2D Shallow Water Equations Research Articles

Computational multicore on two-layer 1D shallow water equations for erodible dambreak

The simulation of erodible dambreak using two-layer shallow water equations and SCHR scheme are elaborated in this paper. The results show that the two-layer SWE model in a good agreement with the data experiment which is performed by Louvain-la-Neuve Université Catholique de Louvain. Moreover, the parallel algorithm with multicore architecture are given in the results. The results show that Computer I with processor Intel(R) Core(TM) i5-2500 CPU Quad-Core has the best performance to accelerate the computational time. Moreover, Computer III with processor AMD A6-5200 APU Quad-Core is observed has higher speedup and efficiency. The speedup and efficiency of Computer III with number of grids 3200 are 3.716050530 times and 92.9% respectively.

A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations

Abstract The high-order numerical solution of the non-linear shallow water equations is susceptible to Gibbs oscillations in the proximity of strong gradients. In this paper, we tackle this issue by presenting a shock capturing model based on the numerical residual of the solution. Via numerical tests, we demonstrate that the model removes the spurious oscillations in the proximity of strong wave fronts while preserving their strength. Furthermore, for coarse grids, it prevents energy from building up at small wave-numbers. When applied to the continuity equation to stabilize the water surface, the addition of the shock capturing scheme does not affect mass conservation. We found that our model improves the continuous and discontinuous Galerkin solutions alike in the proximity of sharp fronts propagating on wet surfaces. In the presence of wet/dry interfaces, however, the model needs to be enhanced with the addition of an inundation scheme which, however, we do not address in this paper.

Interpolation of water surface profile in unsteady open channel flow using the adjoint method based on 2D shallow water equations

Real-time interpolation of high-water stage data obtained at separate water gauges was investigated to provide visual information for evacuation decision-making by residents near a river. The adjoint method was applied to a set of shallow water equations in a generalized coordinate system in which the hydraulic effect of tree communities is included as semi-permeable blocks. The cost function to be minimized was assumed as the summation of four misfits for water level and discharge from observation and for Manning's roughness (n) and tree community permeability (K) from standard values. The Lagrange multipliers method was adopted to determine the uncertain quantities, upstream discharge and spatial distributions of n and K, for minimizing the total cost under the constraint of the shallow water equations. The method was applied to a flood event observed in 2011 over a 20-km reach of the Tone River, where four water gauges are installed. The computation results successfully explained the time series of river discharge observed at stations located in the river reach, as well as the longitudinal profile of the trace water levels measured at 500-m intervals just after the flood event.

The Simple Finite Volume Lax-Wendroff Weighted Essentially Nonoscillatory Schemes for Shallow Water Equations with Bottom Topography

A Lax-Wendroff-type procedure with the high order finite volume simple weighted essentially nonoscillatory (SWENO) scheme is proposed to simulate the one-dimensional (1D) and two-dimensional (2D) shallow water equations with topography influence in source terms. The system of shallow water equations is discretized using the simple WENO scheme in space and Lax-Wendroff scheme in time. The idea of Lax-Wendroff time discretization can avoid part of characteristic decomposition and calculation of nonlinear weights. The type of simple WENO was first developed by Zhu and Qiu in 2016, which is more simple than classical WENO fashion. In order to maintain good, high resolution and nonoscillation for both continuous and discontinuous flow and suit problems with discontinuous bottom topography, we use the same idea of SWENO reconstruction for flux to treat the source term in prebalanced shallow water equations. A range of numerical examples are performed; as a result, comparing with classical WENO reconstruction and Runge-Kutta time discretization, the simple Lax-Wendroff WENO schemes can obtain the same accuracy order and escape nonphysical oscillation adjacent strong shock, while bringing less absolute truncation error and costing less CPU time for most problems. These conclusions agree with that of finite difference Lax-Wendroff WENO scheme for shallow water equations, while finite volume method has more flexible mesh structure compared to finite difference method.

