Classical Shallow Water Equations Research Articles

Nonlinear Optical Pulses in Media with Asymmetric Gain.

A generic novel model governing optical pulse propagation in a nonlinear dispersive amplifying medium with asymmetric (linear spectral slope) gain is introduced. We examine the properties of asymmetric optical pulses formed in such gain-skewed media, both theoretically and numerically. We derive a dissipative optical modification of the classical shallow water equations that highlights an analogy between this phenomenon and hydrodynamic wave breaking. These findings provide insight into the nature of asymmetric optical pulses capable of accumulating large nonlinear phase without wave breaking, a crucial aspect in the design of nonlinear fiber amplifiers.

A hybrid shallow water approach with unstructured triangular grids for urban flood modeling

Classic shallow water equations (SWEs) and their various simplified approaches provide the theoretical basis for urban flood modeling. Aiming at the limitations of the local inertial approach (LInA), this paper proposes a hybrid approach that combines the LInA and classic SWEs to eliminate the errors caused by LInA. Results show that the hybrid approach can reduce the error from 7% to 1.5%, with slightly reduced computation efficiency in the dam break case. In the urban flood case, the hybrid approach can reduce the relative error on flood depth from 50% to 20% in a few grids, while the relative error on flood velocity is improved from 60% to below 30%. The hybrid approach proposed here can provide a practical strategy to eliminate the errors caused by the inherent defects of the simplified approach and still maintain the computation efficiency.

Reappraisal of Whitham’s 1967 theory for wave–meanflow interaction in shallow water

The modulation equations for Stokes waves in shallow water coupled to wave-generated meanflow, derived in Whitham (1967), based on an averaged Lagrangian are re-visited. Firstly, it is shown that they can be recast into two coupled classical shallow water equations, with modified gravity having the sign of the Whitham index: sign(ω0′′ω2). Secondly, it is shown that the amplitude of the meanflow and amplitude of the wave are, in general, independent. Thirdly, the implications of the coalescing characteristics, whose unfolding is associated with the Benjamin–Feir instability, are studied.

Topography-based local spherical Voronoi grid refinement on classical and moist shallow-water finite-volume models

Abstract. Locally refined grids for global atmospheric models are attractive since they are expected to provide an alternative to solve local phenomena without the requirement of a global high-resolution uniform grid, whose computational cost may be prohibitive. Spherical centroidal Voronoi tessellation (SCVT), as used in the atmospheric Model for Prediction Across Scales (MPAS), allows a flexible way to build and work with local refinement. In addition, the Andes Range plays a key role in the South American weather, but it is hard to capture its fine-structure dynamics in global models. This paper describes how to generate SCVT grids that are locally refined in South America and that also capture the sharp topography of the Andes Range by defining a density function based on topography and smoothing techniques. We investigate the use of the mimetic finite-volume scheme employed in the MPAS dynamical core on this grid considering the nonlinear classic and moist shallow-water equations on the sphere. We show that the local refinement, even with very smooth transitions from different resolutions, generates spurious numerical inertia–gravity waves that may even numerically destabilize the model. In the moist shallow-water model, wherein physical processes such as precipitation and cloud formation are included, our results show that the local refinement may generate spurious rain that is not observed in uniform-resolution SCVT grids. Fortunately, the spurious waves originate from small-scale grid-related numerical errors and can therefore be mitigated using fourth-order hyperdiffusion. We exploit a grid geometry-based hyperdiffusion that is able to stabilize spurious waves and has very little impact on the total energy conservation. We show that, in some cases, the clouds are better represented in a variable-resolution grid when compared to a respective uniform-resolution grid with the same number of cells, while in other cases, grid effects can affect the cloud and rain representation.

Numerical ability of hyperbolic flux solvers to compute 2D shear layers in turbulent shallow flows

