Two-dimensional Shallow Water Flow Research Articles

Dynamic-wave cellular automata framework for shallow water flow modeling

A new dynamic-wave cellular automata framework for two-dimensional (2D) shallow water flow (SWF) modeling is herein proposed. During the last decade, the concept of cellular automata (CA) has been widely adopted to develop various 2D shallow water flow models. Such CA models use simple transition rules that only consider the conservation of mass and neglect the conservation of momentum, as a result, the models behave like the non-inertia wave approximation which are not suitable for modeling strong discontinuous flows (i.e., transcritical flows and wet-dry interfaces). To overcome this limitation, the present study proposes a new CA framework (SWFCA) for 2D shallow water flow modeling, in which the Bernoulli hydraulic head is adopted to replace water level as the key factor to determine water movements. A novel methodology for linking water depths and velocities in the framework is developed. Model verification is carried out through four cases of regular flows and six benchmark cases with strong discontinuous flows. Model efficiency assessment is also conducted to analyze the numerical performance under various flow conditions. The outcomes indicate that the proposed SWFCA model and a non-inertia wave CA-based model (WCA2D) both provide satisfactory results on regular flows. In modeling strong discontinuous flows, the SWFCA model performs better than the WCA2D model and provides the same accuracy as the finite volume model with a HLLC scheme (FV-HLLC). In the aspect of moving wet-dry interfaces and flows under the partially wet condition, the SWFCA model can achieve higher accuracy than the FV-HLLC model. The numerical efficiency of the SWFCA model relates to the flow conditions involved. As the portion of strong discontinuous flows in the computational domain is increasing, the SWFCA model can be up to 305.5% and 121.0%-1282.2% faster than the WCA2D model and the FV-HLLC model, respectively. This novel CA-based framework has been proved its accuracy and efficiency for shallow water flow modeling. Therefore, it has considerable potentials as a useful tool for real-time flood inundation modeling.

A distributed hydrodynamic model for urban storm flood risk assessment

Hydrodynamic models have been increasingly applied to urban flood simulations, but still encounter challenges because of complicated urban landscapes, buildings and structures, as well as insufficient sewer data. In order to improve the applicability in real-life urban environments, a distributed hydrodynamic model for urban storm flood risk assessment (DHMUrban) is developed in this study. It is based on the two-dimensional shallow water flow equations simplified by neglecting the convection terms in the momentum equations. The sharp variation of ground surface often happening in urban areas is specially treated with the broad-crested weir flow equation. The rainfall, infiltration and sewer drainage are considered at each overland cell. Two connection approaches between the overland model and the sewer model are proposed depending on the availability of sewer data. The overland cells and the sewer nodes are connected physically in areas with complete sewer data, and connected conceptually through a drainage relationship in areas where sewer data are insufficient or unavailable. The present model has been evaluated in five cases, showing comparable reliability and stability. It has been applied to a real-life urban catchment in Shenzhen City, and demonstrated to be capable of presenting well the ground surface flooding, adapting adequately to the real-life urban areas with different data availability and providing practical and reliable solutions for urban storm flood risk assessments.

A greedy non-intrusive reduced order model for shallow water equations

In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compared with a traditional nonlinear POD (NPOD) model by evaluating the accuracy and robustness for test problems representative of riverine flows. Different greedy algorithms are studied in order to determine a near-optimal distribution of interpolation points for the RBF approximation. A new power-scaled residual greedy (psr-greedy) algorithm is proposed to address some of the primary drawbacks of the existing greedy approaches. The relative performances of these greedy algorithms are studied with numerical experiments using realistic two-dimensional (2D) shallow water flow applications involving coastal and riverine dynamics.