Coupled method for the numerical simulation of 1D shallow water and Exner transport equations in channels with variable cross-section

This work is focused on the a numerical finite volume scheme for the resulting coupled shallow water-Exner system in 1D applications with arbitrary geometry. The mathematical expression modeling the the hydrodynamic and morphodynamic components of the physical phenomenon are treated to deal with cross-section shape variations and empirical solid discharge estimations. The resulting coupled system of equations can be rewritten as a nonconservative hyperbolic system with three moving waves and one stationary wave to account for the source terms discretization. But, even for the simplest solid transport models as the Grass law, to find a linearized Jacobian matrix of the system can be a challenge if one considers arbitrary shape channels. Moreover, the bottom channel slope variations depends on the erosion-deposition mechanism considered to update the channel cross-section profile. In this paper a numerical finite volume scheme is proposed, based on an augmented Roe solver (first order accurate in time and space) and dealing with solid transport flux variations caused by the channel geometry changes. Channel crosssection variations lead to the appearance of a new solid flux source term which should be discretized properly. Comparison of the numerical results for several analytical and experimental cases demonstrate the effectiveness, exact wellbalanceness and accuracy of the scheme.

Data assimilation at local scale to improve CFD simulations of atmospheric dispersion: application to 1D shallow-water equations and method comparisons

Data assimilation at local scale to improve CFD simulations of atmospheric dispersion: application to 1D shallow-water equations and method comparisons

Data assimilation at local scale to improve CFD simulations of atmospheric dispersion: application to 1D shallow-water equations and method comparisons

Atmospheric dispersion modelling requires meteorological inputs over local domains with possibly complex topographies. These local wind fields may be difficult to simulate with CFD models, in particular because of their sensitivity to geometrical features and to model inputs, especially the boundary conditions which are generally provided by larger-scale models or measurements. Using data assimilation, a few measurements inside the domain could add information to the imprecise boundary conditions and thus greatly enhance the precision of the dispersion simulations. Three data assimilation techniques (3DVar, the back and forth nudging algorithm, and the iterative ensemble Kalman smoother) have been adapted to local scale simulations by taking boundary conditions into account instead of initial conditions for which they are usually applied. Their performances have been evaluated at small scales, with a simple representation of the atmosphere into two layers, using 1D solution of the shallow-water equations.

Evaluation of wind induced currents modeling along the Southern Caspian Sea

To improve our understanding of the Caspian Sea hydrodynamics, its circulation is simulated with special focus on wind-driven currents of its southern basin. The hydrodynamic models are forced with a newly developed fine resolution wind field to increase the accuracy of current modeling. A 2D shallow water equation model and a 3D baroclinic model are applied separately to examine the performance of each model for specific applications in the Caspian Sea. The model results are validated against recent field measurements including AWAC and temperature observations in the southern continental shelf region. Results show that the 2D model is able to well predict the depth-averaged current speed in storm conditions in narrow area of southern coasts. This finding suggests physical oceanographers apply 2D modeling as a more affordable method for extreme current speed analysis at the continental shelf region. On the other hand the 3D model demonstrates a better performance in reproducing monthly mean circulation and hence is preferable for surface circulation of Caspian Sea. Monthly sea surface circulation fields of the southern basin reveal a dipole cyclonic-anticyclonic pattern, a dominant eastward current along the southern coasts which intensifies from May to November and a dominant southward current along the eastern coasts in all months except February when the flow is northward. Monthly mean wind fields exhibit two main patterns including a north-south pattern occurring at warm months and collision of two wind fronts especially in the cold months. This collision occurs on a narrow region at the southern continental shelf regions. Due to wind field complexities, it leads to a major source of uncertainty in predicting the wind-driven currents. However, this source of uncertainty is significantly alleviated by applying a fine resolution wind field.

GPU driven finite difference WENO scheme for real time solution of the shallow water equations

The shallow water equations are applicable to many common engineering problems involving modelling of waves dominated by motions in the horizontal directions (e.g. tsunami propagation, dam breaks). As such events pose substantial economic costs, as well as potential loss of life, accurate real-time simulation and visualization methods are of great importance. For this purpose, we propose a new finite difference scheme for the 2D shallow water equations that is specifically formulated to take advantage of modern GPUs. The new scheme is based on the so-called Picard integral formulation of conservation laws combined with Weighted Essentially Non-Oscillatory reconstruction. The emphasis of the work is on third order in space and second order in time solutions (in both single and double precision). Further, the scheme is well-balanced for bathymetry functions that are not surface piercing and can handle wetting and drying in a GPU-friendly manner without resorting to long and specific case-by-case procedures. We also present a fast single kernel GPU implementation with a novel boundary condition application technique that allows for simultaneous real-time visualization and single precision simulations even on large ( > 2000 × 2000) grids on consumer-level hardware - the full kernel source codes are also provided online at https://github.com/pparna/swe_pifweno3.

