Numerical resolution of well-balanced shallow water equations with complex source terms

This paper presents a well-balanced numerical scheme for simulating frictional shallow flows over complex domains involving wetting and drying. The proposed scheme solves, in a finite volume Godunov-type framework, a set of pre-balanced shallow water equations derived by considering pressure balancing. Non-negative reconstruction of Riemann states and compatible discretization of slope source term produce stable and well-balanced solutions to shallow flow hydrodynamics over complex topography. The friction source term is discretized using a splitting implicit scheme. Limiting value of the friction force is derived to ensure stability. This new numerical scheme is validated against four theoretical benchmark tests and then applied to reproduce a laboratory dam break over a domain with irregular bed profile.     

The LHLLC scheme for Two-Layer and Two-Phase transcritical flows over a mobile bed with avalanching,wetting and drying

A numerical method of the Godunov type is presented for solving either Two-Phase or Two-Layer forms of Debris Flow Models (DFMs) describing shallow-water flow and sediment dynamics. DFMs explicitly link sediment concentrations to the momentum balance, and thus can be applied to cases involving high sediment concentrations, as in debris flows, in addition to low concentration test cases typically found in surface waters. In this paper, Two-Phase and Two-Layer DFMs are presented in a common mathematical framework to illuminate key similarities and differences and lay a foundation for a general purpose DFM solver. The proposed solver termed LHLLC is shown to achieve good accuracy over a wide range of test cases. Importantly, numerical diffusion of sediment profiles is minimized, particularly on steep slopes, the scheme is shown to preserve stationary solutions involving wet/dry interfaces, and the scheme accounts for gravity-driven slumping (avalanching) which cannot be resolved by classical DFMs.     

Balancing the source terms in a SPH model for solving the shallow water equations

A shallow flow generally features complex hydrodynamics induced by complicated domain topography and geometry. A numerical scheme with well-balanced flux and source term gradients is therefore essential before a shallow flow model can be applied to simulate real-world problems. The issue of source term balancing has been exhaustively investigated in grid-based numerical approaches, e.g. discontinuous Galerkin finite element methods and finite volume Godunov-type methods. In recent years, a relatively new computational method, smooth particle hydrodynamics (SPH), has started to gain popularity in solving the shallow water equations (SWEs). However, the well-balanced problem has not been fully investigated and resolved in the context of SPH. This work aims to discuss the well-balanced problem caused by a standard SPH discretization to the SWEs with slope source terms and derive a corrected SPH algorithm that is able to preserve the solution of lake at rest. In order to enhance the shock capturing capability of the resulting SPH model, the Monotone Upwind-centered Scheme for Conservation Laws (MUSCL) is also explored and applied to enable Riemann solver based artificial viscosity. The new SPH model is validated against several idealized benchmark tests and a real-world dam-break case and promising results are obtained.     

A mass conservative scheme for simulating shallow flows over variable topographies using unstructured grid

Most available numerical methods face problems, in the presence of variable topographies, due to the imbalance between the source and flux terms. Treatments for this problem generally work well for structured grids, but most of them are not directly applicable for unstructured grids. On the other hand, despite of their good performance for discontinuous flows, most available numerical schemes (such as HLL flux and ENO schemes) induce a high level of numerical diffusion in simulating recirculating flows. A numerical method for simulating shallow recirculating flows over a variable topography on unstructured grids is presented. This mass conservative approach can simulate different flow conditions including recirculating, transcritical and discontinuous flows over variable topographies without upwinding of source terms and with a low level of numerical diffusion. Different numerical tests cases are presented to show the performance of the scheme for some challenging problems.     