A well-balanced and positivity-preserving SPH method for shallow water flows in open channels

A well-balanced and positivity-preserving meshless method based on smoothed particle hydrodynamics (SPH) is developed to simulate one-dimensional (1D) and two-dimensional (2D) shallow water (SW) flows in open channels with irregular geometries. A new form of the characteristic equations that govern the water-surface level and water velocity is introduced to specify the numerical inflow/outflow boundary conditions. An additional condition, derived from temporal discretization to determine the time-step size, forces the water depth to be positive. A 1D finite volume shallow water (FVSW) model based on the first-order Godunov upwind method is built to conduct a comparison of the 1D meshless-based and mesh-based SW models. Six benchmark cases – still water, single trapezoidal and rectangular and prismatic and non-prismatic channels, and a dendritic channel network – are employed to validate the proposed models and compared with the exact and mesh-based numerical solutions. A real-world case of the Chicago Area Waterways System (CAWS) is investigated to highlight the performance of the proposed 1D model for a practical hydraulic system.

A robust second-order surface reconstruction for shallow water flows with a discontinuous topography and a Manning friction

A second-order surface reconstruction (SR) method for the shallow water equations with a discontinuous bottom topography and a Manning friction source term is presented. We redefine the water surface level at the cell interface by using the minimum difference between the bottom level and the original water surface level. The reconstructed water surface level is used to define the intermediate bottom level and the intermediate water height at the cell interface. We propose an explicit-implicit method to address the friction source term. The new second-order SR scheme together with the explicit-implicit method can preserve a special steady-state solution of the system and can maintain the positivity of the water depth. We also extend the new scheme to two-dimensional shallow water flows. To demonstrate the robustness and effectiveness of the new scheme, we use several classical numerical experiments for the shallow water flows over a complex bottom topography.

A new collision operator for lattice Boltzmann shallow water model: a convergence and stability study

This work presents a new lattice Boltzmann model for steady and unsteady two-dimensional shallow water flows. Compared to previous lattice Boltzmann based shallow water models, the proposed method uses a consistent characteristic speed in the pressure term and in the viscosity. To preserve the isotropy of viscosity, the relaxation rates for the different cumulants have to be decoupled. This is only possible by using multiple relaxation rates. The recovery of the correct viscosity is investigated by a convergence study based on the decay of a Taylor Green Vortex. Results from shallow water models using the proposed collision operator are then compared to those derived from standard BGK approach and from a continuous model.

A fractional-order infiltration model to improve the simulation of rainfall/runoff in combination with a 2D shallow water model

Abstract In this work, a distributed two-dimensional (2D) shallow water (SW) flow model is combined with a fractional-order version of the Green-Ampt (FOGA) infiltration law to improve rainfall/runoff simulation in real catchments. The surface water model is based on a robust finite volume method on triangular grids that can handle flow over dry bed and multiple wet/dry fronts. When supplied with adequate infiltration laws, this model can provide useful information in surface hydrology. The classical Green-Ampt law is generalized by using a Caputo fractional derivative of order less than or equal to 1 in Darcy's law. The novelty of this combination is that, on the one hand, the distributed SW simulation provides a detailed surface water distribution and, on the other hand, the FOGA model offers the possibility to model infiltration rates not monotonically decreasing. In order to obtain the best results, a non-uniform order of the fractional derivative depending on the cumulative infiltration and the existence of available surface water is proposed for realistic cases. This allows significant improvement of previous published numerical results in the literature for several storm events in catchments where the infiltration process occurs.

Optimal design of interception for flood control: An integrated simulation approach

This study develops a computational framework that integrates a flood simulation model, a response surface method, and an optimization technique for an optimal flood control design. The objective function was the minimization of the difference between simulated flooding depths and pre-specified allowable flooding depths, with simulated flooding depths constrained to less than allowable flooding depths in the control regions. The design variables were the design capacities of hydraulic structures. The flood simulation model was based on two-dimensional shallow water flow equations for open channel flow, rainfall, overland flow flooding, and river interactions. The response surface method was combined with the central composite design method to construct approximate functions for the objective and the constraint functions in terms of the design variables. Sequential quadratic programming was used to obtain the optimal design variables with the minimum value of the approximated objective function. Actual flood data from Typhoon Morakot for the Linbian River was used to verify the flood simulation model. The integrated computational framework was then used to calculate the optimal upstream interception strengths for controlling downstream flooding under hydrologic conditions associated with a 100-year return period storm. Numerical examples demonstrated that the proposed computational framework can efficiently provide optimal designs for practical flood control problems.