Modeling and simulation of ocean wave propagation using lattice Boltzmann method

In this paper, we present on modeling and simulation of ocean wave propagation from the deep sea to the shoreline. This requires high computational cost for simulation with large domain. We propose to couple a 1D shallow water equations (SWE) model with a 2D incompressible Navier-Stokes equations (NSE) model in order to reduce the computational cost. The coupled model is solved using the lattice Boltzmann method (LBM) with the lattice Bhatnagar-Gross-Krook (BGK) scheme. Additionally, a special method is implemented to treat the complex behavior of free surface close to the shoreline. The result shows the coupled model can reduce computational cost significantly compared to the full NSE model.

Implicit finite volume simulation of 2D shallow water flows in flexible meshes

In this work, an implicit method for solving 2D hyperbolic systems of equations is presented, focusing on the application to the 2D shallow water equations. It is based on the first order Roe’s scheme, in the framework of finite volume methods. A conservative linearization is done for the flux terms, leading to a non-structured matrix for unstructured meshes thus requiring iterative methods for solving the system. The validation is done by comparing numerical and exact solutions in both unsteady and steady cases. In order to test the applicability of the implicit scheme to real world situations, a laboratory scale tsunami simulation is carried out and compared to the experimental data. The implicit schemes have the advantage of the unconditional stability, but a quality loss in the transient solution can appear for high CFL numbers. The properties of the scheme are well suited for the simulation of unsteady shallow water flows over irregular topography using all kind of meshes.

Physical habitat simulation for a fish community using the ANFIS method

This study presents physical habitat simulations for the fish community in a reach of the Dal River, Korea. The study reach is 2.3km long, located downstream from the Goesan Dam. The reach is gravel-bed, including a bend. A riffle is present at the apex of the bend, and pools are located before and after the riffle. Field monitoring revealed that five fish species are dominant, namely Zacco platypus, Coreoleuciscus splendidus, Zacco temminckii, Pungfungia herzi, and Acheilognathus yamatsute, and account for 78% of the total fish community. These five fishes, which include both lotic and lentic fishes, were selected as target species. The Center for Computational Hydroscience and Engineering 2D (CCHE2D) model, a 2D shallow water equation solver, and the Adaptive Neuro Fuzzy Inference System (ANFIS) method were used for hydraulic and habitat simulations, respectively. The Mahalanobis distance method was used for identifying outliers in the monitoring data. It was shown that the ANFIS method combined with data quality assessment predicts the Composite Suitability Index (CSI) better than the method without it. The CSI distributions in the study reach are presented for various flows. It was found that the distributions of the combined CSI predicted by the ANFIS method are higher than those predicted by the conventional multiplicative aggregation method with Habitat Suitability Curves (HSCs). In addition, it was revealed that the discharge that yields the maximum Weighted Usable Area (WUA) for the whole fish community is more than twice the discharge yielding the maximum WUA for a single target species. This increased discharge leads to a 58% increase in the maximum WUA for the fish community.

