Classical Shallow Water Equations Research Articles

A coupled 1D–2D hydrodynamic model for flood simulation in flood detention basin

A coupled 1D–2D hydrodynamic model linking the channel and flood detention basin for flood simulation with complex topography and irregular boundary was presented in this paper. The 1D Saint-Venant equations were used for governing flows in channel, and the four-point implicit Preissmann scheme was used for discretization. To simulate floods in flood detention basin, the 2D shallow equations were adopted as governing equations, and a well-balanced, unstructured finite-volume method was proposed for the numerical resolution. The 2D model is based on a new formulation of the classical shallow water equations in hyperbolic conservation form. The numerical fluxes are computed by HLLC algorithm, and the MUSCL–Hancock predictor–corrector scheme is used to achieve high-accuracy and high-resolution results. A simple and efficient method was proposed to reflect the coupled hydraulic connection between the channel and the flood detention basin. The novelties of the proposed model include (1) a robust method for wetting/drying treatment and (2) an efficient way to link the 1D and 2D models. The 1D, 2D, and coupled models are all tested through some benchmark cases, and numerical results validated the accuracy of the models. Furthermore, the coupled 1D–2D model was used for a real flood simulation in Jiakouwa flood detention basin, China. The flood-risk information including flood arrival time and maximal water depth was mapped using GIS. Those flood-risk maps can be used as an important decision-making basis of flood control and rescue for the flood control departments at all levels.

The Riemann problem for shallow-water equations on a step: stationary solutions, the quasi-two-layer model and numerics

For the problem of a steady-state flow over a fixed step, in the shallow-water approximation the dependence of the flow regime on the flow direction and the ratio of the fluid depth on the right of the step to the step height are found. The range of admissible values of the fluid flow as a function of the fluid mass flow and the step height is determined. We propose an arbitrary discontinuity decay model for shallow-water equations on a step boundary, which is based on the two-layer representation of fluid flows taking into account the flow peculiarities in the vicinity of a step. A distinctive feature of the suggested model is the separation of a studied flow into two layers in calculating flow quantities near each step, and thus improving approximation of depth-averaged solutions to the initial three-dimensional Euler equations. We are solving the shallow-water equations for one layer, introducing the fictitious lower layer only as an auxiliary structure in setting up the appropriate Riemann problems for the upper layer. Thus, the solution to the problem reduces to an arbitrary discontinuity decay problem for classical shallow-water equations on a plane boundary. The proposed model realizes all analytically possible configurations in the case of presence of the step boundary, qualitatively takes into account the dissipation of translational mechanical energy due to turbulence. A finite-volume numerical method for studying shallow-water flows over the step is proposed. This method uses the quasi-two-layer model of hydrodynamic flows over a stepwise boundary with an advanced consideration of the flow features near the step. The results of computations showing the efficiency of the algorithm developed are discussed.

Numerical modelling of free-surface shallow flows over irregular topography with complex geometry

A well-balanced Godunov-type finite volume algorithm is developed for modelling free-surface shallow flows over irregular topography with complex geometry. The algorithm is based on a new formulation of the classical shallow water equations in hyperbolic conservation form. Unstructured triangular grids are used to achieve the adaptability of the grid to the geometry of the problem and to facilitate localised refinement. The numerical fluxes are calculated using HLLC approximate Riemann solver, and the MUSCL-Hancock predictor–corrector scheme is adopted to achieve the second-order accuracy both in space and in time where the solutions are continuous, and to achieve high-resolution results where the solutions are discontinuous. The novelties of the algorithm include preserving well-balanced property without any additional correction terms and the wet/dry front treatments. The good performance of the algorithm is demonstrated by comparing numerical and theoretical results of several benchmark problems, including the preservation of still water over a two-dimensional hump, the idealised dam-break flow over a frictionless flat rectangular channel, the circular dam-break, and the shock wave from oblique wall. Besides, two laboratory dam-break cases are used for model validation. Furthermore, a practical application related to dam-break flood wave propagation over highly irregular topography with complex geometry is presented. The results show that the algorithm can correctly account for free-surface shallow flows with respect to its effectiveness and robustness thus has bright application prospects.