Stabilizing local boundary conditions for two-dimensional shallow water equations

In this article, we present a sub-critical two-dimensional shallow water flow regulation. From the energy estimate of a set of one-dimensional boundary stabilization problems, we obtain a set of po...

Development of a finite volume two-dimensional model and its application in a bay with two inlets: Mobile Bay, Alabama

The purpose of this study was to develop a two-dimensional shallow water flow model using the finite volume method on a combined unstructured triangular and quadrilateral grid system to simulate coastal, estuarine and river flows. The intercell numerical fluxes were calculated using the classical Osher-Solomon's approximate Riemann solver for the governing conservation laws to be able to handle wetting and drying processes and to capture a tidal bore like phenomenon. The developed model was validated with several benchmark test problems including the two-dimensional dam-break problem. The model results were well agreed with results of other models and experimental results in literature. The unstructured triangular and quadrilateral combined grid system was successfully implemented in the model, thus the developed model would be more flexible when applying in an estuarine system, which includes narrow channels. Then, the model was tested in Mobile Bay, Alabama, USA. The developed model reproduced water surface elevation well as having overall Predictive Skill of 0.98. We found that the primary inlet, Main Pass, only covered 35% of the fresh water exchange while it covered 89% of the total water exchange between the ocean and Mobile Bay. There were also discharge phase difference between MP and the secondary inlet, Pass aux Herons, and this phase difference in flows would act as a critical role in substances’ exchange between the eastern Mississippi Sound and the northern Gulf of Mexico through Main Pass and Pass aux Herons in Mobile Bay.

Lattice Boltzmann simulation of the open channel flow connecting two cascaded hydropower stations

This paper examines the feasibility and the efficiency of a multiple-relaxation-time lattice Boltzmann model (MRT-LBM) for simulating open channel flows in engineering practice. A MRT-LBM scheme for 2-D shallow water flows taking into account of the bed slope and the friction is proposed. The scheme's reliability is verified by benchmark problems and the simulation capability is improved by implementing the scheme on a graphic processing unit (GPU). We use the method to analyze the flow characteristics in the connecting open channel of two cascaded hydropower stations. The flow fields and parameters such as the water depth, the flow rate, and the side-weir discharge, under different operating conditions, are analyzed. The factors affecting the accuracy and the efficiency are discussed. The results are found to be reasonable and may be used as a guidance in the project design. It is shown that the GPU-implemented MRT-LBM on a fine mesh can efficiently simulate two-dimensional shallow water flows in engineering practice.

A New Total Variation Diminishing Predictor Corrector Approach for Two-Dimensional Shallow Water Flow

This paper presents a new efficient and robust hydrodynamic model for simulation of unsteady shallow water flow. The governing equations of shallow water flows in two dimensional forms are solved using a new total variation diminishing (TVD) MacCormack predictor corrector scheme. In this numerical technique an additional TVD term is added after the traditional predictor corrector steps. The advantage of the present TVD term is that it is very simple and gives accurate results at the same time removing the numerical oscillations. Further, application of semi implicit treatment of the friction slope term helps in flow simulation even with very low water depth. Finally the model is used to analyze a wide variety of hydraulic problems including quiescent water above irregular bed, steady flow over irregular bed, steady flow over irregular bed with a shock, dam break flow over dry bed and dam break flow over wet bed. For each of the cases numerical results are compared with available analytical solution and known experimental data. The agreements between the results are satisfactory.

Two-dimensional shallow water flow modeling based on an improved unstructured finite volume algorithm

An improved finite volume algorithm is proposed for modeling the free surface flows. The effects of variable reconstruction and treatments for partially wetted cells on numerical oscillations are demonstrated through a complex field-scale simulation. Results show that the combination of a vertex-based slope limiting approach and an updating strategy that the continuity and momentum equations should be simultaneously updated for partially wetted cells is essential to prevent unphysical oscillations. In this situation, the bed slope terms in wetted cells are exactly discretized, whereas the divergence-form-based approach is adopted for partially wetted cells. This new hybrid method provides an alternative way that precisely preserves the well-balanced property. A practical application of realistic dam-break flood propagation is presented. It is found that the model is free of numerical oscillation and can accurately predict dam-break flows over complicated topography with wetting and drying.

