• Bulletin of the Korean Mathematical Society
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• v.61 no.4
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• pp.969-998
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• 2024

We present numerical schemes to deal with nonconservative terms in the two-dimensional shallow water equations with variable topography. Relying on the dimensional splitting technique, we construct Godunov-type schemes. Such schemes can be categorized into two classes, namely the partly and fully splitting ones, depending on how deeply the scheme employs the splitting method. An upwind scheme technique is employed for the evolution of the velocity component for the partly splitting scheme. These schemes are shown to possess interesting properties: They can preserve the positivity of the water height, and they are well-balanced.

• Journal of Korea Water Resources Association
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• v.46 no.2
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• pp.139-153
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• 2013

For analyzing shallow-water flows over the uneven bottom, the HLLL scheme and the divergence form for bed slope source term (DFB) technique, respectively were applied to the flux gradient and the bottom gradient source terms in a finite-volume model for the shallow water equations. And also the model incorporated the volume/free-surface relationship (VFR) to consider the partially submerged cells (PSC). It was identified that a simpler version of the weighted surface-depth gradient method in the MUSCL was equivalent to the original one in the accuracy for 1D steady flows. It was verified that the flux gradient term and the bottom gradient source term were well-balanced exactly by the VFR for the 1D PSC. The VFR for the triangular PSC settled the problem which the governing equations were not well-balanced by the DFB technique for the 2D PSC. There were good agreements in simulations and experiments for 2D dam-break flows over a triangular sill and a round bump. In addition, the partial dam-break flow was successfully simulated for flooding of roughnesses in an irregular bottom as well as a sloping one. Therefore, this model is expected to be applied to the real river with uneven topography.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.53-65
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• 2003

Here we present a study of small-amplitude, shallow water waves on sloping beds. The beds considered in this analysis are linear and quadratic in nature. First we start with stating the relevant governing equations and boundary conditions for the theory of water waves. Once the complete prescription of the water-wave problem is available based on some assumptions (like inviscid, irrotational flow), we normalize it by introducing a suitable set of non-dimensional variables and then we scale the variables with respect to the amplitude parameter. This helps us to characterize the various types of approximation. In the process, a summary of equations that represent different approximations of the water-wave problem is stated. All the relevant equations are presented in rectangular Cartesian coordinates. Then we derive the equations and boundary conditions for small-amplitude and shallow water waves. Two specific types of bed are considered for our calculations. One is a bed with constant slope and the other bed has a quadratic form of surface. These are solved by using separation of variables method.

• Journal of Korea Water Resources Association
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• v.48 no.11
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• pp.957-967
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• 2015

It was reviewed for the 2D numerical simulations to evaluate the effects of flood control by detention basin, even if stage-discharge relationships for the side weir were not known. A 2D depth-integrated numerical model was constructed by the application of the finite volume method to the shallow water equations as a numerical method and the introduction of an approximate Riemann solver for the accurate calculation of fluxes. Results by the model were compared with those by the laboratory test for the cases of free overflow and submerged flow over a side weir between the channel and storage. The difference between simulated and measured discharge coefficients for the case of free overflow is very small. In addition, the results by simulations were in good agreement with those by experiments for the submerged flow over a side weir and its mechanism was reproduced well. Through this study the discharge coefficients of side weirs can be accurately determined by the 2D numerical model and a considerable degree of accuracy can be achieved to evaluate the effect of flood defenses by detention basins. Thus, it will be expected to apply this model practically to the plan of detention basins, the evaluation of design alternatives, or the management of the existing ones.

• International Journal of Ocean System Engineering
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• v.3 no.2
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• pp.68-76
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• 2013

In this study, a 3D numerical model was used to predict nonlinear wave forces on a cylindrical pile installed in a shallow water region. The model was based on solving the viscous and incompressible Navier-Stokes equations for a two-phase flow (water and air) model and the volume of fluid method for treating the free surface of water. A new application was developed based on the cut-cell method to allow easy installation of complicated obstacles (e.g., bottom geometry and cylindrical pile) in a computational domain. Free-surface elevation, water particle velocities, and inline wave forces were calculated, and the results show good agreement with experimental data obtained by the Danish Hydraulic Institute. The simulation results revealed that the proposed model can, without the use of empirical formulas (i.e., Morison equation) and additional wave analysis models, reliably predict non-linear wave forces on an offshore wind turbine foundation installed in a shallow water region.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.43 no.5
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• pp.587-599
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• 2023

