Dispersion and blockage effects in the flow over a sill

Liapidevskii, V. Yu.; Gavrilova, K.

doi:10.1007/s10808-008-0005-7

Cited by 14 publications

(3 citation statements)

References 7 publications

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“…This model is hyperbolic if η < 3 2 g λ + 1 τ . This system is similar to the one proposed by Liapidevskii and Gavrilova (2008) [23] where a different approach based on the averaging of instantaneous variables was used. Due to the Noether theorem, system (14) admits the energy conservation law:…”

Section: First Lagrangianmentioning

confidence: 99%

A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves

2017

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A new numerical method for solving the Serre-Green-Naghdi (SGN) equations describing dispersive waves on shallow water is proposed. From the mathematical point of view, the SGN equations are the Euler-Lagrange equations for a 'master' lagrangian submitted to a differential constraint which is the mass conservation law. One major numerical challenge in solving the SGN equations is the resolution of an elliptic problem at each time instant. This is the most time-consuming part of the numerical method. The idea is to replace the 'master' lagrangian by a one-parameter family of 'augmented' lagrangians, depending on a greater number of variables, for which the corresponding Euler-Lagrange equations are hyperbolic. In such an approach, the 'master' lagrangian is recovered by the augmented lagrangian in some limit (for example, when the corresponding parameter is large). The choice of such a family of augmented lagrangians is proposed and discussed. The corresponding hyperbolic system is numerically solved by a Godunov type method. Numerical solutions are compared with exact solutions to the SGN equations. It appears that the computational time in solving the hyperbolic system is much lower than in the case where the elliptic operator is inverted. The new method is applied, in particular, to the study of 'Favre waves' representing non-stationary undular bores produced after reflection of the fluid flow with a free surface at an immobile wall.

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Section: First Lagrangianmentioning

confidence: 99%

A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves

2017

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“…The idea goes back to the seminal work of Cattaneo [10], who proposed a hyperbolic approximation of the heat equation. More recent work on the topic also regards the hyperbolic approximation of the dissipative continuum mechanics [16,17] as well as of nonlinear dispersive systems [1,4,11,41,44].…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic approximation of the BBM equation

Gavrilyuk

Shyue

2022

Nonlinearity

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It is well known that the Benjamin–Bona–Mahony (BBM) equation can be seen as the Euler–Lagrange equation for a Lagrangian expressed in terms of the solution potential. We approximate the Lagrangian by a two-parameter family of Lagrangians depending on three potentials. The corresponding Euler–Lagrange equations can be then written as a hyperbolic system of conservations laws. The hyperbolic BBM system has two genuinely nonlinear eigenfields and one linear degenerate eigenfield. Moreover, it can be written in terms of Riemann invariants. Such an approach conserves the variational structure of the BBM equation and does not introduce the dissipation into the governing equations as it usually happens for the classical relaxation methods. The state-of-the-art numerical methods for hyperbolic conservation laws such as the Godunov-type methods are used for solving the ‘hyperbolized’ dispersive equations. We find good agreement between the corresponding solutions for the BBM equation and for its hyperbolic counterpart.

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“…A numerical scheme developed in [8] is based on a different approach: the idea is to replace the dispersive Green-Naghdi equations by approximate hyperbolic equations. This approach was also applied in [9,10].…”

Section: Introductionmentioning

confidence: 99%

Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations

Дородницын

Kaptsov²,

Meleshko

2020

Preprint

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The paper is devoted to the Lie group properties of the one-dimensional Green-Naghdi equations describing the behavior of fluid flow over uneven bottom topography. The bottom topography is incorporated into the Green-Naghdi equations in two ways: in the classical Green-Naghdi form and in the approximated form of the same order. The study is performed in Lagrangian coordinates which allows one to find Lagrangians for the analyzed equations. Complete group classification of both cases of the Green-Naghdi equations with respect to the bottom topography is presented. Applying Noether's theorem, the obtained Lagrangians and the group classification, conservation laws of the one-dimensional Green-Naghdi equations with uneven bottom topography are obtained. Difference schemes which preserve the symmetries of the original equations and the conservation laws are constructed. Analysis of the developed schemes is given. The schemes are tested numerically on the example of an exact traveling-wave solution.

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Dispersion and blockage effects in the flow over a sill

Cited by 14 publications

References 7 publications

A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves

A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves

Hyperbolic approximation of the BBM equation

Symmetries, conservation laws, invariant solutions and difference schemes of the one-dimensional Green-Naghdi equations

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