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1

Modelling of Flood Wave Propagation with Wet-dry Front by One-dimensional Diffusive Wave Equation

Gąsiorowski D.

Archives of Hydro-Engineering and Environmental Mechanics

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2014

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Vol. 61, nr 3-4

111–-125

EN

A full dynamic model in the form of the shallow water equations (SWE) is often useful for reproducing the unsteady flow in open channels, as well as over a floodplain. However, most of the numerical algorithms applied to the solution of the SWE fail when flood wave propagation over an initially dry area is simulated. The main problems are related to the very small or negative values of water depths occurring in the vicinity of a moving wet-dry front, which lead to instability in numerical solutions. To overcome these difficulties, a simplified model in the form of a non-linear diffusive wave equation (DWE) can be used. The diffusive wave approach requires numerical algorithms that are much simpler, and consequently, the computational process is more effective than in the case of the SWE. In this paper, the numerical solution of the one-dimensional DWE based on the modified finite element method is verified in terms of accuracy. The resulting solutions of the DWE are compared with the corresponding benchmark solution of the one-dimensional SWE obtained by means of the finite volume methods. The results of numerical experiments show that the algorithm applied is capable of reproducing the reference solution with satisfactory accuracy even for a rapidly varied wave over a dry bottom.

2

Kinetic-induced moment systems for the Saint-Venant equations

Gil Montoya D. C., Struckmeier J.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2013

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Vol.17, No 1-2

63--90

EN

Based on the relation between kinetic Boltzmann-like transport equations and nonlinear hyperbolic conservation laws, we derive kinetic-induced moment systems for the spatially one-dimensional shallow water equations (the Saint-Venant equations). Using Chapman-Enskog-like asymptotic expansion techniques in terms of the relaxation parameter of the kinetic equation, the resulting moment systems are asymptotically closed without the need for an additional closure relation. Moreover, the new second order moment equation for the (asymptotically) third order system may act as a monitoring function to detect shock and rarefaction waves, which we confirm by a number of numerical experiments.

3

Solution of the Dike-break Problem Using Finite Volume Method and Splitting Technique

Gąsiorowski D.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2011

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Vol. 15, No 3-4

251-270

EN

In this paper, an approach using the finite volume method (FVM) for the solution of two-dimensional shallow water equations is described. Such equations are frequently used to simulate dam-break and dike-break induced flows. The applied numerical algorithm of the FVM is based on a wave-propagation algorithm, which ensures a stable solution and, simultaneously, minimizes numerical errors. Dimensional decomposition according to the coordinate directions was used to split two-dimensional shallow water equations into one-dimensional equations. Additionally, splitting was also applied with respect to the physical processes. The applied dimensional and physical splitting, together with the wave-propagation algorithm led to an effective algorithm and ensured proper incorporation of source terms into the scheme of the finite volume method. A detailed description of an approximation for numerical fluxes and source terms is presented. The obtained numerical results are compared with analytical solutions, laboratory experiments and other results available in the literature.

4

Balance errors in numerical solutions of shallow water equations

Gąsiorowski D.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2007

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Vol. 11, No 4

329-340

EN

An analysis of the conservative properties of shallow water equations is presented, focused on the consistency of their numerical solution with the conservation laws of mass and momentum. Two different conservative forms are considered, solved by an implicit box scheme. Theoretical analysis supported with numerical experiments is carried out for a rectangular channel and arbitrarily assumed flow conditions. The improper conservative form of the dynamic equation is shown not to guarantee a correct solution with respect to the conservation of momentum. Consequently, momentum balance errors occur in the numerical solution. These errors occur when artificial diffusion is simultaneously generated by a numerical algorithm.

5

Finite-volume solvers for a multilayer Saint-Venant system

Audusse E., Bristeau M. O.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 3

311-320

EN

We consider the numerical investigation of two hyperbolic shallow water models. We focus on the treatment of the hyperbolic part. We first recall some efficient finite volume solvers for the classical Saint-Venant system. Then we study their extensions to a new multilayer Saint-Venant system. Finally, we use a kinetic solver to perform some numerical tests which prove that the 2D multilayer Saint-Venant system is a relevant alternative to 3D hydrostatic Navier-Stokes equations.

6

Numerical simulation of extreme flooding in a built-up area

Szydłowski M.

Archives of Hydro-Engineering and Environmental Mechanics

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2005

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Vol. 52, nr 4

321-333

EN

Two numerical simulations of extreme (flash) flood propagation in an urban area are presented. The simulations are performed to recognize some specific features of flow in a built-up area. As the mathematical model of free surface unsteady water flow the shallow water equations are assumed. In order to solve the equations, a numerical scheme based on finite volume method is applied. For approximation of mass and momentum fluxes the Roe method is used. The calculations are examined against the experimental data. The measured variations of water depth at some control points of flooded area are available due to physical modelling. The experiments of model city flooding events were carried out at the hydraulic laboratory of ENEL-CESI in Milan (Italy) in a framework of EC IMPACT project. The aim of these experiments was to simulate a flood in the area where a building group representing a simplified city configuration was introduced.

7

Numerical simulation of rapidly varied water flow in the "Wild River" type water slide

Burzyński K., Szydłowski M.

Archives of Hydro-Engineering and Environmental Mechanics

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2003

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Vol. 50, nr 1

37-57

EN

The numerical analysis of the water flow along the 'Wild River' type water slide is presented. As the mathematical model of the free surface flow shallow water equations are assumed. In order to solve the equations, when transient, rapidly varied flow is present, the numerical scheme based on finite volume method is applied. The numerical simulation of water slide flow is computed on unstructured, triangular mesh. The results of calculation are examined against flow parameters observed on the real object installed in water park in Sopot. Generally good agreement between measured and calculated results was observed. Moreover, the calculations are compared to experimental data available due to physical modelling. As the similarity between physical phenomena of flow within water slide and in the river valley after dam-break event is observed, the investigation was realized within the framework of the State Committee for Scientific Research 6P06S04121 project.

8

Two-dimensional shallow water model for rapidly and gradually varied flow

Szydłowski M.

Archives of Hydro-Engineering and Environmental Mechanics

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2001

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Vol. 48, nr 1

35-61

EN

The numerical solution of full shallow water equation (SWE) including the eddy viscosity terms is presented. In the first part of the paper the solution of the homogeneous part of SWE for discontinuous, rapidly varied flow is reported. The method presented here is based on Roe idea of numerical fluxes of mass and momentum. The numerical solution of SWE on unstructured, triangular mesh is reported and the influence of geometry approximation is examined. The imposing of the boundary condition on a triangular numerical mesh is described in detail. The consistent with finite volume method (FVM) approximation of the viscous part of SWE is presented. The procedure similar to the finite element method (FEM) is used to calculate the function derivatives inside the finite volumes. The specific difficulties of source terms numerical integration are studied and some methods to avoid these problems are presented. To integrate the bottom friction term the splitting technique is implemented. The computed results are compared to analytical solution of Saint-Venant equations, experimental data and results available in the literature. Good agreement between these results is observed.