Numerical treatment of a discontinuous top surface in 2D shallow water mixed flow modeling

SummaryThis paper presents a numerical strategy based on shallow water equations (SWE) coupled with the 2D Preissmann slot model to handle a ceiling step discontinuity in finite volume schemes for mixed flow modeling. In practice, a typical situation would be a closed structure, such as a bridge or culvert, which induces a sudden vertical flow constriction and may even run partly or totally full in high flow conditions. In such case, both the inlet and outlet of the structure involve a discontinuity in the top elevation. This special singularity is topologically represented by inserting a fictitious cell between 2 adjacent computational cells characterized by sharply different ceiling elevation. The 2D SWE are solved by means of a well‐balanced quasi‐conservative Godunov‐type numerical scheme based on the Slope Limiter Centered (SLIC) scheme. The flow variables at each boundary of the fictitious cell are reconstructed by adopting the cross‐sectional shape of the adjoining cell. Accordingly, the dynamic effect of the structure deck on the flow is suitably modeled, and the C‐property for a stationary solution is rigorously satisfied, even when the closed structure is partially full. The capability of the numerical scheme is verified by comparison with both novel analytical solutions of 1D Riemann problems with a ceiling step discontinuity and experimental data of steady and unsteady mixed flows available in literature. Finally, a real‐scale application to a multiple arch bridge is presented. The results show that the method is robust and effective in predicting the 2D features induced by a crossing structure on the flow dynamics.

A 1D–2D Shallow Water Equations solver for discontinuous porosity field based on a Generalized Riemann Problem

A novel augmented Riemann Solver capable of handling porosity discontinuities in 1D and 2D Shallow Water Equation (SWE) models is presented. With the aim of accurately approximating the porosity source term, a Generalized Riemann Problem is derived by adding an additional fictitious equation to the SWEs system and imposing mass and momentum conservation across the porosity discontinuity. The modified Shallow Water Equations are theoretically investigated, and the implementation of an augmented Roe Solver in a 1D Godunov-type finite volume scheme is presented. Robust treatment of transonic flows is ensured by introducing an entropy fix based on the wave pattern of the Generalized Riemann Problem. An Exact Riemann Solver is also derived in order to validate the numerical model. As an extension of the 1D scheme, an analogous 2D numerical model is also derived and validated through test cases with radial symmetry. The capability of the 1D and 2D numerical models to capture different wave patterns is assessed against several Riemann Problems with different wave patterns.

Experimental calibration and validation of sewer/surface flow exchange equations in steady and unsteady flow conditions

The linkage between sewer pipe flow and floodplain flow is recognised to induce an important source of uncertainty within two-dimensional (2D) urban flood models. This uncertainty is often attributed to the use of empirical hydraulic formulae (the one-dimensional (1D) weir and orifice steady flow equations) to achieve data-connectivity at the linking interface, which require the determination of discharge coefficients. Because of the paucity of high resolution localised data for this type of flows, the current understanding and quantification of a suitable range for those discharge coefficients is somewhat lacking. To fulfil this gap, this work presents the results acquired from an instrumented physical model designed to study the interaction between a pipe network flow and a floodplain flow. The full range of sewer-to-surface and surface-to-sewer flow conditions at the exchange zone are experimentally analysed in both steady and unsteady flow regimes. Steady state measured discharges are first analysed considering the relationship between the energy heads from the sewer flow and the floodplain flow; these results show that existing weir and orifice formulae are valid for describing the flow exchange for the present physical model, and yield new calibrated discharge coefficients for each of the flow conditions. The measured exchange discharges are also integrated (as a source term) within a 2D numerical flood model (a finite volume solver to the 2D Shallow Water Equations (SWE)), which is shown to reproduce the observed coefficients. This calibrated numerical model is then used to simulate a series of unsteady flow tests reproduced within the experimental facility. Results show that the numerical model overestimated the values of mean surcharge flow rate. This suggests the occurrence of additional head losses in unsteady conditions which are not currently accounted for within flood models calibrated in steady flow conditions.

Validation of 2D shock capturing flood models around a surcharging manhole

This work offers a detailed validation of finite volume (FV) flood models in the case where horizontal floodplain flow is affected by sewer surcharge flow via a manhole. The FV numerical solution of the 2D shallow water equations is considered based on two approximate Riemann solvers, HLLC and Roe, on both quadrilateral structured and triangular unstructured mesh-types. The models are validated against a high resolution experimental data-set obtained using a physical model of a sewer system linked to a floodplain via a manhole. It was verified that the sensitivity of the models is inversely proportional to the surcharged flow/surface inflow ratio, and therefore requires more calibration from the user especially when concerned with localised modelling of sewer-to-floodplain flow. Our findings provide novel evidence that shock capturing FV-based flood models are applicable to simulate localised sewer-to-floodplain flow interaction.

