Finite-volume schemes for shallow-water equations

We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C2 test-functions, are confined in a compact set in H−1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.

Well-balanced central-upwind scheme for a fully coupled shallow water system modeling flows over erodible bed

A well-balanced high-resolution shape-preserving central scheme to solve one-dimensional sediment transport equations

We consider the Saint Venant system (shallow water equations), i.e. an approximation of the incompressible Euler equations widely used to describe river flows, flooding phenomena or erosion problems. We focus on problems involving dry-wet transitions and propose a solution technique using the Spectral Element Method (SEM) stabilized with a variant of the Entropy Viscosity Method (EVM) that is adapted to treat dry zones.KeywordsShallow Water EquationEntropy InequalitySpectral Element MethodEuler SystemIncompressible Euler EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper presents a well-balanced two-dimensional (2D) finite volume model to simulate the propagation, runup and rundown of long wave. Non-staggered grid is adopted to discretize the governing equation and the intercell flux is computed using a central upwind scheme, which is a Riemann-problem-solver-free method for hyperbolic conservation laws. The nonnegative reconstruction method for water depth is implemented in the present model to treat the appearance of wet/dry fronts, and the friction term is solved by a semi-implicit scheme to ensure the stability of the model. The Euler method is applied to update flow variable to the new time level. The model is verified against two experimental cases and good agreements are observed between numerical results and observed data.

The source of water flow in rivers or irrigation canals has a relatively small flow rate of water, which can be used as a pico hydropower plant. One of the main components of a pico hydropower plant is a water wheel, a hybrid pico hydropowerplant is a combination of hydroelectric power and solar power. By utilizing the power of the water flow in a small river or ditch in Wukirsari hamlet, we designed a waterwheel that will convert the energy of motion into electrical energy. The solar panel system designer uses a 50 wp solar cell, a 300 W inverter, a 10 Ampere solar charger controller, and a 12 Volt, 32 Ah battery. The no-load test results on the performance of the single pinwheels and the performance of the double pinwheels at a water level of 37 cm get an average power of 3.2 W and 2.11 W. The resulting voltage is 6.72 V, the current is 0.46 A for a single mill, while for the double pinwheel the voltage is 5,26 , and the current is 0.38 A. From the experimental results, it is better if the single pinwheel and the double pinwheel are used. In a series of no-load solar panels fromPower test results, at 11.00 GMT+7 the highest power value was 33.25 W. The electricity generated by the pico hydro hybrid was used by the people of Wukirsari Hamlet as street lighting. Key words: water wheel, hybrid power, solar panels

The use of multiple-level non-uniform rectangular mesh in coupled flow and sediment transport modeling is preferred to achieve high accuracy in important region without increasing computational cost greatly. Here, a robust coupled hydrodynamic and non-equilibrium sediment transport model is developed on non-uniform rectangular mesh to simulate dam break flow over movable beds. The enhanced shallow water and sediment transport equations are adopted to consider the mass and momentum exchange between the flow phase and sediment phase. The flux at the interface is calculated by the positivity preserving central upwind scheme, which belongs to Godunov-type Riemann-problem-solver-free central schemes and is less expensive than other popular Riemann solvers while still capable of tracking wet/dry fronts accurately. The nonnegative water depth reconstruction method is used to achieve second-order accuracy in space. The model was first verified against two laboratory experiments of dam break flow over irregular fixed bed. Then the quantitative performance of the model was further investigated by comparing the computational results with measurement data of dam break flow over movable bed. The good agreements between the measurements and the numerical simulations are found for the flow depth, velocity and bed changes.

