Two-dimensional Saint-Venant Equations Research Articles

Two-dimensional mathematical modeling of 2013 and 2020 Amur River floods

Predicting river flooding of the territory where people live and engage in economic activities is urgent. The most problematic area in the Russian Far East is the territory through which the Amur River and its tributaries flow. The article considers the calculations of two Amur River floods: 2013 – catastrophic flood and 2020 – low flood. The simulation was carried out using a system of two-dimensional Saint-Venant equations using the Stream 2D CUDA program. The solution of the system of equations by numerical methods is based on the original author’s methodology. Channel depth maps and WordDEMTM (Airbus Defense and Space, Intelligence) data at 24 m resolution were used as a digital elevation model. Calculations of river floods in 2013 and 2020 were performed on a built-in and calibrated mathematical model, which matches the observational data well.

Stability Analysis of an Upwind Difference Splitting Scheme for Two-Dimensional Saint–Venant Equations

The paper is devoted to the construction and study of a numerical method for solving two-dimensional Saint–Venant equations. These equations have important applied significance in modern hydraulic engineering and are suitable for describing waves in the atmosphere, rivers and oceans, and for modeling tides. The issues of formulation of the mixed problems for these equations are studied. The system of equations is reduced to a symmetrical form by transforming dependent variables. Then, the matrices of coefficients are represented as the sums of two symmetric semidefinite matrices. This transformation allows constructing an upwind difference scheme in spatial directions to determine the numerical solution of the initial boundary value problem. The stability of the proposed difference scheme in energy norms is rigorously proved. The results of numerical experiments conducted for a model problem are provided to confirm the stability of the proposed method.

Heterogeneous Parallel Implementation of Large-Scale Numerical Simulation of Saint-Venant Equations

Large-scale floods are one of the major events that impact the national economy and people’s livelihood every year during the flood season. Predicting the factors of flood evolution is a worldwide problem. We use the two-dimensional Saint-Venant equations as an example and for high-performance computing in modelling the flood behavior. Discretization of the two-dimensional Saint-Venant equations with initial and boundary conditions with the finite difference method in the explicit leapfrog scheme is carried out. Afterwards, we employed a large-scale heterogeneous parallel solution on the “SunRising-1” supercomputer system using MPI, OpenMP, Pthread, and OpenCL runtime libraries. On this basis, we applied communication/calculation overlapping and the local memory acceleration to optimize the performance. Finally, various performance tests of the parallel scheme are carried out from different perspectives. We have found this method is efficient and recommend this approach be used in solving systems of partial differential equations similar to the Saint-Venant equations.

Use of Accumulation Basins to Reduce Silting of Reservoirs of Hydroelectric Power Plants Located on Mountain Rivers

The effectiveness of the use of accumulation basins ordinarily used for river sediment to reduce the rate of silting of the reservoirs of hydroelectric power plants situated on mountain rivers is considered. The investigation is performed by methods of numerical hydrodynamic simulation based on the two-dimensional Saint-Venant equations and a new model of sediment transport that takes into account the fractional composition of the sediment. Using as an example a concrete reservoir located on a mountain river, it is shown that it is extraordinarily important to take into account the heterogeneity of the soil in the process of solving these types of problems. A combination of the annual hydraulic flushing of reservoirs and mechanical removal of soil from the accumulation basins together assure the required useful volume throughout the entire service life of the hydroelectric plant, though it requires the annual removal of up to 40% of the sediment discharge of the river.

Calibration of Manning’s roughness in non-instrumented rural basins using a distributed hydrological model

The purpose of this research work was to calibrate the Manning's roughness coefficient in rural non-instrumented basins using a distributed hydrological model. The process consisted of the selection of several basins with vegetal cover of forests and grasslands, and its subsequent experimental numerical study, in which the hydrological response hydrograph of each reference basin was obtained from the HEC-HMS software and the hydrograph to be calibrated was the Iber software, which is a hydrodynamic model based on the two-dimensional Saint Venant equations, solved by the finite volume method. Once the calibration process was carried out, the roughness coefficients with the best fit for each basin were identified with increases in precipitation, identifying the limitations of these values and the standard model used.

Flood frequency analysis and generation of flood hazard indicator maps in a semi-arid environment, case of Ourika watershed (western High Atlas, Morocco)

The development of the statistical models and flood risk modeling approaches have seen remarkable improvements in their productivities. Their application in arid and semi-arid regions, particularly in developing countries, can be extremely useful for better assessment and planning of flood risk in order to reduce the catastrophic impacts of this phenomenon.This study focuses on the Setti Fadma region (Ourika basin, Morocco) which is potentially threatened by floods and is subject to climatic and anthropogenic forcing. The study is based on two main axes: (i) the extreme flow frequency analysis, using 12 probability laws adjusted by Maximum Likelihood method and (ii) the generation of the flood risk indicator maps are based on the solution proposed by the Nays2DFlood solver of the Hydrodynamic model of two-dimensional Saint-Venant equations. The study is used as a spatial high-resolution digital model (Lidar) in order to get the nearest hydrological simulation of the reality.The results showed that the GEV is the most appropriate law of the extreme flows estimation for different return periods. Taking into consideration the mapping of 100-year flood area, the study revealed that the fluvial overflows extent towards the banks of Ourika and consequently, affects some living areas, cultivated fields and the roads that connects the valley to the city of Marrakech. The aim of this study is to propose new technics of the flood risk management allowing a better planning of the flooded areas.