A non-homogeneous Riemann solver for shallow water equations in porous media

The purpose of the current research is to develop an accurate and efficient solver for shallow water flows in porous media. The hydraulics is modeled by the two-dimensional shallow water flows with variable horizontal porosity. The variation of porosity in the water flows can be attributed to the variation of bed properties of the water system. As an example of porous shallow water flows is the passage of water discharge over vegetated areas in a river. Driving force of the phase separation and the mixing is the gradient of the porosity. For the numerical solution procedure, we propose a non-homogeneous Riemann solver in the finite volume framework. The proposed method consists of a predictor stage for the discretization of gradient terms and a corrector stage for the treatment of source terms. The gradient fluxes are discretized using a modified Roe’s scheme using the sign of the Jacobian matrix in the coupled system. A well-balanced discretization is used for the treatment of source terms. The efficiency of the solver is evaluated by several test problems for shallow water flows in porous media. The numerical results demonstrate high resolution of the proposed non-homogeneous Riemann solver and confirm its capability to provide accurate simulations for porous shallow water equations under flow regimes with strong shocks.

Two-dimensional solutions of the shallow water equations in the neighbourhood of a shoreline

Two-dimensional shallow water flows when the boundary is smooth at the initial instant of time are considered. Three different flow configurations when a wave reaches the shore: a) with a finite inclination, b) with zero inclination (a long tongue) and c) with a vertical to the shoreline (breaking of the wave) are examined. Theorems in the existence and uniqueness of the solutions of the initial-boundary value problems are proved. The solutions are constructed in the form of series that converge in the neighbourhood of the shoreline. The laws of motion of the shoreline and the instants of time up to which a continuous flow pattern is maintained and after which another flow configuration emerges are found.

Modelling sediment deposition and phosphorus retention in a river floodplain

Phosphorus (P) is one of the major limiting nutrient in many freshwater ecosystems. During the last decade, attention has been focused on the fluxes of suspended sediment and particulate P through freshwater drainage systems because of severe eutrophication effects in aquatic ecosystems. Hence, the analysis and prediction of phosphorus and sediment dynamics constitute an important element for ecological conservation and restoration of freshwater ecosystems. In that sense, the development of a suitable prediction model is justified, and the present work is devoted to the validation and application of a predictive soluble reactive phosphorus (SRP) uptake and sedimentation models, to a real riparian system of the middle Ebro river floodplain. Both models are coupled to a fully distributed two-dimensional shallow-water flow numerical model. The SRP uptake model is validated using data from three field experiments. The model predictions show a good accuracy for SRP concentration, where the linear regressions between measured and calculated values of the three experiments were significant (r2 ≥ 0.62; p ≤ 0.05), and a Nash–Sutcliffe coefficient (E) that ranged from 0.54 to 0.62. The sedimentation model is validated using field data collected during two real flooding events within the same river reach. The comparison between calculated and measured sediment depositions showed a significant linear regression (p ≤ 0.05; r2 = 0.97) and an E that ranged from 0.63 to 0.78. Subsequently, the complete model that includes flow dynamics, solute transport, SRP uptake and sedimentation is used to simulate and analyse floodplain sediment deposition, river nutrient contribution and SRP uptake. According to this analysis, the main SRP uptake process appears to be the sediment sorption. The analysis also reveals the presence of a lateral gradient of hydrological connectivity that decreases with distance from the river and controls the river matter contribution to the floodplain. Copyright © 2014 John Wiley & Sons, Ltd.