Pressure Coefficient in Dam-Break Flows of Dry Granular Matter

AbstractThe propagation of dry granular flows, such as rock and snow avalanches, can be described by depth-averaged models. Different from classical shallow-water equations, these models take into account the anisotropy of normal stresses inside the flowing pile through using an earth pressure coefficient in the pressure term. A new regularization function for calculating the pressure coefficient in the Savage-Hutter-type models at the early stages of dam-break flows and collapses is proposed. In such circumstances the flow lines are significantly curved with respect to the basal surface and a special treatment of the earth-pressure coefficient is required for obtaining a satisfactory agreement with experimental data. The comparison between numerical simulations and laboratory experimental data shows an apparent improvement in describing the early stages of dam-break waves over rough beds. The comparison with experiments over smooth bed surface exhibits minor evidence of improvement. Nonetheless, in this ...

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

In this article the weakly compressible two-phase diffuse interface method (DIM) for the simulation of complex two-dimensional non-hydrostatic free surface flows proposed by Dumbser in [24] is extended to three-dimensional unstructured tetrahedral meshes. As in the 2D case, a reduced version of the Baer–Nunziato model for compressible multiphase flows is used. The physical model is closed by the Tait equation of state for water and can be implemented easily into existing compressible codes based on high resolution shock capturing finite volume schemes. Since the proposed model is fully three-dimensional, it includes the fluid accelerations in gravity direction and hence does not assume a hydrostatic pressure distribution, like the classical shallow water equations. Furthermore, the 3D two-phase model can naturally deal also with breaking waves. To solve the system of conservation laws of mass and momentum coupled with the non-conservative evolution equation of the fluid volume fraction, a high order path-conservative one-step WENO finite volume scheme is applied, together with a new generalized Osher-type Riemann solver at the element interfaces. The accurate Riemann solver in combination with a high order finite volume approach leads to a simple but sharp resolution of the free surface.A thorough comparison of experimental reference data with the computational results obtained for a large set of three-dimensional test cases shows the suitability of the present approach for the accurate simulation of complex three-dimensional free surface flows. The use of a compressible flow model allows the method to simulate both, low speed and high speed free surface flow problems, which makes the approach applicable to a very wide class of environmental and industrial free surface flow problems.

Water waves over a rough bottom in the shallow water regime

This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem of homogenization theory. In this setting we derive a new model system of equations, consisting of the classical shallow water equations coupled with nonlocal evolution equations for a periodic corrector term. We also exhibit a new resonance phenomenon between surface waves and a periodic bottom. This resonance, which gives rise to secular growth of surface wave patterns, can be viewed as a nonlinear generalization of the classical Bragg resonance. We justify the derivation of our model with a rigorous mathematical analysis of the scaling limit and the resulting error terms. The principal issue is that the shallow water limit and the homogenization process must be performed simultaneously. Our model equations and the error analysis are valid for both the two- and the three-dimensional physical problems.

A Shallow Water model for the numerical simulation of overland flow on surfaces with ridges and furrows

We introduce a new Shallow Water model for the numerical simulation of overland flow with furrow effects without representing them explicitly. The model is obtained by adding an anisotropic friction term that takes into account these effects to the classical Shallow Water equations. We validate the model with numerical tests, and we compare it with the classic Shallow Water model in which the furrows are explicitly and precisely described.

A Boussinesq-type model incorporating random wave-breaking

A recent Boussinesq-type model is herein modified to account for breaking waves in shallow water. The model is based on a system of equations in terms of surface elevation and depth-averaged horizontal velocities, in two horizontal dimensions for fully dispersive and weakly nonlinear random waves over any finite water depth. The formulation involves five terms in each momentum equation, including the classical shallow-water equation terms, and only one frequency dispersion term. This work extends the model by including depth-induced wave-breaking in one horizontal dimension, based on the eddy viscosity and surface roller criteria. The modified model was applied to simulate the propagation and wave-breaking of regular and random waves using a simple explicit finite difference scheme. The simulation results were compared with experimental data and with results from one of the most widespread commercial Boussinesq wave models, indicating good agreement in most cases.

