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Article

Zhanhong Wan, Saihua Huang, Zhilin Sun and Zhenjiang You

The present work is devoted to the numerical study of the stability of shallow jet. The effects of important parameters on the stability behavior for large scale shallow

Abstract

Purpose

The present work is devoted to the numerical study of the stability of shallow jet. The effects of important parameters on the stability behavior for large scale shallow jets are considered and investigated. Connections between the stability theory and observed features reported in the literature are emphasized. The paper aims to discuss these issues.

Design/methodology/approach

A linear stability analysis of shallow jet incorporating the effects of bottom topography, bed friction and viscosity has been carried out by using the shallow water stability equation derived from the depth averaged shallow water equations in conjunction with both Chézy and Manning resistance formulae. Effects of the following main factors on the stability of shallow water jets are examined: Rossby number, bottom friction number, Reynolds number, topographic parameters, base velocity profile and resistance model. Special attention has been paid to the Coriolis effects on the jet stability by limiting the rotation number in the range of Ro∈[0, 1.0].

Findings

It is found that the Rossby number may either amplify or attenuate the growth of the flow instability depending on the values of the topographic parameters. There is a regime where the near cancellation of Coriolis effects due to other relevant parameters influences is responsible for enhancement of stability. The instability can be suppressed by the bottom friction when the bottom friction number is large enough. The amplification rate may become sensitive to the relatively small Reynolds number. The stability region using the Manning formula is larger than that using the Chézy formula. The combination of these effects may stabilize or destabilize the shallow jet flow. These results of the stability analysis are compared with those from the literature.

Originality/value

Results of linear stability analysis on shallow jets along roughness bottom bed are presented. Different from the previous studies, this paper includes the effects of bottom topography, Rossby number, Reynolds number, resistance formula and bed friction. It is found that the influence of Reynolds number on the stability of the jet is notable for relative small value. Therefore, it is important to experimental investigators that the viscosity should be considered with comparison to the results from inviscid assumption. In contrast with the classical analysis, the use of multi-parameters of the base velocity and topographic profile gives an extension to the jet stability analysis. To characterize the large scale motion, besides the bottom friction as proposed in the related literature, the Reynolds number Re, Rossby number Ro, the topographic parameters and parameters controlling base velocity profile may also be important to the stability analysis of shallow jet flows.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 2
Type: Research Article
ISSN: 0961-5539

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Article

Alia Al-Ghosoun, Ashraf S. Osman and Mohammed Seaid

The purpose of this study is twofold: first, to derive a consistent model free-surface runup flow problems over deformable beds. The authors couple the nonlinear…

Abstract

Purpose

The purpose of this study is twofold: first, to derive a consistent model free-surface runup flow problems over deformable beds. The authors couple the nonlinear one-dimensional shallow water equations, including friction terms for the water free-surface and the two-dimensional second-order solid elastostatic equations for the bed deformation. Second, to develop a robust hybrid finite element/finite volume method for solving free-surface runup flow problems over deformable beds. The authors combine the finite volume for free-surface flows and the finite element method for bed elasticity.

Design/methodology/approach

The authors propose a new model for wave runup by static deformation on seabeds. The model consists of the depth-averaged shallow water system for the water free-surface coupled to the second-order elastostatic formulation for the bed deformation. At the interface between the water flow and the seabed, transfer conditions are implemented. Here, hydrostatic pressure and friction forces are considered for the elastostatic equations, whereas bathymetric forces are accounted for in the shallow water equations. As numerical solvers, the authors propose a well-balanced finite volume method for the flow system and a stabilized finite element method for elastostatics.

Findings

The developed coupled depth-averaged shallow water system and second-order solid elastostatic system is well suited for modeling wave runup by deformation on seabeds. The derived coupling conditions at the interface between the water flow and the bed topography resolve well the condition transfer between the two systems. The proposed hybrid finite volume element method is accurate and efficient for this class of models. The novel technique used for wet/dry treatment accurately captures the moving fronts in the computational domain without generating nonphysical oscillations. The presented numerical results demonstrate the high performance of the proposed methods.

Originality/value

Enhancing modeling and computations for wave runup problems is at an early stage in the literature, and it is a new and exciting area of research. To the best of our knowledge, solving wave runup problems by static deformation on seabeds using a hybrid finite volume element method is presented for the first time. The results of this research study, and the research methodologies, will have an important influence on a range of other scientists carrying out research in related fields.

