Dispersive Shallow Water Wave Modelling. Part I: Model Derivation on a Globally Flat Space

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

doi:10.4208/cicp.oa-2016-0179a

Cited by 23 publications

(44 citation statements)

References 71 publications

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“…For the derivation of Eqs. (2.1), (2.2) we refer to the first part of the present series of papers [92]. The depthintegrated pressure P(u, H) is defined as…”

Section: Mathematical Modelmentioning

confidence: 99%

“…Notice that it is slightly different from the (fully-)conservative form given in Part I [92]. Conservative equations § (2.13), (2.14) can be supplemented by the energy conservation equation which can be used to check the accuracy of simulation (in conservative case, i.e.…”

Section: Conservative Form Of Equationsmentioning

confidence: 99%

“…The present manuscript is the continuation of our series of papers devoted to the long wave modelling. In the first part of this series we derived the so-called base model [92], which encompasses a number of previously known models (but, of course, not all of nonlinear dispersive systems). The governing equations of the base model are :=ū + U is the modified horizontal velocity and U = U (H,ū) is the closure relation to be specified later.…”

Section: Introductionmentioning

confidence: 99%

“…The governing equations of the base model are :=ū + U is the modified horizontal velocity and U = U (H,ū) is the closure relation to be specified later. Depending on the choice of this variable various models can be obtained (see [92,Section §2.4]). Variables P andp are related to the fluid pressure.…”

Section: Introductionmentioning

confidence: 99%

“…The base model (1.1), (1.2) admits an elegant conservative form [92]: (1.4) where I ∈ Mat 2×2 (R) is the identity matrix and the operator ⊗ denotes the tensorial product. We note that the pressure function P(H,ū) incorporates the familiar hydrostatic pressure part gH 2 2 well-known from the Nonlinear Shallow Water Equations (NSWE) [11,43].…”

Section: Introductionmentioning

confidence: 99%

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Dispersive Shallow Water Wave Modelling. Part II: Numerical Simulation on a Globally Flat Space

Khakimzyanov¹,

Dutykh²,

Gusev³

et al. 2018

CiCP

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In this paper we describe a numerical method to solve numerically the weakly dispersive fully nonlinear SERRE-GREEN-NAGHDI (SGN) celebrated model. Namely, our scheme is based on reliable finite volume methods, proven to be very efficient for the hyperbolic part of equations. The particularity of our study is that we develop an adaptive numerical model using moving grids. Moreover, we use a special form of the SGN equations where non-hydrostatic part of pressure is found by solving a linear elliptic equation. Moreover, this form of governing equations allows to determine the natural form of boundary conditions to obtain a well-posed (numerical) problem.

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“…For the derivation of Eqs. (2.1), (2.2) we refer to the first part of the present series of papers [92]. The depthintegrated pressure P(u, H) is defined as…”

Section: Mathematical Modelmentioning

confidence: 99%

Section: Conservative Form Of Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Dispersive Shallow Water Wave Modelling. Part II: Numerical Simulation on a Globally Flat Space

Khakimzyanov¹,

Dutykh²,

Gusev³

et al. 2018

CiCP

Self Cite

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The one-dimensional Green–Naghdi equations with a time dependent bottom topography and their conservation laws

2020

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This paper deals with the one-dimensional Green–Naghdi equations describing the behavior of fluid flow over an uneven bottom topography depending on time. Using Matsuno’s approach, the corresponding equations are derived in Eulerian coordinates. Further study is performed in Lagrangian coordinates. This study allowed us to find the general form of the Lagrangian corresponding to the analyzed equations. Then, Noether’s theorem is used to derive conservation laws. As some of the tools in the application of Noether’s theorem are admitted generators, a complete group classification of the Green–Naghdi equations with respect to the bottom depending on time is performed. Using Noether’s theorem, the found Lagrangians, and the group classification, conservation laws of the one-dimensional Green–Naghdi equations with uneven bottom topography depending on time are obtained.

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Numerical modeling of the long surface wave impact on a partially immersed structure in a coastal zone: Solitary waves over a flat slope

Gusev,

Khakimzyanov,

Skiba

et al. 2023

Physics of Fluids

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This paper describes the numerical simulation of the solitary wave impact on a partially immersed and fixed structure located over a flat coastal slope. This topic is related to the need for assessment of the possible impact of long waves, such as tsunamis, on partially immersed structures in coastal waters. Numerical algorithms on a movable grid adapting to the motion of the shore point are developed for a fully nonlinear dispersive model and a dispersionless shallow water model. Their validation is carried out by comparing the obtained solutions with the data from laboratory experiments and with the results obtained using a fully nonlinear potential flow model. The study shows that the difference between the maximum wave impact on the body at the foot of the slope and near the shore can be up to 6 times. In many cases, the maximum horizontal component of the wave force occurs under the influence of the wave reflected from the shore, indicating the need to consider the influence of the shore-reflected wave when assessing the impact of long waves on structures located in coastal waters. Furthermore, the need to use runup algorithms instead of reflective boundary conditions (vertical wall) has been identified for gentler slopes, where the differences in the wave impact for these two configurations can be 2–3 times.

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Dispersive Shallow Water Wave Modelling. Part I: Model Derivation on a Globally Flat Space

Cited by 23 publications

References 71 publications

Dispersive Shallow Water Wave Modelling. Part II: Numerical Simulation on a Globally Flat Space

Dispersive Shallow Water Wave Modelling. Part II: Numerical Simulation on a Globally Flat Space

The one-dimensional Green–Naghdi equations with a time dependent bottom topography and their conservation laws

Numerical modeling of the long surface wave impact on a partially immersed structure in a coastal zone: Solitary waves over a flat slope

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