Two-dimensional numerical modeling of dam-break flows over natural terrain using a central explicit scheme

A two-dimensional (2D) numerical model has been developed to solve shallow water equations for simulation of dam-break flows. The spatial derivatives are discretized using a well-balanced explicit central upwind conservative scheme. The scheme is Riemann solver free and guarantees the positivity of the flow depth over complex topography if the Courant number is kept less than 0.25. The time integration is performed by Euler’s scheme. The model is verified against analytical results for water surface elevation and discharge for three benchmark test cases. A good agreement between analytical solutions and computed results is observed. The property of well-balancing in still water over an uneven bottom is also confirmed. The model is then validated by simulating a laboratory experiment in which a dam break flow propagates over a triangular obstacle. The model performance was found to be satisfactory. A dam break laboratory experimental test case on a frictionless horizontal bottom is also simulated for 2D validation of the model, and good agreement between simulation and the experimental data is observed. The suitability of the proposed model for real life applications is demonstrated by simulating the Malpasset dam-break event, which occurred in 1959 in France. The computed arrival time of the flood wave front and the maximum flow depths at various observation points matched well with the measurements on a 1/400 scale physical model. The overall performance indicates that this model can be applied for simulation of dam-break waves in real life cases.     

Comparison between TVD-MacCormack and ADI-type solvers of the shallow water equations

A total variation diminishing (TVD) modification of the MacCormack scheme is developed for simulating shallow water dynamics on a uniform Cartesian grid. Results obtained using conventional and deviatoric forms of the conservative non-linear shallow water equations (SWEs) are compared for cases where the bed has a varying topography. The comparisons demonstrate that the deviatoric form of the SWEs gives more accurate results than the conventional form, in the absence of numerical balancing of the flux-gradient and source terms. A further comparison is undertaken between the TVD-MacCormack model and an alternating direction implicit (ADI) model for cases involving steep-fronted shallow flows. It is demonstrated that the ADI model is unable to predict trans-critical flows correctly, and artificial viscosity has to be introduced to remove spurious oscillations. The TVD-MacCormack model reproduces all flow regimes accurately. Finally, the TVD-MacCormack model is used to predict a laboratory-scale dyke break undertaken at Delft University of Technology. The predictions agree closely with the experimental data, and are in excellent agreement with results from an alternative Godunov-type model.     

Numerical Modeling of Flow Kinematics in a Segment of Inerodable Channel

A numerical model of a riffle segment of a large river is applied to examine the kinematic flow structure at different phases of water regime. A finite-difference scheme on an irregular triangular grid is proposed for solving shallow-water equations. The results of field measurements and calculations are presented.     

Calculating complex‐valued P‐wave first‐arrival traveltimes in attenuative vertical transversely isotropic media with an irregular surface

The complex‐valued first‐arrival traveltime can be used to describe the properties of both velocity and attenuation as seismic waves propagate in attenuative elastic media. The real part of the complex‐valued traveltime corresponds to phase arrival and the imaginary part is associated with the amplitude decay due to energy absorption. The eikonal equation for attenuative vertical transversely isotropic media discretized with rectangular grids has been proven effective and precise to calculate the complex‐valued traveltime, but less accurate and efficient for irregular models. By using the perturbation method, the complex‐valued eikonal equation can be decomposed into two real‐valued equations, namely the zeroth‐ and first‐order traveltime governing equations. Here, we first present the topography‐dependent zeroth‐ and first‐order governing equations for attenuative VTI media, which are obtained by using the coordinate transformation from the Cartesian coordinates to the curvilinear coordinates. Then, we apply the Lax–Friedrichs sweeping method for solving the topography‐dependent traveltime governing equations in order to approximate the viscosity solutions, namely the real and imaginary parts of the complex‐valued traveltime. Several numerical tests demonstrate that the proposed scheme is efficient and accurate in calculating the complex‐valued P‐wave first‐arrival traveltime in attenuative VTI media with an irregular surface.     

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.     

ASSESSMENT OF MACCORMACK SCHEME IN UNSTEADY SEDIMENT LADEN FLOW COMPUTATION

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Well-balanced high-order centred schemes for non-conservative hyperbolic systems. Applications to shallow water equations with fixed and mobile bed

This paper concerns the development of high-order accurate centred schemes for the numerical solution of one-dimensional hyperbolic systems containing non-conservative products and source terms. Combining the PRICE-T method developed in [Toro E, Siviglia A. PRICE: primitive centred schemes for hyperbolic system of equations. Int J Numer Methods Fluids 2003;42:1263–91] with the theoretical insights gained by the recently developed path-conservative schemes [Castro M, Gallardo J, Parés C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products applications to shallow-water systems. Math Comput 2006;75:1103–34; Parés C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300–21], we propose the new PRICE-C scheme that automatically reduces to a modified conservative FORCE scheme if the underlying PDE system is a conservation law. The resulting first-order accurate centred method is then extended to high order of accuracy in space and time via the ADER approach together with a WENO reconstruction technique. The well-balanced properties of the PRICE-C method are investigated for the shallow water equations. Finally, we apply the new scheme to the shallow water equations with fix bottom topography and with variable bottom solving an additional sediment transport equation.     