100 Years of Competition between Reduction in Channel Capacity and Streamflow during Floods in the Guadalquivir River (Southern Spain)

Reduction in channel capacity can trigger an increase in flood hazard over time. It represents a geomorphic driver that competes against its hydrologic counterpart where streamflow decreases. We show that this situation arose in the Guadalquivir River (Southern Spain) after impoundment. We identify the physical parameters that raised flood hazard in the period 1997–2013 with respect to past years 1910–1996 and quantify their effects by accounting for temporal trends in both streamflow and channel capacity. First, we collect historical hydrological data to lengthen records of extreme flooding events since 1910. Next, inundated areas and grade lines across a 70 km stretch of up to 2 km wide floodplain are delimited from Landsat and TerraSAR-X satellite images of the most recent floods (2009–2013). Flooded areas are also computed using standard two-dimensional Saint-Venant equations. Simulated stages are verified locally and across the whole domain with collected hydrological data and satellite images, respectively. The thoughtful analysis of flooding and geomorphic dynamics over multi-decadal timescales illustrates that non-stationary channel adaptation to river impoundment decreased channel capacity and increased flood hazard. Previous to channel squeezing and pre-vegetation encroachment, river discharges as high as 1450 m3·s−1 (the year 1924) were required to inundate the same areas as the 790 m3·s−1 streamflow for recent floods (the year 2010). We conclude that future projections of one-in-a-century river floods need to include geomorphic drivers as they compete with the reduction of peak discharges under the current climate change scenario.

Simulating Flood Waves in Residential Areas

An approach to numerical simulation of flooding of residential areas is proposed. It involves modeling of all buildings and other structures in the flood plain, which can automatically be done with the help of modern software. Two-dimensional Saint-Venant equations on unstructured adaptive meshes are used without the need to introduce additional parameters and coefficients. The main advantage of the method is high accuracy and realism of modeling and graphical representation of flood waves and dam-break waves on residential areas. This method is validated against experimental data and real objects.

Quasi-2D model for computation of supercritical free surface flow in sudden expansion

This work deals with the mathematical modeling of a supercritical flow at an abrupt channel expansion. A simplified approach for the study of the characteristics and features of the flow in the presence of the cross shock waves is then proposed. Based on the two-dimensional Saint-Venant equations and using certain simplifying assumptions in practice, this 2D shallow water model is reconverted to an equivalent 1D flow problem by inverting space time variables. The numerical solution of the 1D equation then makes it possible to reconstitute the solution of the 2D flow field according to translating planes. The results obtained show a satisfactory agreement with the experiments carried out for this problem. The profiles of the waterline along the wall and the flow axis are accurately reconstituted thus allowing the good design of the channel walls and the positioning of the oblique shock waves.

Conditions for tidal bore formation in convergent alluvial estuaries

Over the last decade there has been an increasing interest in tidal bore dynamics. However most studies have been focused on small-scale bore processes. The present paper describes the first quantitative study, at the estuary scale, of the conditions for tidal bore formation in convergent alluvial estuaries. When freshwater discharge and large-scale spatial variations of the estuary water depth can be neglected, tide propagation in such estuaries is controlled by three main dimensionless parameters: the nonlinearity parameter ε0, the convergence ratio δ0 and the friction parameter ϕ0. In this paper we explore this dimensionless parameter space, in terms of tidal bore occurrence, from a database of 21 estuaries (8 tidal-bore estuaries and 13 non tidal-bore estuaries). The field data point out that tidal bores occur for convergence ratios close to the critical convergence δc. A new proposed definition of the friction parameter highlights a clear separation on the parameter plane (ϕ0,ε0) between tidal-bore estuaries and non tidal-bore estuaries. More specifically, we have established that tidal bores occur in convergent estuaries when the nonlinearity parameter is greater than a critical value, εc, which is an increasing function of the friction parameter ϕ0. This result has been confirmed by numerical simulations of the two-dimensional Saint Venant equations. The real-estuary observations and the numerical simulations also show that, contrary to what is generally assumed, tide amplification is not a necessary condition for tidal bore formation. The effect of freshwater discharge on tidal bore occurrence has been analyzed from the database acquired during three long-term campaigns carried out on the Gironde/Garonne estuary. We have shown that in the upper estuary the tidal bore intensity is mainly governed by the local dimensionless tide amplitude ε. The bore intensity is an increasing function of ε and this relationship does not depend on freshwater discharge. However, freshwater discharge damps the tidal wave during its propagation and thus reduces ε and consequently limits the tidal bore development in the estuary. To take into account this process in the tidal-bore scaling analysis, it is necessary to introduce a fourth external parameter, the dimensionless river discharge Q0.