Supercritical flow simulation at a right channel junction. Comparison between a uniform and a sparse mesh

This technical note studies the supercritical junction flow occurring at the right-angled confluence of four equal-width channels in which two upstream channels carry flow towards the junction. The note seeks to investigate, firstly, the pertinence of a 2D shallow water equation model to reproduce the typical flow structures at the junction. The second objective is to build a computationally efficient model with more resolution applied at critical point i.e. junction and less in areas where the flow is primarily 1D i.e. channels and compare such model, named, ‘sparse’ with a uniformly meshed model regarding solution accuracy and computational efficiency. The results indicate that the sparse model is able to reproduce typical flow structures appearing at the channel junction in an adequate manner. The discharge distribution is fairly well predicted. The jump angles are almost the same in the two models as well as the location and size of the recirculation zones and the flow depth super-elevation areas. However, the two models diverge in the prediction of very small depths in the recirculation zone where the sparse model overestimates the depths. As regards, computational efficiency, the sparse model is found to be 61% more efficient than the uniform mesh model.

Calibration of the 1D shallow water equations: a comparison of Monte Carlo and gradient-based optimization methods

The calibration of parameters in complex systems usually requires a large computational effort. Moreover, it becomes harder to perform the calibration when non-linear systems underlie the physical process, and the direction to follow in order to optimize an objective function changes depending on the situation. In the context of shallow water equations (SWE), the calibration of parameters, such as the roughness coefficient or the gauge curve for the outlet boundary condition, is often required. In this work, the SWE are used to simulate an open channel flow with lateral gates. Due to the uncertainty in the mathematical modeling that these lateral discharges may introduce into the simulation, the work is focused on the calibration of discharge coefficients. Thus, the calibration is performed by two different approaches. On the one hand, a classical Monte Carlo method is used. On the other hand, the development and application of an adjoint formulation to evaluate the gradient is presented. This is then used in a gradient-based optimizer and is compared with the stochastic approach. The advantages and disadvantages are illustrated and discussed through different test cases.

Three-dimensional flood routing of a dam break based on a high-precision digital model of a dense urban area

Dense buildings are the major factor affecting urban flood routing. Currently, the study of urban dam-break flood routing primarily focuses on a simplified terrain model and 2D shallow water equations, which ignore the effects of dense urban buildings. Furthermore, the complex interactions between the dam-break wave and the wall surfaces of buildings are not reflected in the results. To tackle these problems, three-dimensional flood routing with a high-precision digital model of an urban area is studied in this paper. Firstly, the vector data of various land types is extracted from a remote sensing image, and the NURBS algorithm based on the TIN algorithm is introduced to construct a three-dimensional terrain model. Coupled with the vector data and the terrain model, a three-dimensional digital model of the urban area is established. Next, a three-dimensional \(k{-}\varepsilon\) turbulence model is proposed for the flood routing simulation. A polyhedral grid with a second-order accuracy and a discrete format is used to divide the digital model of the urban area, and the governing equations are solved using the PISO algorithm. Finally, the superiority of the 3D mathematical model and the computational efficiency of the polyhedral mesh model are validated according to the urban flood routing experiments of Testa and Soares-Frazao. An urban reservoir, located in SZ City, China, is modelled to show that a dam-break flood in an urban area exhibits significant three-dimensional characteristics. Moreover, due to the surrounding buildings, the flood exhibits complex three-dimensional turbulence phenomena, including collision, reflection and vortices.

2D Versus 1D Models for Shallow Water Equations

In this paper we present a general framework to construct 1D width averaged models when the flow is constrained -e.g. by topography- to be almost 1D. We start from two dimensional shallow water equations, perform an asymptotic expansion of the fluid elevation and velocity field in the spirit of wave diffusive equations and establish a set of 1D equations made of a mass, momentum and energy equations which are close to the one usually used in hydraulic engineering. We show that in some special cases, like the U-shaped river bed, that our set of equations reduces to the classical 1d shallow water equations. Out of these configurations, there is an O (1) deviation of our model from the classical one.