Numerical modelling of fluid flows in the framework of a two-dimensional shallow-water model with the use of adaptive grids

Барахнин, В. Б.; Khakimzyanov, G. S.

doi:10.1515/rnam.1997.12.2.95

Cited by 4 publications

(5 citation statements)

References 4 publications

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“…This mapping is based on the one-to-one nondegenerate transformation of the coordinates. The equations considered, the initial and boundary conditions are written in the new coordinates [1][2][3]5].…”

Section: Computational Algorithmmentioning

confidence: 99%

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Numerical modelling of fluid oscillations caused by an instantaneous slope of the reservoir base

Барахнин¹,

Khakimzyanov²

1998

Russian Journal of Numerical Analysis and Mathematical Modelling

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We consider the numerical solution of a problem of fluid oscillations caused by an instantaneous slope of the reservoir base. The calculations are carried out in the framework of a potential flow model, nonlinear and nonlinear dispersive shallow-water models. Using the computational experiments, we specify the limits of applicability of the used models and algorithms, give the quantitative calculation results.The dynamics of vessels with fluid that has a free surface have intensively been investigated over nearly fifty years. Of considerable importance in this field are the monographs [4,6-8]. The analytic and numerical methods of solving the problems of free fluid oscillations in vessels of different forms are described in these monographs. The problems are often considered in a linear approximation, i.e. under the assumption that the oscillations are small. The nonlinear statements of problems of this class are investigated rather seldom.The study of the fluid oscillations caused by an instantaneous slope of the reservoir base is of great importance for designing toxic and radioactive substances storage. The study of this phenomenon requires direct numerical modelling because of the substantial nonlinearity of the process. If a vessel has large sizes and the fluid layer is not thin, the effect of viscosity can be disregarded and mathematical modelling can be based on models of an ideal fluid.It is difficult to specify what model is most suitable for the numerical study of the process considered. In this work we are guided by the concept of a computational experiment. According to this concept different models and computational algorithms are used to describe the same phenomenon, which leads to more reliable results.We use the models: two-and three-dimensional models of potential fluid flows (PF) as well as one-and two-dimensional (areal) approximate long-wave models. The potential flow model describes more adequately the phenomenon studied but involves larger computational costs. Therefore in order to reduce computer time it is advisable in some cases to use a nonlinear shallow water model in a first approximation (NL model) or some nonlinear dispersive model (NLD model), for example, the ZheleznyakPelinovskii model [10]. The latter model is interest because the assumption that the wave amplitude is small is not used to obtain it. The one-dimensional analogue of this model is also obtained in [9].The aim of this work is to compare the results of the numerical modelling of the phenomenon studied, which are obtained in the framework of the potential flow model, the Zheleznyak-Pelinovskii nonlinear dispersive model, and the shallow water model in

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Section: Computational Algorithmmentioning

confidence: 99%

“…The shallow-water equations are solved by an explicit finite difference predictor-corrector scheme with self-adjusting approximate viscosity [1,2].…”

Section: Computational Algorithmmentioning

confidence: 99%

Numerical modelling of fluid oscillations caused by an instantaneous slope of the reservoir base

Барахнин¹,

Khakimzyanov²

1998

Russian Journal of Numerical Analysis and Mathematical Modelling

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“…The predictor-corrector scheme is described in detail in our work [3]. Therefore we describe here only the scheme that is used to numerically solve the elliptic equation …”

Section: Finite Difference Algorithmmentioning

confidence: 99%

“…=^1A 2 /4 for the nodes of types 5-8 (comer ones). Then we can write the finite difference equations for the node q. as(3…”

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confidence: 99%

On the algorithm for one nonlinear dispersive shallow-water model

Барахнин¹,

Khakimzyanov²

1997

Russian Journal of Numerical Analysis and Mathematical Modelling

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We consider the finite difference algorithm for modelling surface waves in the framework of one nonlinear dispersive model. The algorithm proposed allows us to use nonstationary adaptive grids adjusting to the complex geometry of the domain and to the peculiarities of the solution. A distinctive feature of the algorithm is to separate the elliptic and hyperbolic parts in the original equations. In order to solve an elliptic equation obtained we construct a finite difference approximation of the scheme of the type Oblique cross' with a selfadjoint and positive definite operator. We estimate the boundaries of the spectrum of this operator. We give an example of the numerical modelling of the oblique run-up of a solitary wave on a vertical wall.

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“…The finite element methods of numerical realization of these models have been extensively developed in the past few years [1]. At the same time the use of conventional finite difference approximations (schemes) especially on curvilinear grids allows one to obtain adequate results at much smaller computational costs [2,6,9]. There are a lot of mathematical models of the above type and a great number of variants of constructing the corresponding computational algorithms.…”

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confidence: 99%

Investigation of computational models of long surface waves in the problem of interaction of a solitary wave with a conic island

Чубаров¹,

Fedotova²,

Shkuropatskii³

1998

Russian Journal of Numerical Analysis and Mathematical Modelling

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Due to the great diversity of mathematical models used for studying the phenomena of about the same character in problems of wave hydrodynamics it is necessary to specify their fields of application in terms of the main characteristic parameters of physical processes simulated. Laboratory experiments with detailed description of the experimental unit itself, the parameters of waves generated, and the observation results are most appropriate for the study of this kind. In this paper we discuss the results of modelling the transformation of long gravitational surface waves in the framework of mathematical models of shallow water theory, mainly nonlinear dispersive (NLD) models of the Boussinesq type. On the whole the work is further development of the approaches to the construction and study of the numerical models, which were earlier discussed by the authors and their colleagues [3,5,7,9].Nonlinear dispersive models of the Boussinesq type are often used to describe the propagation and transformation of surface waves of finite length and of moderate amplitude for the cases of one and two horizontal space variables. The finite element methods of numerical realization of these models have been extensively developed in the past few years [1]. At the same time the use of conventional finite difference approximations (schemes) especially on curvilinear grids allows one to obtain adequate results at much smaller computational costs [2,6,9]. There are a lot of mathematical models of the above type and a great number of variants of constructing the corresponding computational algorithms. Therefore the problem of comparative analysis of the developed computational models of wave hydrodynamics and the determination of the fields of application of the models and the algorithms in terms of the characteristic parameters of a physical phenomenon studied have become very urgent. Of special importance are the results of this study when sufficiently representative analytical and laboratory test materials are available.The finite difference algorithms developed and studied by the authors are based on the familiar sufficiently simple NLD model, which was first proposed by Peregrine [11]. One of the algorithms for approximating this model was described in Rygg's work [12]. It was successfully used for solving the problem of transformation of surface waves, which propagate in a canal with its model 'quasi-one-dimensional' bottom structure.The investigations made in this work partially use the ideas of [12]. This refers mainly to the regularization of the original model that facilitates the construction of the efficient difference scheme.The modification of the Peregrine model is described in Section 1. Section 2 mainly deals with the construction and study of computational algorithm. In Section 3 we

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Numerical modelling of fluid flows in the framework of a two-dimensional shallow-water model with the use of adaptive grids

Cited by 4 publications

References 4 publications

Numerical modelling of fluid oscillations caused by an instantaneous slope of the reservoir base

Numerical modelling of fluid oscillations caused by an instantaneous slope of the reservoir base

On the algorithm for one nonlinear dispersive shallow-water model

Investigation of computational models of long surface waves in the problem of interaction of a solitary wave with a conic island

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