Climate change and urbanization, among various factors, are expected to exacerbate the risk of flood disasters in urban areas. This prompts the construction of appropriate modeling tools capable of addressing full-scale urban floods for hazard and risk assessment. In this view, sub-grid porosity models based on the classic shallow water equations (SWE) appear to be a promising approach for full-scale applications in urban environments with reduced computational cost with respect to classic SWE models on high-resolution grids. The present work focuses on the recently proposed two-dimensional binary single porosity (BSP) model, which is a porosity flooding model written in differential form and based on the use of a binary indicator function to locate obstacles and buildings. Several applications (synthetic, experimental, and real-world cases) show that (i) the BSP results tend to the classic SWE solution for sufficiently refined mesh and that (ii) the BSP model can be successfully applied to realistic conditions with complicated terrain and obstacle distribution on coarser grids. Clearly, the adoption of medium/coarse grids makes the BSP model inherently less accurate than the classic SWE model on high-resolution grids, but the corresponding reduction of computational cost makes the use of the BSP model promising in full-scale urban flood applications when (i) multiple simulations are needed to perform stochastic or scenario analysis, (ii) no detailed information of local flow characteristics is required, and/or (iii) for complementing classic SWE models in a nesting cascade.

The shallow water equations have wide applications in ocean, atmospheric modeling and hydraulic engineering, also they can be used to model flows in rivers and coastal areas. In this article we obtained exact solutions of three equations of shallow water by using $\frac{{G'}} {G} $-expansion method. Hyperbolic and triangular periodic solutions can be obtained from the $\frac{{G'}} {G} $-expansion method.

A diffuse interface method for complex three-dimensional free surface flows

Based on the Godunov‐type cell‐centered finite volume method, this paper presents a two‐dimensional well‐balanced shallow water model for simulating flows over arbitrary topography with wetting and drying. The central upwind scheme is used for the computation of mass and momentum fluxes on interface. The novel aspect of the present model is a robust and accurate nonnegative water depth reconstruction method which is implemented in the unstructured mesh to achieve second‐order accuracy in space and to track the moving wet/dry fronts of the flow over irregular terrain. By defining the bed elevation and primary flow variables at the cell center in the nonstaggered grid system, all computational cells are either fully wet or dry to avoid the problem of being partially wetted. The developed model is capable of being well balanced and preserving the computed water depth to be nonnegative under a certain CFL restriction, which makes it robust and stable. The present model is validated against three benchmark tests and two laboratory dam‐break cases. Finally, the good agreement between the numerical results by the established model and measured data of the Malpasset dam break event on a 1/400 scale physical model demonstrates the capability of the model for the real‐life applications.

The exact Riemann Solver to the Shallow Water equations for natural channels with bottom steps

This paper aims to develop a robust two-dimensional coupled numerical model based on an unstructured mesh, which can simulate rapidly varying flows over an erodible bed involving wet–dry fronts that is a complex yet practically important problem. Using a modified spatial reconstruction based on the finite volume method, the well-balanced property is preserved, which is important for accurate and efficient simulation of morphological problems. In the present study, the central-upwind scheme is extended to simulation of bed erosion and sediment transport for the first time. It is demonstrated that the proposed scheme shows good accuracy and high efficiency. A modified shallow water system is adopted to improve the model. The shallow water equations, sediment transport equation and bed evolution equation are coupled in the governing system. Multiple test cases are employed to demonstrate the robustness, accuracy, and efficiency of the current model. Furthermore, with a field scale dam-break test case, the efficiency and accuracy of the central-upwind method is verified in comparison with other popular Riemann solvers. The effects of the additional source terms in the adopted modified shallow water equations are also investigated by comparing the numerical results with a laboratory study available in the literature. The proposed scheme can efficiently track wetting and drying interfaces while preserving stability in simulating the bed erosion near the wet-dry fronts. The added terms in shallow water equations can improve the accuracy of the simulation when intense sediment-exchange exists; the central-upwind method adopted in the current study shows great accuracy and efficiency compared with other popular solvers; the developed model is robust, efficient and accurate to deal with various challenging cases.

C3: A finite volume-finite difference hybrid model for tsunami propagation and runup

High order finite volume WENO schemes for the shallow water flows through channels with irregular geometry