A new well-balanced Hermite weighted essentially non-oscillatory scheme for shallow water equations

Hermite weighted essentially non-oscillatory (HWENO) methods were introduced in the literature, in the context of Euler equations for gas dynamics, to obtain high-order accuracy schemes characterized by high compactness (e.g. Qiu and Shu, J. Comput. Phys. 2003; 193:115). For example, classical fifth-order weighted essentially non-oscillatory (WENO) reconstructions are based on a five-cell stencil whereas the corresponding HWENO reconstructions are based on a narrower three-cell stencil. The compactness of the schemes allows easier treatment of the boundary conditions and of the internal interfaces. To obtain this compactness in HWENO schemes both the conservative variables and their first derivatives are evolved in time, whereas in the original WENO schemes only the conservative variables are evolved. In this work, an HWENO method is applied for the first time to the shallow water equations (SWEs), including the source term due to the bottom slope, to obtain a fourth-order accurate well-balanced compact scheme. Time integration is performed by a strong stability preserving the Runge–Kutta method, which is a five-step and fourth-order accurate method. Besides the classical SWE, the non-homogeneous equations describing the time and space evolution of the conservative variable derivatives are considered here. An original, well-balanced treatment of the source term involved in such equations is developed and tested. Several standard one-dimensional test cases are used to verify the high-order accuracy, the C-property and the good resolution properties of the model. Copyright © 2010 John Wiley & Sons, Ltd.

The Riemann Problem for the one-dimensional, free-surface Shallow Water Equations with a bed step: Theoretical analysis and numerical simulations

In this paper, the solution of the Riemann Problem for the one-dimensional, free-surface Shallow Water Equations over a bed step is analyzed both from a theoretical and a numerical point of view. Particular attention has been paid to the wave that is generated at the location of the bed discontinuity. Starting from the classical Shallow Water Equations, considering the bed level as an additional variable, and adding to the system an equation imposing its time invariance, we show that this wave is a contact wave, across which one of the Riemann invariants, namely the energy, is not constant. This is due to the fact that the relevant problem is nonconservative. We demonstrate that, in this type of system, Riemann Invariants do not generally hold in contact waves. Furthermore, we show that in this case the equations that link the flow variables across the contact wave are the Generalized Rankine–Hugoniot relations and we obtain these for the specific problem. From the numerical point of view, we present an accurate and efficient solver for the step Riemann Problem to be used in a finite-volume Godunov-type framework. Through a two-step predictor–corrector procedure, the solver is able to provide solutions with any desired accuracy. The predictor step uses a well-balanced Generalized Roe solver while the corrector step solves the exact nonlinear system of equations that consitutes the problem by means of an iterative procedure that starts from the predictor solution. In order to show the effectiveness and the accuracy of the proposed approach, we consider several step Riemann Problems and compare the exact solutions with the numerical results obtained by using a standard Roe approach far from the step and the novel two-step algorithm for the fluxes over the step, achieving good results.

Unstructured finite volume discretization of two‐dimensional depth‐averaged shallow water equations with porosity

AbstractThis paper deals with the numerical discretization of two‐dimensional depth‐averaged models with porosity. The equations solved by these models are similar to the classic shallow water equations, but include additional terms to account for the effect of small‐scale impervious obstructions which are not resolved by the numerical mesh because their size is smaller or similar to the average mesh size. These small‐scale obstructions diminish the available storage volume on a given region, reduce the effective cross section for the water to flow, and increase the head losses due to additional drag forces and turbulence. In shallow water models with porosity these effects are modelled introducing an effective porosity parameter in the mass and momentum conservation equations, and including an additional drag source term in the momentum equations. This paper presents and compares two different numerical discretizations for the two‐dimensional shallow water equations with porosity, both of them are high‐order schemes. The numerical schemes proposed are well‐balanced, in the sense that they preserve naturally the exact hydrostatic solution without the need of high‐order corrections in the source terms. At the same time they are able to deal accurately with regions of zero porosity, where the water cannot flow. Several numerical test cases are used in order to verify the properties of the discretization schemes proposed. Copyright © 2009 John Wiley & Sons, Ltd.