Topographic eddies in temporally varying oceanic flows

Abstract

In this paper we examine the behaviour of oceanic unsteady flow impinging on isolated topography by means of numerical simulation. The ocean model is quasigeostrophic and forced by an oscillatory mean flow. The fluid domain is of the channel type and open-boundary numerical conditions are used to represent downstream and upstream flow.

In certain cases, vortex shedding, either cyclonic or anticyclonic, is observed in the lee of obstacles. Such shedding can be explained as the consequence of both an enhanced process of vorticity dissipation over the topography which locally affects the balance of potential vorticity on the advective timescale, and a periodic dominance of advective effects which sweep the fluid particles trapped on the seamount. For refined resolution and smallest viscosity the model will predict flows in which the shed eddies are coherent structures with closed streamlines.

The model suggests a mechanism by which topographically generated eddies may be swept away from a seamount in the ocean.     

On the primary variable switching technique for simulating unsaturated–saturated flows

Primary variable switching appears as a promising numerical technique for variably saturated flows. While the standard pressure-based form of the Richards equation can suffer from poor mass balance accuracy, the mixed form with its improved conservative properties can possess convergence difficulties for dry initial conditions. On the other hand, variable switching can overcome most of the stated numerical problems. The paper deals with variable switching for finite elements in two and three dimensions. The technique is incorporated in both an adaptive error-controlled predictor–corrector one-step Newton (PCOSN) iteration strategy and a target-based full Newton (TBFN) iteration scheme. Both schemes provide different behaviors with respect to accuracy and solution effort. Additionally, a simplified upstream weighting technique is used. Compared with conventional approaches the primary variable switching technique represents a fast and robust strategy for unsaturated problems with dry initial conditions. The impact of the primary variable switching technique is studied over a wide range of mostly 2D and partly difficult-to-solve problems (infiltration, drainage, perched water table, capillary barrier), where comparable results are available. It is shown that the TBFN iteration is an effective but error-prone procedure. TBFN sacrifices temporal accuracy in favor of accelerated convergence if aggressive time step sizes are chosen.     

Storm surge and tidal interaction in the Tjeldsund channel, northern Norway

The aim of this study is to investigate tide–surge interaction in narrow channels with complex and relatively shallow topography. A high-resolution depth-integrated tidal and storm surge model has been implemented for the Tjeldsund channel which is an important sailing lane in northern Norway. A horizontal grid resolution down to 50 m is applied in order to represent the complex bottom topography and the formation of jets and small-scale eddies. Two typically storm surge events in December 2004 have been examined in detail. The tide–surge interaction is found to influence the generation of higher harmonics and the formation of eddies in the current field. In some cases, the magnitude of storm surge currents may reach the same magnitude as the tidal currents enhancing the formation of jets and eddies.     

Well-balanced numerical modelling of non-uniform sediment transport in alluvial rivers

The last two decades have witnessed the development and application of well-balanced numerical models for shallow flows in natural rivers.However,until now there have been no such models for flows with non-uniform sediment transport.This paper presents a 1D well-balanced model to simulate flows and non-capacity transport of non-uniform sediment in alluvial rivers.The active layer formulation is adopted to resolve the change of bed sediment composition.In the framework of the finite volume Slope Llmiter Centred(SLIC) scheme,a surface gradient method is incorporated to attain well-balanced solutions to the governing equations.The proposed model is tested against typical cases with irregular topography,including the refilling of dredged trenches,aggradation due to sediment overloading and flood flow due to landslide dam failure.The agreement between the computed results and measured data is encouraging.Compared to a non-well-balanced model,the well-balanced model features improved performance in reproducing stage,velocity and bed deformation.It should find general applications for non-uniform sediment transport modelling in alluvial rivers,especially in mountain areas where the bed topography is mostly irregular.     