Current computational power allows the modelling of complex time dependent flows. In particular, shallow water flows are frequently simulated to achieve solutions for different problems in the field of civil and environmental engineering. The mesh resolution and related computational cost are primarily associated to the scale of the flow structures to be investigated. Nonetheless, turbulence also plays an important role on computational cost since it participates in the generation, shedding and support of such flow structures. Different mathematical models can be considered to numerically simulate the turbulence of unsteady shallow flows depending on the degree of turbulent scale resolution required. Depending on the adopted approach, the level of accuracy required (i.e. the range of scales that must be resolved with a low diffusive and dispersive error) is different. Such accuracy, namely the dispersive and dissipative characteristic, is directly related with the numerical scheme used to discretize the equations. In finite volume schemes, the range of scales of turbulent motion that a numerical model can accurately resolve strongly depends on the Riemann solver used (via its intrinsic numerical diffusion), apart from the order of accuracy and degrees of freedom of the method. In this work, we aim at the analysis of two well-known Riemann solvers in the framework of the classical shallow water equations (i.e. considering the full convective terms and neglecting dissipation): the ARoe and HLLS solvers. An important difference between the ARoe and HLLS solvers is the numerical diffusion inherent to each of them. This artificial diffusion combined with the mesh resolution determine the cut-off scale resolved by each numerical technique. For this purpose, we assess the suitability of each solver by means of the analysis of the kinetic energy cascade of the numerical solution using a double shear layer configuration. This analysis is combined with the examination of the analytical expression of he approximate solution for a shear wave, provided by the aforementioned solvers. The study herein presented allows to assess whether or not all the relevant turbulent flow structures are resolved and if the phenomenon of interest is thus accurately modeled. The numerical results evidence that a diffusive profile appears at the shear line during the first steps of the simulation, determining the duration of the linear regime prior to the turbulent motion. The strength of this profile, shown to be higher for the HLLS solver, is associated to the numerical diffusion of the solver. The analysis of the energy cascade also agrees with this observation.

Well-posedness and derivative blow-up for a dispersionless regularized shallow water system

We study local-time well-posedness and breakdown for solutions of regularized Saint-Venant equations (regularized classical shallow water equations) recently introduced by Clamond and Dutykh. The system is linearly non-dispersive, and smooth solutions conserve an H1-equivalent energy. No shock discontinuities can occur, but the system is known to admit weakly singular shock-profile solutions that dissipate energy. We identify a class of small-energy smooth solutions that develop singularities in the first derivatives in finite time.

An extended linear shallow-water equation

An extension to the classical shallow-water equation (SWE) is derived that exactly satisfies the bed condition and can be regarded as an approximation to wave scattering at the next order in the small parameter $(h/\unicode[STIX]{x1D706})^{2}$ (depth to wavelength ratio squared). In the frequency domain, the extended SWE shares the same simple structure as the standard SWE with coefficients modified by terms relating to the bed variation. In three dimensions the governing equation demonstrates that variable topography gives rise to anisotropic effects on wave scattering not present in the standard SWE, with consequences for the design of water wave metamaterials. Numerical examples illustrate that approximations to wave scattering using the extended SWE are significantly improved in comparison with the standard SWE.

Numerical simulations of hydraulic jumps with the Shear Shallow Water model

An extension and numerical approximation of the shear shallow water equations model recently proposed in [21] is considered in this work. The model equations are able to describe the oscillatory nature of turbulent hydraulic jumps and as such correct the deficiency of the classical shallow water equations in describing such phenomena. The model equations, orig- inally developed for horizontal flow or flows occurring over small constant slopes, are straight- forwardly extended here for modeling flows over non-constant slopes and numerically solved by a second-order well-balanced finite volume scheme. Further, a new set of exact solutions to the extended model equations are derived and several numerical tests are performed to validate the numerical scheme and its ability to predict the oscillatory nature of hydraulic jumps under different conditions.

Modelling urban floods using a finite element staggered scheme with an anisotropic dual porosity model

In porosity models for urban flooding, artificial porosity is used as a statistical descriptor of the urban medium. Buildings are treated as subgrid-scale features and, even with the use of relatively coarse grids, their effects on the flow are accounted for. Porosity models are attractive for large-scale applications due to limited computational demand with respect to solving the classical Shallow Water Equations on high-resolution grids. In the last decade, effective schemes have been developed that allowed accounting for a wealth of sub-grid processes; unfortunately, they are known to suffer from oversensitivity to mesh design in the case of anisotropic porosity fields, which are typical of urban layouts. In the present study, a dual porosity approach is implemented into a two-dimensional Finite Element numerical scheme that uses a staggered unstructured mesh. The presence of buildings is modelled using an isotropic porosity in the continuity equation, to account for the reduced water storage, and a tensor formulation for conveyance porosity in the momentum equations, to account for anisotropy and effective flow velocity. The element-by-element definition of porosities, and the use of a staggered grid in which triangular cells convey fluxes and continuity is balanced at grid nodes, allow avoiding undesired mesh-dependency. Tested against refined numerical solutions and data from a laboratory experiment, the model provided satisfactory results. Model limitations are discussed in view of applications to more complex, real urban layouts.