Shallow water equations on a movable bottom

Nonlinear dispersion Zheleznyak-Pelinovskii equations describing surface water waves are derived in the paper for the case of an unsteady bottom surface. It is shown that the obtained equations and the well-known Green-Naghdi equations are different forms of the same system of shallow water equations of the second approximation taking into account the bottom variations. The issue of wave generation by a movable bottom is rather topical nowadays. It is related to mathematical simulations of long surface waves (tsunami) caused by the sea bottom fluctuations due to landslides or the formation of extended clefts (3, 4, 7). This class also includes the problems of motion of bodies over a reservoir bottom, which may cause waves on the water surface (2). The simulation of surface waves using complete hydrodynamic models requires much computation time (9, 10). Thus, wave patterns appearing in the motion of an underwater landslide were studied numerically in (9). The specifics of simulation of such waves is determined by a large time period of the landslide movement from the shore to the deep part of the reservoir, by the sufficiently high speed of its movement, and, as a consequence, by the necessity to use extended calculation domains and grids with a large number of nodes. As a result, computation of a single variant by the complete model takes several hours. On the other hand, approximate models require a few minutes of computation and the general picture of the wave modes is satisfactory described even by simplest models. However, it has been noted that a detailed simulation of a long-time phe- nomenon requires models capable to reproduce the variance and the inhomogeneity of the process in the vertical direction. For example, this was shown in (7, 14) where various approximate and complete hydrodynamic equations were used for the study of the generation of waves by a solid rigid body moving along a slope. Moreover, experimental physical data were used in (7) for the study of the formation of waves. Comparative analysis shows that in the case of a long thin landslide at the ini- tial stage of the process all models, from the classic shallow water equations to the complete model of ideal liquid flows, sufficiently well describe the most noticeable wave-forming characteristics observed in experiments. Even the linear shallow wa- ter model gives a pattern sufficiently close to experimental data at the initial stage � Institute of Computational Technologies, Siberian Branch of the Russian Academy of Sciences,

Gravity-driven surges of a viscoplastic fluid: An experimental study

We present a new experimental setup dedicated to understanding the dynamics of gravity-driven free-surface surges made of viscoplastic fluids. It consists of a 3-m-long and 0.4-m-wide inclined channel whose bottom is constituted by an upward-moving conveyor belt with controlled velocity. This setup allows to create gravity-driven surges that are stationary in the laboratory frame. Series of experiments have been conducted using a kaolin slurry characterized by a Herschel-Bulkley rheology with a yield stress of the order of 10 Pa. The shape of the free-surface was monitored by means of a laser sheet projected with a low incidence angle and ultrasonic height sensors. The obtained stationary surges systematically present steep fronts followed by a zone of uniform fluid height. We report on the evolution of surge characteristics (uniform height, front shape) as a function of belt velocity. These results are then compared to the predictions of a theoretical model based on thin-layer approximation. Searching for traveling wave solutions to the classical shallow-water equations, we obtain an ordinary differential equation for the shape of the waves which, in the case of a Herschel-Bulkley fluid, is integrated numerically. The agreement between free-surface shapes computed from the model and those extracted from the experiments is good, including in the vicinity of the surge fronts. Lastly, we discuss perspectives offered by our setup for the study of viscoplastic surges with reference, in particular, to natural debris flows.

Interaction of Shallow Water Waves

In this paper, we consider the Riemann problem and interaction of elementary waves for a nonlinear hyperbolic system of conservation laws that arises in shallow water theory. This class of equations includes as a special case the equations of classical shallow water equations. We study the bore and dilatation waves and their properties, and show the existence and uniqueness of the solution to the Riemann problem. Towards the end, we discuss numerical results for different initial data along with all possible interactions of elementary waves. It is noticed that in contrast to the p‐system, the Riemann problem is solvable for arbitrary initial data, and its solution does not contain vacuum state.