Material transportation and fluid-melt activity in the subduction channel: Numerical modeling

The subduction channel is defined as a planar to wedge-like area of variable size,internal structure and composition,which forms between the upper and lower plates during slab subduction into the mantle.The materials in the channel may experience complex pressure,temperature,stress and strain evolution,as well as strong fluid and melt activity.A certain amount of these materials may subduct to and later exhume from100 km depth,forming high to ultra-high pressure rocks on the surface as widely discovered in nature.Rock deformation in the channel is strongly assisted by metamorphic fluids activities,which change composition and mechanical properties of rocks and thus affect their subduction and exhumation histories.In this study,we investigate the detailed structure and dynamics of both oceanic and continental subduction channels,by conducting highresolution petrological-thermomechanical numerical simulations taking into account fluid and melt activities.The numerical results demonstrate that subduction channels are composed of a tectonic rock melange formed by crustal rocks detached from the subducting slab and the hydrated mantle rocks scratched from the overriding plate.These rocks may either extrude sub-vertically upward through the mantle wedge to the crust of the upper plate,or exhume along the subduction channel to the surface near the suture zone.Based on our numerical results,we first analyze similarities and differences between oceanic and continental subduction channels.We further compare numerical models with and without fluid and melt activity and demonstrate that this activity results in strong weakening and deformation of overriding lithosphere.Finally,we show that fast convergence of orogens subjected to fluid and melt activity leads to strong deformation of the overriding lithosphere and the topography builds up mainly on the overriding plate.In contrast,slow convergence of such orogens leads to very limited deformation of the overriding lithosphere and the mountain building mainly occurs on the subducting plate.     

三维复杂山谷地形SV波垂直输入地震反应分析

本文基于显式有限元法研究了地震波垂直入射时三维复杂山谷地形对地震地面运动的影响,在数值分析中应用了三维化二维的解法和黏弹性人工边界的处理方法,实现了地震波垂直输入下三维复杂场地地震动数值模拟,并验证了该方法的合理性.以四川桃坪地区一山谷地形作为研究对象,基于地表高程数据分别建立了二维和三维场地模型,对比研究表明:在复杂地形情况下考虑二、三维模型时具有明显差异,三维模型能更真实地反映地形变化对地震动的影响,复杂地形条件下有必要考虑三维实际场地模型.本文对边界自由场的处理方法也可用于处理三维复杂场地地震动斜入射问题,为三维复杂地形场地地震效应研究提供参考.     

鄱阳湖二维水动力和水质耦合数值模拟

针对大型通江湖泊水位变化剧烈,地形起伏多变,岸线复杂,湖泊内部窄小洪道与大面洲滩和洼地连接复杂,湖泊洲滩出露、淹没频繁交替等湖泊水情和地貌特征,基于二维浅水方程和对流扩散方程组构建了鄱阳湖二维水动力和水质耦合模拟模型.模型采用非结构网格有限体积法进行离散,以HLLC算法计算单元界面的水量、动量和物质输运通量.水陆边界通...     

A symmetric formulation and SUPG scheme for the shallow-water equations

Shallow-water flows with supercritical and subcritical subregions often exhibit numerical difficulties because of their associated hydraulic jumps (shock waves), steep layers, and fictitious oscillations. Analogous problems in gas dynamics have led to the recent development of a promising class of Petrov-Galerkin methods specifically designed for hyperbolic/incompletely parabolic systems, and are written in a symmetric conservation form. One of the major difficulties in the application of this class of methods to shallow water problems has been the unavailability of a suitable symmetric form of the governing equations. In the present work, this issue is addressed by introducing the total energy of the water column to motivate a change of variables which symmetrizes the shallow-water conservation system. Then, the one-dimensional case is considered and a time-accurate, streamline-upwind Petrov-Galerkin (SUPG) scheme is developed based on the proposed symmetric form. Numerical results illustrate the method and permit comparison with other schemes.