As climate change increasesthe frequency and risk of flooding in major cities around theworld, the importance ofsimulation technology that can quickly and accurately analyze high-resolution 2D flooding information in large-scale areasis emerging. The physically-based approaches based on the Shallow Water Equations (SWE) often requires huge computer resources hindering high-resolution flood prediction. This study investigated the theoretical background of Weighted Cellular Automata 2D (WCA2D), which simulates spatio-temporal changes offlooding using transition rules and weight-based system, and assessed feasibility to simulate pluvial flooding in the urbancatchment, theOncheon-cheon catchmentinBusan, SouthKorea.Inaddition,the computation performancewas compared by applying versions using OpenComputing Language (OpenCL) andOpenMulti-Processing (OpenMP) parallel computing techniques. Simulationresultsshowed that the maximuminundation depthmap by theWCA2Dmodel cansimilarly reproduce historical inundation maps. Also, it can precisely simulate spatio-temporal changes of flooding extent in the urban catchment with complex topographic characteristics. For computation efficiency, parallel computing schemes, theOpenCLandOpenMP, improved the computation by about 8~14 and 5~6 folds respectively, compared to the sequential computation.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.17 no.1
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• pp.1-8
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• 2005

For the simulation of tsunami propagation an active dispersion-correction two-dimensional finite element model has been developed based on a shallow-water wave equation. This model employs an arbitrary triangular mesh and an explicit time integration scheme. However, the physical dispersion effects as included in the Boussinesq equations can be taken into account in the computation. The validity of the dispersion-correction scheme developed in this study is verified through the comparison of numerical solutions calculated using the new scheme with analytical ones considering dispersion effect of waves. As a result, the present model is shown to be considerably accurate.

• Journal of Korea Water Resources Association
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• v.38 no.3 s.152
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• pp.179-188
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• 2005

This study on the HwaBuk Multipurpose Dam showed that two- and three- dimensional numerical model experiments, as well as hydraulic model experiments, can be useful analysis tools for engineers. A commercially available RMA2, which solves the shallow water equations, and FLOW-3D, which solves the Reynolds averaged Navier-Stokes equations, were used to simulate the hydraulic model setup. Numerical simulation results on the following were compared with the hydraulic model results: the flow in the reservoir basin and the approaching channel; the discharge in the overflow weir; the water surface profiles in the rollway, chute, and stilling basin; and the pressure distributions in the rollway. It was shown that there is a reasonably good agreement between the numerical model and the hydraulic model for the most of computations. There were, however, some differences between the numerical simulation results and hydraulic model results for the hydraulic jump in the stilling basin because of air entrainment effect.

• Proceedings of the Korea Water Resources Association Conference
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• 2005.05b
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• pp.414-418
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• 2005

A finite element model is developed for the numerical simulation of bed elevation change in a curved channel. The SU/PG (Streamline-Upwind/Petrov-Galerkin) method is used to solve 2D shallow water equations and the BG (Bubnov-Galerkin) method is used for the Exner equation. For the time derivative terms, the Crank-Nicolson scheme is used. The developed model is a decoupled model in a sense that the bed elevation does not change simultaneously with the flow during the computational time step. The total load formula with is used for the sediment transport model. The slip conditions are described along the lateral boundaries. The effects of gravity force due to geometry change and the secondary flows in a curved channel are considered in the model. For the verification, the model is applied to two laboratory experiments. The first is $140^{\circ}$ bended channel data at Delft Hydraulics Laboratory and the second is $140^{\circ}$ bended channel data at Laboratory of Fluid Mechanics of the Delft University of Technology. The finite element grid is constructed with linear quadrilateral elements. It is found that the computed results are in good agreement with measured data, showing a point bar at the inner bank and a pool at the outer bank.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.26 no.5B
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• pp.447-457
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• 2006

This paper presents a numerical model for the simulations of 2D depth-averaged flows. The shallow water equations are solved numerically by the Characteristic Dissipative Galerkin (CDG) finite element method. For validation, the developed model is applied to the hydraulic jump. The computed results are compared with the analytical solution, revealing good agreement. In addition, flow in a contracting channel showing standing waves is simulated. The calculated water surface profile appears to be qualitatively consistent with the observed data. The foregoing results indicate that the model is capable of simulating the abrupt change in flow field. Next, the model is applied to the flow in a $180^{\circ}$ curved channel. The simulated results show that the velocity near the inner bank is faster than that near the outer bank and the water depth near the inner bank is shallower than that near the outer bank. However, the simulated results show that the velocity distribution across the channel is almost uniform in the bend except the reach close to the end of the bend. This is due to the limitation of the governing equations in which the transverse convection of momentum by the secondary flows along a channel bend is not taken into account.