AMass Conservation AlgorithmFor Adaptive Unrefinement Meshes Used By Finite Element Methods

The AdaptiveHydraulics (ADH) model is an adaptivefinite element method to simulate three-dimensional Navier-Stokes flow, unsaturated and saturated groundwater flow, overland flow, and two-or three-dimensional shallow-water flow and transport. In the shallow-water flow and transport, especially involving multispecies transport, the water depth (h), the product of water depth and velocities (uh and vh), as well as water depth and chemical concentration (hc) are dependent variables of fluid-motion simulations and are often solved at various times. It is important for the numerical model to predict accurate water depth, velocity fields, and chemical distribution, as well as conserve mass, especially forwater quality applications. Solution accuracydepends highly on mesh resolution. Adaptive mesh refinement (AMR), particularly the h-refinement, is often used to add new nodes in the region where they are needed and to remove others where they are no longer required during the simulation. The AMR is proven to optimize the performance of a computed solution. However, mass withgain or loss can occur when elements are merged due to removing a node at mesh coarsening. Therefore, we develop and implement the mass-conservative unrefinement algorithm to ensure the mass conserved in a merged element in which a node has been removed. This study describes the use of the Galerkin finite element method to redistribute mass to nodes comprising a merged element. The algorithmwas incorporated into the ADH code. This algorithm minimizes mass error during the unrefinement process to conserve mass during the simulation for two-dimensional shallow-water flow and transport. The implementation neither significantly increases the computational time nor memory usage. The simulation was runwithvarious numbersof processors.The resultsshowedgoodscalingof solutiontimeasthenumberof processors increases.

Improved Riemann solvers for complex transport in two-dimensional unsteady shallow flow

The numerical solution of advection–reaction–diffusion transport problems in two-dimensional shallow water flow is split in three subproblems in order to analyze them separately. In the first part, the advection component is solved with the help of an extended Jacobian matrix for the coupled system of flow and advection conservation laws and focusing on the correct definitions of the approximate or weak solutions. Considering that one of the conserved quantities is the solute volume, nonphysical solutions for the solute concentration may appear in complex situations and a solute fix is proposed. This is formulated for first and second order schemes. In the second part of this work, the solution of problems with volumetric reaction terms is studied and the results of single-step as well as multi-step pointwise and upwind approaches are compared in order to establish their relative performance. The upwind treatment is done in 2D cases dividing cell volumes to transform reacting terms in singular source terms. The third part is concerned with the diffusion term. The focus of this part is put on the interference between numerical and physical diffusion. A simple form to estimate the magnitude of the numerical diffusion is proposed and it is shown to improve the accuracy of the results in first and second order approaches.

A robust well-balanced finite volume model for shallow water flows with wetting and drying over irregular terrain

An unstructured Godunov-type finite volume model is developed for the numerical simulation of geometrically challenging two-dimensional shallow water flows with wetting and drying over convoluted topography. In the framework of sloping bottom model, a modified formulation of shallow water equations is used to preserve mass conservation during flooding and recession. The key ingredient of the model is the use of this combination of the sloping bottom model and the modified shallow water equations to provide a robust technique for wet/dry fronts tracking and, together with centered discretization of the bed slope source term, to exactly preserve the static flow on irregular topographies. The variable reconstruction technique ensures nonnegative reconstructed water depth and reasonable reconstructed velocity, and the friction terms are solved by semi-implicit scheme that does not invert the direction of velocity components. The robustness and accuracy of the proposed model are assessed by comparing numerical and reference results of extensive test cases. Moreover, the results of a dam-break flooding over real topography are presented to show the capability of the model on field-scale application.

A Discontinuous Galerkin Method for Two-Dimensional Shock Wave Modeling

A numerical scheme based on discontinuous Galerkin method is proposed for the two-dimensional shallow water flows. The scheme is applied to model flows with shock waves. The form of shallow water equations that can eliminate numerical imbalance between flux term and source term and simplify computation is adopted here. The HLL approximate Riemann solver is employed to calculate the mass and momentum flux. A slope limiting procedure that is suitable for incompressible two-dimensional flows is presented. A simple method is adapted for flow over initially dry bed. A new formulation is introduced for modeling the net pressure force and gravity terms in discontinuous Galerkin method. To validate the scheme, numerical tests are performed to model steady and unsteady shock waves. Applications include circular dam break with shock, shock waves in channel contraction, and dam break in channel with bend. Numerical results show that the scheme is accurate and efficient to model two-dimensional shallow water flows with shock waves.