Finite-volume schemes for shallow-water equations

Shallow-water equations are widely used to model water flow in rivers, lakes, reservoirs, coastal areas, and other situations in which the water depth is much smaller than the horizontal length scale of motion. The classical shallow-water equations, the Saint-Venant system, were originally proposed about 150 years ago and still are used in a variety of applications. For many practical purposes, it is extremely important to have an accurate, efficient and robust numerical solver for the Saint-Venant system and related models. As their solutions are typically non-smooth and even discontinuous, finite-volume schemes are among the most popular tools. In this paper, we review such schemes and focus on one of the simplest (yet highly accurate and robust) methods: central-upwind schemes. These schemes belong to the family of Godunov-type Riemann-problem-solver-free central schemes, but incorporate some upwinding information about the local speeds of propagation, which helps to reduce an excessive amount of numerical diffusion typically present in classical (staggered) non-oscillatory central schemes. Besides the classical one- and two-dimensional Saint-Venant systems, we will consider the shallow-water equations with friction terms, models with moving bottom topography, the two-layer shallow-water system as well as general non-conservative hyperbolic systems.

Wave–Current Interaction: A 2DH Model for Turbulent Jet and Bottom-Friction Dissipation

A correct representation of the non-linear interactions between waves and currents is one of the key points when studying the morphological evolution of nearshore environments, in particular close to river mouths or tidal inlets. Undoubtedly, the numerical modelling of similar phenomena can be very complex and computationally demanding, given the size of the domains. In the present paper, a two‐dimensional horizontal (2DH) numerical model is applied to investigate the hydrodynamics of a turbulent jet current interacting with frontal waves, preparatory to the study of morphodynamical processes. The purpose is to reproduce accurately the turbulence of the current flow, which develops in both vertical and horizontal planes, even with the simplifications of depth-averaged velocities. Moreover, the bottom shear stress induces a mechanism of dissipation, which acts both on the jet hydrodynamics and on the wave field. Significant attention is given to this process, which turns out to be crucial in shallow waters. The present model, based on classic shallow-water equations and wave action balance, is applied to a literature test. Comparisons with theoretical and numerical outcomes are shown, the latter obtained with a quasi-3D model.

Weighted interior penalty discretization of fully nonlinear and weakly dispersive free surface shallow water flows

In this paper, we further investigate the use of a fully discontinuous Finite Element discrete formulation for the study of shallow water free surface flows in the fully nonlinear and weakly dispersive flow regime. We consider a decoupling strategy in which we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects. This source term can be computed through the resolution of elliptic second-order linear sub-problems, which only involve second order partial derivatives in space. We then introduce an associated Symmetric Weighted Internal Penalty discrete bilinear form, allowing to deal with the discontinuous nature of the elliptic problem's coefficients in a stable and consistent way. Similar discrete formulations are also introduced for several recent optimized fully nonlinear and weakly dispersive models. These formulations are validated again several benchmarks involving h-convergence, p-convergence and comparisons with experimental data, showing optimal convergence properties.

Shallow water dynamics on linear shear flows and plane beaches

Long waves in shallow water propagating over a background shear flow towards a sloping beach are being investigated. The classical shallow-water equations are extended to incorporate both a background shear flow and a linear beach profile, resulting in a non-reducible hyperbolic system. Nevertheless, it is shown how several changes of variables based on the hodograph transform may be used to transform the system into a linear equation which may be solved exactly using the method of separation of variables. This method can be used to investigate the run-up of a long wave on a planar beach including the development of the waterline.

Dual integral porosity shallow water model for urban flood modelling

With CPU times 2 to 3 orders of magnitude smaller than classical shallow water-based models, the shallow water equations with porosity are a promising tool for large-scale modelling of urban floods. In this paper, a new model formulation called the Dual Integral Porosity (DIP) model is presented and examined analytically and computationally with a series of benchmark tests. The DIP model is established from an integral mass and momentum balance whereby both porosity and flow variables are defined separately for control volumes and boundaries, and a closure scheme is introduced to link control volume- and boundary-based flow variables. Previously developed Integral Porosity (IP) models were limited to a single set of flow variables. A new transient momentum dissipation model is also introduced to account for the effects of sub-grid scale wave action on porosity model solutions, effects which are validated by fine-grid solutions of the classical shallow-water equations and shown to be important for achieving similarity in dam-break solutions. One-dimensional numerical test cases show that the proposed DIP model outperforms the IP model, with significantly improved wave propagation speeds, water depths and discharge calculations. A two-dimensional field scale test case shows that the DIP model performs better than the IP model in mapping the floods extent and is slightly better in reproducing the anisotropy of the flow field when momentum dissipation parameters are calibrated.