Details

Engineering Computations, vol. ahead-of-print no. ahead-of-print
Type: Research Article
ISSN: 0264-4401

Keywords

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Article

Thomas Rowan and Mohammed Seaid

The purpose of this paper is to present a new numerical model for shallow water flows over heterogeneous sedimentary layers. It is already several years since the…

Abstract

Purpose

The purpose of this paper is to present a new numerical model for shallow water flows over heterogeneous sedimentary layers. It is already several years since the single-layered models have been used to model shallow water flows over erodible beds. Although such models present a real opportunity for shallow water flows over movable beds, this paper is the first to propose a multilayered solver for this class of flow problems.

Design/methodology/approach

Multilayered beds formed with different erodible soils are considered in this study. The governing equations consist of the well-established shallow water equations for the flow, a transport equation for the suspended sediments, an Exner-type equation for the bed load and a set of empirical equations for erosion and deposition terms. For the numerical solution of the coupled system, the authors consider a non-homogeneous Riemann solver equipped with interface-tracking tools to resolve discontinuous soil properties in the multilayered bed. The solver consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms.

Findings

This paper reveals that modeling shallow water flows over multilayered sedimentary topography can be achieved by using a coupled system of partial differential equations governing sediment transport. The obtained results demonstrate that the proposed numerical model preserves the conservation property, and it provides accurate results, avoiding numerical oscillations and numerical dissipation in the approximated solutions.

Originality/value

A novel implementation of sediment handling is presented where both averaged and separate values for sediment species are used to ensure speed and precision in the simulations.

Details

Engineering Computations, vol. 37 no. 2
Type: Research Article
ISSN: 0264-4401

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Article

Chunchen Xia, Zhixian Cao, Gareth Pender and Alistair Borthwick

The purpose of this paper is to present a fully conservative numerical algorithm for solving the coupled shallow water hydro-sediment-morphodynamic equations governing…

Abstract

Purpose

The purpose of this paper is to present a fully conservative numerical algorithm for solving the coupled shallow water hydro-sediment-morphodynamic equations governing fluvial processes, and also to clarify the performance of a conventional algorithm, which redistributes the variable water-sediment mixture density to the source terms of the governing equations and accordingly the hyperbolic operator is rendered similar to that of the conventional shallow water equations for clear water flows.

Design/methodology/approach

The coupled shallow water hydro-sediment-morphodynamic equations governing fluvial processes are arranged in full conservation form, and solved by a well-balanced weighted surface depth-gradient method along with a slope-limited centred scheme. The present algorithm is verified for a spectrum of test cases, which involve complex flows with shock waves and sediment transport processes with contact discontinuities over irregular topographies. The computational results of the conventional algorithm are compared with those of the present algorithm and evaluated by available referenced data.

Findings

The fully conservative numerical algorithm performs satisfactorily over the spectrum of test cases, and the conventional algorithm is confirmed to work similarly well.

Originality/value

A fully conservative numerical algorithm, without redistributing the water-sediment mixture density, is proposed for solving the coupled shallow water hydro-sediment-morphodynamic equations. It is clarified that the conventional algorithm, involving redistribution of the water-sediment mixture density, performs similarly well. Both algorithms are equally applicable to problems encountered in computational river modelling.

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Article

Kazuo Kashiyama and Mutsuto Kawahara

An interpolation is presented for preparation of input data of water depth in finite element analysis for shallow water problems. The algorithm, computer program for…

Abstract

An interpolation is presented for preparation of input data of water depth in finite element analysis for shallow water problems. The algorithm, computer program for interpolation of water depth and example are shown.

Details

Engineering Computations, vol. 2 no. 4
Type: Research Article
ISSN: 0264-4401

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Article

Yuri N. Skiba and Denis M. Filatov

The purpose of this paper is to suggest a new approach to the numerical simulation of shallowwater flows both in plane domains and on the sphere.

Abstract

Purpose

The purpose of this paper is to suggest a new approach to the numerical simulation of shallowwater flows both in plane domains and on the sphere.

Design/methodology/approach

The approach involves the technique of splitting of the model operator by geometric coordinates and by physical processes. Specially chosen temporal and spatial approximations result in one‐dimensional finite difference schemes that conserve the mass and the total energy. Therefore, the mass and the total energy of the whole two‐dimensional split scheme are kept constant too.