Gas-dynamic analogy for vortex free-boundary flows

The classical shallow-water equations describing the propagation of long waves in flow without a shear of the horizontal velocity along the vertical coincide with the equations describing the isentropic motion of a polytropic gas for a polytropic exponent γ = 2 (in the theory of fluid wave motion, this fact is called the gas-dynamic analogy). A new mathematical model of long-wave theory is derived that describes shear free-boundary fluid flows. It is shown that in the case of one-dimensional motion, the equations of the new model coincide with the equations describing nonisentropic gas motion with a special choice of the equation of state, and in the multidimensional case, the new system of long-wave equations differs significantly from the gas motion model. In the general case, it is established that the system of equations derived is a hyperbolic system. The velocities of propagation of wave perturbations are found.

Depth-integrated flow modelling taking into account bottom curvature

Successfully modelling flows over a spillway and on strongly vertically curved bottoms is a challenge for any depth-integrated model. This type of computation requires the use of axes properly inclined along the mean flow direction in the vertical plane and a modelling of curvature effects. The proposed generalized model performs such computations by means of suitable curvilinear coordinates in the vertical plane, leading to a fully integrated approach. This means that the flows in the upstream reservoir, on the spillway, in the stilling basin and in the downstream river reach are all handled in a single simulation. The velocity profile is generalized in comparison with the uniform one usually assumed in the classical shallow water equations. The pressure distribution is modified as a function of the bottom curvature and is thus not purely hydrostatic. Representative test cases, as well as the application of the extended model to the design of a large hydraulic structure in Belgium, lead to satisfactory validation results

Modeling 3D Supercritical Flow with Extended Shallow-Water Approach

Despite the three-dimensional (3D) nature of the flow, the classical shallow-water equations are often used to simulate supercritical flow in channel transitions. A closer comparison with experimental data, however, often shows large discrepancies in the height and pattern of the shock waves that increase with the Froude number. An extension to the classical shallow-water approach is derived considering higher-order distribution functions for pressure and horizontal and vertical velocities, therefore taking nonhydrostatic pressure distribution and vertical momentum into account. The approach is applied to highly supercritical flow in a channel contraction ( F0 =4.0) , a channel junction ( F0 ≈4.5 for both branches), and a channel expansion ( F0 =8.0) . Specific problems of such flows—wetting and drying of computational cells and wave breaking due to steep free-surface gradients—are discussed and solved numerically. The solutions with the extended approach are compared both with experimental data and class...

Particular solutions and Riemann problem for modified shallow water equations

In this article gas-dynamic analogy for shallow water equations is generalized in the case when initial conditions depend on vertical coordinate. Simple parameterization of advective term allowing full theoretical analysis of solutions of simple waves and Riemann problem for modified shallow water equations is suggested. The simple wave solutions obtained have permitted to find dimensionless parameter, which restricts limits of applicability of classical shallow water equations and neglecting advective impulse transfer. Solution of the initial discontinuity decay problem for modified shallow water equations is found.

Generalized Boltzmann equation for shallow water flows

This invited review paper introduces the Boltzmann-based approach for the numerical modelling of surface water flows to hydroinformaticians. The paper draws upon earlier work by our group as well as others. This review formulates the generalized Boltzmann equation for 1D and 2D shallow water flows and shows that the statistical moments of these generalized equations provide the classical continuity and momentum equations in shallow waters. The connection between the generalized Boltzmann equation and classical shallow water equations provides a framework for formulating new computational approaches to surface water flows. To illustrate, a first-order explicit scheme based on the generalized Boltzmann equation for 1D shallow waters in frictionless and horizontal channels is formulated. The resulting scheme is applied to the classical dam break problem. Comparison with the analytical solution shows that the Boltzmann-based scheme is highly accurate and free of spurious oscillations, illustrating the potential of the method for surface water problems and other applications.

Solitary Wave Runup on Plane Slopes

This study deals with the runup of solitary waves on a uniform plane beach connected to an open ocean of constant depth. The waves are nonbreaking during the runup process. A nonlinear solution to the classical shallow water equation, that describes the wave characteristics on the beach, is obtained analytically by using a hodograph transformation. It was found that the nonlinear theory agreed well with experimental results. The maximum runup predicted by the nonlinear theory is larger than that predicted earlier and the correction is on the order of the offshore relative wave height for a given slope. This correction for nonbreaking waves on beaches decreases as the beach slope steepens.