATIONS IN THE THIN LAYER APPROXIMATIONThe spherical system of coordinates Oλθr is used with the origin at the center of a sphere of radius R rotating with a constant velocity Ω. We designated the longitude by λ, and the addition to the latitude ϕ < ϕ < we denoted by θ = -ϕ, while r is the radial coordinate. It is assumed that the water layer is limited from below by an impenetrable mobile bottom r = Rh(λ, θ, t) and from above by a free surface r = R + η(λ, θ, t). As external forces, we considered only the force of Newtonian attraction g directed to the center of a rotating sphere. Considering the thickness H = η + h of the water layer as small in comparison with R, we assume that the value of g = |g| and the water density ρ are constant in the entire layer, ρ ≡ 1. The mathe matical models of the long wave hydrodynamics are derived here from Euler's equations written with sin gling out the radial direction:(1)where the vectors U = (U 1 , U 2 ) = ( , ) and V = (V 1 , V 2 ) are composed correspondingly from the contravariant and covariant components of the "horizontal" com ponent of the velocity vector; in this case, V 1 = (Ω + U 1 )r 2 sin 2 θ, V 2 = r 2 U 2 , the radial velocity component is designated by ...

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“…Here we continue the recent investigations in [3][4][5][6]. It is necessary to acknowledge [7][8][9][10] and others as the first studies on this theme.…”

supporting

confidence: 59%

“…Therefore, a mod ification of the models of these chains is required. Cer tain steps in this direction are made in [14] in which, in particular, it was shown that the modified models of this group admit the writing of the momentum bal ance equation in a laconic quasi conservative form.…”

Section: Hierarchy Of Shallow Water Modelsmentioning

confidence: 99%

Hierarchy of nonlinear models of the hydrodynamics of long surface waves

2015

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“…The procedure of finding the quantities E and q E follows a similar outline as the derivations in [26] for a class of Boussinesq systems and [27] for the KdV equation. Expressions for energy functionals associated to Boussinesq systems have also been developed in [28]. The analytical results are put to use in the study of undular bores.…”

Section: Introductionmentioning

confidence: 99%

Mechanical balance laws for fully nonlinear and weakly dispersive water waves

Kalisch¹,

Khorsand²,

Mitsotakis³

2016

Physica D: Nonlinear Phenomena

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The Serre-Green-Naghdi system is a coupled, fully nonlinear system of dispersive evolution equations which approximates the full water wave problem. The system is known to describe accurately the wave motion at the surface of an incompressible inviscid fluid in the case when the fluid flow is irrotational and two-dimensional. The system is an extension of the well known shallow-water system to the situation where the waves are long, but not so long that dispersive effects can be neglected.In the current work, the focus is on deriving mass, momentum and energy densities and fluxes associated with the Serre-Green-Naghdi system. These quantities arise from imposing balance equations of the same asymptotic order as the evolution equations. In the case of an even bed, the conservation equations are satisfied exactly by the solutions of the Serre-Green-Naghdi system. The case of variable bathymetry is more complicated, with mass and momentum conservation satisfied exactly, and energy conservation satisfied only in a global sense. In all cases, the quantities found here reduce correctly to the corresponding counterparts in both the Boussinesq and the shallow-water scaling.One consequence of the present analysis is that the energy loss appearing in the shallow-water theory of undular bores is fully compensated by the emergence of oscillations behind the bore front. The situation is analyzed numerically by approximating solutions of the Serre-Green-Naghdi equations using a finite-element discretization coupled with an adaptive Runge-Kutta time integration scheme, and it is found that the energy is indeed conserved nearly to machine precision. As a second application, the shoaling of solitary waves on a plane beach is analyzed. It appears that the Serre-Green-Naghdi equations are capable of predicting both the shape of the free surface and the evolution of kinetic and potential energy with good accuracy in the early stages of shoaling.2010 Mathematics Subject Classification. 35L65, 76B15, 76B25.

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“…Since the structure of the equations of the full nonlinear dispersive model holds true for the weakly dispersion model on a sphere, as well as the model of weakly dispersive flows over weakly deformable bottom [4], the algorithm for the numerical solution of the full nonlinear dispersive equations can be transferred unchanged also to this approximate models.…”

Section: Discussionmentioning

confidence: 99%

“…In [3,4] with using the unified approach to a construction of long-wave approximations, the hierarchy of hydrodynamic models, which possess the physically meaningful properties inherited, has been built. The hierarchical chains of shallow water models of first-and secondorder long-wave approximation enclosed in each other are constructed on a rotating sphere and on a plane.…”

Section: Introductionmentioning

confidence: 99%