Findings

Explicit expressions for the schemes of arbitrary approximation orders in space are given. The schemes are shown to be mass‐ and energy‐conserving, and hence absolutely stable because the square root of the total energy is the norm of the solution. The schemes of the first four approximation orders are then tested by simulating nonlinear solitary waves generated by a model topography. In the analysis, the primary attention is given to the study of the time‐space structure of the numerical solutions.

Originality/value

The approach can be used for the numerical simulation of shallowwater flows in domains of both Cartesian and spherical geometries, providing the solution adequate from the physical and mathematical standpoints in the sense of keeping its mass and total energy constant even when fully discrete shallowwater models are applied.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 19 no. 8
Type: Research Article
ISSN: 0961-5539

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Article

J.G. Zhou and I.M. Goodwill

Describes the solution of the shallow water flow equations in strongly conservative form using a finite volume method. A SIMPLE‐like scheme is developed to treat the…

Abstract

Describes the solution of the shallow water flow equations in strongly conservative form using a finite volume method. A SIMPLE‐like scheme is developed to treat the velocity depth coupling. The method is applied to flow in a sharply curved channel and the results compared with published data. An error analysis is included which indicates that the method proposed is suitable for solving two‐dimensional steady state problems in open channel flow.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 7 no. 1
Type: Research Article
ISSN: 0961-5539

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Article

T. Kodama and M. Kawahara

A finite element method dealing with an open boundary condition for theanalysis of long wave problem is presented. The key feature of the method isthat spurious reflective…

Abstract

A finite element method dealing with an open boundary condition for the analysis of long wave problem is presented. The key feature of the method is that spurious reflective waves which occurred for the initial transient state on the open boundary can be eliminated by introducing a subdomain technique. For the numerical outflow boundary condition, the progressive wave condition, based on the shallow water long wave theory, is successfully employed. This method is quite suitable for practical analysis because of its adaptability for the arbitrary configuration of the open boundary and shape of elements adjacent to the open boundary. This method is numerically verified for flow in a one dimensional channel and the two dimensional tidal current in Tokyo Bay. The numerical results are compared with analytical solutions and observed data obtained by field measurements. These results are all in close agreement.

Details

Engineering Computations, vol. 11 no. 1
Type: Research Article
ISSN: 0264-4401

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Article

P. Glaister

A shock capturing scheme is presented for the equations of isentropicflow based on upwind differencing applied to a locally linearized set ofRiemann problems. This…

Abstract

A shock capturing scheme is presented for the equations of isentropic flow based on upwind differencing applied to a locally linearized set of Riemann problems. This includes the two‐dimensional shallow water equations using the familiar gas dynamics analogy. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency, leading to arithmetic averaging. This is in contrast to usual ‘square root’ averages found in this type of Riemann solver where the computational expense can be prohibitive. The scheme is applied to a two‐dimensional dam‐break problem and the approximate solution compares well with those given by other authors.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 4 no. 5
Type: Research Article
ISSN: 0961-5539

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Article

Shooka Karimpour Ghannadi and Vincent H. Chu

The purpose of this paper is to evaluate the performance of a numerical method for the solution to shallow-water equations on a staggered grid, in simulations for shear…

Abstract

Purpose

The purpose of this paper is to evaluate the performance of a numerical method for the solution to shallow-water equations on a staggered grid, in simulations for shear instabilities at two convective Froude numbers.

Design/methodology/approach

The simulations start from a small perturbation to a base flow with a hyperbolic-tangent velocity profile. The subsequent development of the shear instabilities is studied from the simulations using a number of flux-limiting schemes, including the second-order MINMOD, the third-order ULTRA-QUICK and the fifth-order WENO schemes for the spatial interpolation of the nonlinear fluxes. The fourth-order Runge-Kutta method advances the simulation in time.

Findings

The simulations determine two parameters: the fractional growth rate of the linear instabilities; and the vorticity thickness of the first nonlinear peak. Grid refinement using 32, 64, 128, 256 and 512 nodes over one wave length determines the exact values by extrapolation and the computational error for the parameters. It also determines the overall order of convergence for each of the flux-limiting schemes used in the numerical simulations.

Originality/value

The four-digit accuracy of the numerical simulations presented in this paper are comparable to analytical solutions. The development of this reliable numerical simulation method has paved the way for further study of the instabilities in shear flows that radiate waves.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

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