A discontinuous Galerkin method for a new class of Green–Naghdi equations on simplicial unstructured meshes

In this paper, we introduce a discontinuous Finite Element formulation on simplicial unstructured meshes for the study of free surface flows based on the fully nonlinear and weakly dispersive Green-Naghdi equations. Working with a new class of asymptotically equivalent equations, which have a simplified analytical structure, we consider a decoupling strategy: we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non-hydrostatic effects and we show that this source term can be computed through the resolution of scalar elliptic second-order sub-problems. The assets of the proposed discrete formulation are: (i) the handling of arbitrary unstructured simplicial meshes, (ii) an arbitrary order of approximation in space, (iii) the exact preservation of the motionless steady states, (iv) the preservation of the water height positivity, (v) a simple way to enhance any numerical code based on the nonlinear shallow water equations. The resulting numerical model is validated through several benchmarks involving nonlinear wave transformations and run-up over complex topographies.

Momentum balance in the shallow water equations on bottom discontinuities

This work investigates the topical problem of balancing the shallow water equations over bottom steps of different heights. The current approaches in the literature are essentially based on mathematical analysis of the hyperbolic system of balance equations and take into account the relevant progresses in treating the non-conservative form of the governing system in the framework of path-conservative schemes. An important problem under debate is the correct position of the momentum balance closure when the bottom elevation is discontinuous. Cases of technical interest are systematically analysed, consisting of backward-facing steps and forward-facing steps, tackled supercritical and subcritical flows; critical (sonic) conditions are also analysed and discussed.The fundamental concept governing the problem and supported by the present computations is that the energy-conserving approach is the only approach that is consistent with the classical shallow water equations formulated with geometrical source terms and that the momentum balance is properly closed if a proper choice of a conventional depth on the bottom step is performed. The depth on the step is shown to be included between the depths just upstream and just downstream of the step. It is also shown that current choices (as given in the literature) of the depth on (or in front of) the step can lead to unphysical configurations, similar to some energy-increasing solutions.

Hydromorpho: A coupled model for unsteady Stokes/Exner equations and numerical results with Feel++ library

We propose to couple the Exner equation with the Stokes equations to model the bedload sediment in geophysical flows . This work is a preliminary study to directly model the hydrodynamic flow by the unsteady Stokes equation instead of the classical shallow water equation. We focus in this proceeding on the coupling applying fluid structure interaction approach to morphodynamical behavior. In other words, we follow the approach of fluid interaction models replacing the structure equation by the Exner equation. The aim of this work is to validate the proposed procedure. These equations are solved by finite element method using the library FEEL++.

Characteristics of finite difference methods for dispersive shallow water equations

AbstractThe paper contains a description of the most important properties of numerical methods for solving nonlinear dispersive hydrodynamic equations and their distinctions from similar properties of finite difference schemes approximating classic dispersion-free shallow water equations.

Well-balanced central schemes for systems of shallow water equations with wet and dry states

In this paper we propose a new well-balanced unstaggered central finite volume scheme for the shallow water equations on variable bottom topographies, with wet and dry states. Based on a special piecewise linear reconstruction of the cell-centered numerical solution and a careful discretization of the system of partial differential equations, the proposed numerical scheme ensures both a well-balanced discretization and the positivity requirement of the water height component. More precisely, the well-balanced requirement is fulfilled by following the surface gradient method, while the positivity requirement of the computed water height component is ensured by following a new technique specially designed for the unstaggered central schemes. The developed scheme is then validated and classical shallow water equation problems on variable bottom topographies with wet and dry states are successfully solved. The reported numerical results confirm the potential and efficiency of the proposed method.

A space-time discontinuous Galerkin method for Boussinesq-type equations

In this work we propose a new high order accurate, fully implicit space-time discontinuous Galerkin (DG) method for advection-diffusion-dispersion equations, i.e. for so-called Korteweg-de-Vries-type equations. In particular, we focus on Boussinesq-type models for free surface flows, which are used for the modeling of water waves that travel in deep water, where the classical shallow water equations are not valid any more. Our method follows the ideas of the local DG method (LDG) for dispersion equations proposed by Yan and Shu (2002), who used an explicit Runge–Kutta method to integrate their scheme in time. However, such explicit time integrators applied to dispersive equations imply a very severe restriction on the time step, which has to be taken proportional to the cube of the mesh spacing, and which therefore can make even one-dimensional computations prohibitively expensive on fine grids. For the scalar case and with some simplifying assumptions, the scheme presented in this paper can be proven to be unconditionally stable in L2 norm. Furthermore, our method is based directly on a space-time finite element formulation, which also provides a natural way to discretize third order dispersive terms that contain a mixed space-time derivative. Such terms appear frequently in the context of Boussinesq-type models for free surface flows. We will show numerical convergence studies for linear scalar advection-diffusion-dispersion equations and furthermore, we will also study the convergence of our method using solitary wave solutions of the nonlinear Boussinesq-type model of Madsen, Murray and Sørensen (1991).