Two-dimensional hydrodynamic flood modelling for populated valley areas of Russian rivers

Abstract. Results of flood modelling for three cities located in different parts of Russia: (1) Veliky Ustyug at the Northern Dvina river (Europe); (2) Mezhdurechensk at the Tom river (Siberia); and (3) Blagoveschensk at the Amur river (Far East) are presented. The two-dimensional hydrodynamic model of flow in channels and on floodplain STREAM_2D on the basis of the numerical solution of two-dimensional Saint–Venant equations on a hybrid curvilinear quadrangular and rectangular mesh was used for the simulations. Verification of the model through a comparison of simulated inundated areas with outlines of flooded zones from satellite images for known hydrologic situations demonstrate close correspondence (relative errors of 7–12% in terms of the area for peaks of the analysed floods). Analyses of embankment influence of large-scale levees on the water flow demonstrate that, in some cases, water levels could rise by more than 1 m and the patterns of the flooding zones could significantly differ.

Modelling the hydraulics of the Carlisle 2005 flood event

The performance of a two-dimensional numerical model of flood hydraulics is tested for a major event in Carlisle, UK, in 2005. This event is associated with a unique data set, with global positioning system surveyed wrack lines and flood extent surveyed 3 weeks after the flood. The simple finite-volume model is used to solve the two-dimensional Saint-Venant equations over an unstructured mesh of 30 000 elements representing channel and floodplain, and allowing detailed hydraulics of flow around bridge piers and other influential features to be represented. The simple finite-volume model is also used to corroborate flows recorded for the event at two gauging stations. Calibration of Manning's n is performed with a two-stage strategy, with channel values determined by calibration of the gauging station models, and floodplain values determined by optimising the fit between model results and observed water levels and flood extent for the 2005 event. The root mean square error for the calibrated model compared with surveyed water levels is of the order of ±0·4 m, the same order of magnitude as the estimated error in the survey data. The study demonstrates the ability of unstructured mesh hydraulic models to represent important hydraulic processes across a range of scales, with potential applications to flood risk management.

Discontinuous Galerkin Finite-Element Method for Simulation of Flood in Crossroads

A numerical solution of the two-dimensional Saint Venant equations is presented for the study of the propagation of the floods through the crossroads of the city. The numerical scheme is a Runge-Kutta discontinuous Galerkin method (RKDG) with a slope limiter. The work studies the robustness and the stability of the method. The study is organized around three aspects: the prediction of the water depths, the location of the right and oblique hydraulic jumps in the crossing, and especially the distribution of the flow discharges in the downstream branches. The objective of this paper was to use the RKDG method in order to simulate supercritical flow in crossroads and to compare these simulations with experimental results and to show the advantage of this RKDG method compared to a second-order finite-volume method. A good agreement between the proposed method and the experimental data was found. The method is then able to simulate the flow patterns observed experimentally and to predict accurately the water depths, the location of the hydraulic jumps, and the discharge distribution in the downstream branches.

Simulation of supercritical flow in crossroads: Confrontation of a 2D and 3D numerical approaches to experimental results

This work deals with the modeling of a flood in an urban environment. Among the various types of urban flood events, it was decided to study specifically the severe surface flooding events, which take place in highly urbanized areas. This work concerns particularly the numerical resolution of the two-dimensional Saint Venant equations for the study of the propagation of flood through the crossroads in the city. A discontinuous finite-element space discretization with a second-order Runge–Kutta time discretization is used to solve the two-dimensional Saint Venant equations. The scheme is well suited to handle complicated geometries and requires a simple treatment of boundary conditions and source terms to obtain high-order accuracy. The explicit time integration, together with the use of orthogonal shape functions, makes the method for the investigated flows computationally more efficient than comparable second-order finite volume methods. The scheme is applied to several supercritical flows in crossroads, which are investigated by Mignot. The experimental results obtained by the author are used to verify the accuracy and the robustness of the method. The results obtained are compared to those obtained by a second-order finite volume method (Rubar20 (2D)) and by FLUENT (3D). A very good agreement between the numerical solution obtained by the Runge–Kutta discontinuous Galerkin (RKDG) method and the experimental measured data were found. The method is then able to simulate the flow patterns observed experimentally and able to predict well the water depths, the discharge distribution in the downstream branches of the crossroad and the location of the hydraulic jumps and other flow characteristics more than the other methods.