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Water Equations (water + equation)

Distribution by Scientific Domains
Engineering97%

Kinds of Water Equations

  • shallow water equation
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  • two-dimensional shallow water equation
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    Selected Abstracts

    Well-balanced finite volume evolution Galerkin methods for the shallow water equations with source terms

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005
    M. Luká, ová-Medvid'ová
    Abstract The goal of this paper is to present a new well-balanced genuinely multi-dimensional high-resolution finite volume evolution Galerkin method for systems of balance laws. The derivation of the method will be illustrated for the shallow water equation with geometrical source term modelling the bottom topography. The results can be generalized to more complex systems of balance laws. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    Reduced-complexity flow routing models for sinuous single-thread channels: intercomparison with a physically-based shallow-water equation model

    EARTH SURFACE PROCESSES AND LANDFORMS, Issue 5 2009
    A. P. Nicholas
    Abstract Reduced-complexity models of fluvial processes use simple rules that neglect much of the underlying governing physics. This approach is justified by the potential to use these models to investigate long-term and/or fundamental river behaviour. However, little attention has been given to the validity or realism of reduced-complexity process parameterizations, despite the fact that the assumptions inherent in these approaches may limit the potential for elucidating the behaviour of natural rivers. This study presents two new reduced-complexity flow routing schemes developed specifically for application in single-thread rivers. Output from both schemes is compared with that from a more sophisticated model that solves the depth-averaged shallow water equations. This comparison provides the first demonstration of the potential for deriving realistic predictions of in-channel flow depth, unit discharge, energy slope and unit stream power using simple flow routing schemes. It also highlights the inadequacy of modelling unit stream power, shear stress or sediment transport capacity as a function of local bed slope, as has been common practice in a number of previous reduced-complexity models. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Solution of non-linear dispersive wave problems using a moving finite element method

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2007
    Abigail Wacher
    Abstract The solution of the fully non-linear time-dependent two-dimensional shallow water equations is considered. Dispersive effects due to the Coriolis forces are taken into account. Such effects are of major importance in geophysical fluid dynamics applications. The recently proposed string gradient weighted moving finite element method is extended for this class of problems. This method simultaneously determines, at each time step, the solution of the governing partial differential equations and an optimal location of the finite element nodes. It has previously been applied to non-dispersive wave problems; here its performance under the demanding conditions of large Coriolis forces, inducing large mesh and field rotation, is studied. Optimal rates of convergence are obtained. Results for some example problems of water hump release are presented. Non-linear and linearized solutions are compared. Copyright © 2006 John Wiley & Sons, Ltd. [source]

    Hybrid finite element/volume method for shallow water equations

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2010
    Shahrouz Aliabadi
    Abstract A hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two-fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non-conservation form and solve the non-linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell-center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix-free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method. Copyright © 2010 John Wiley & Sons, Ltd. [source]

    Shoreline tracking and implicit source terms for a well balanced inundation model

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2010
    Giovanni FranchelloArticle first published online: 31 JUL 200
    Abstract The HyFlux2 model has been developed to simulate severe inundation scenario due to dam break, flash flood and tsunami-wave run-up. The model solves the conservative form of the two-dimensional shallow water equations using the finite volume method. The interface flux is computed by a Flux Vector Splitting method for shallow water equations based on a Godunov-type approach. A second-order scheme is applied to the water surface level and velocity, providing results with high accuracy and assuring the balance between fluxes and sources also for complex bathymetry and topography. Physical models are included to deal with bottom steps and shorelines. The second-order scheme together with the shoreline-tracking method and the implicit source term treatment makes the model well balanced in respect to mass and momentum conservation laws, providing reliable and robust results. The developed model is validated in this paper with a 2D numerical test case and with the Okushiri tsunami run up problem. It is shown that the HyFlux2 model is able to model inundation problems, with a satisfactory prediction of the major flow characteristics such as water depth, water velocity, flood extent, and flood-wave arrival time. The results provided by the model are of great importance for the risk assessment and management. Copyright © 2009 John Wiley & Sons, Ltd. [source]

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

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2010
    L. Cea
    Abstract This 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. [source]

    Practical evaluation of five partly discontinuous finite element pairs for the non-conservative shallow water equations

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2010
    Richard Comblen
    Abstract This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second-order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non-conforming linear elements for both velocities and elevation (P,P), is presented, giving optimal rates of convergence in all test cases. P,P1 and P,P1 mixed formulations lack convergence for inviscid flows. P,P2 pair is more expensive but provides accurate results for all benchmarks. P,P provides an efficient option, except for inviscid Coriolis-dominated flows, where a small lack of convergence is observed. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    An approximate-state Riemann solver for the two-dimensional shallow water equations with porosity

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2010
    P. Finaud-Guyot
    Abstract PorAS, a new approximate-state Riemann solver, is proposed for hyperbolic systems of conservation laws with source terms and porosity. The use of porosity enables a simple representation of urban floodplains by taking into account the global reduction in the exchange sections and storage. The introduction of the porosity coefficient induces modified expressions for the fluxes and source terms in the continuity and momentum equations. The solution is considered to be made of rarefaction waves and is determined using the Riemann invariants. To allow a direct computation of the flux through the computational cells interfaces, the Riemann invariants are expressed as functions of the flux vector. The application of the PorAS solver to the shallow water equations is presented and several computational examples are given for a comparison with the HLLC solver. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Simple efficient algorithm (SEA) for shallow flows with shock wave on dry and irregular beds

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 11 2008
    Alireza Zia
    Abstract An explicit Godunov-type solution algorithm called SEA (simple efficient algorithm) has been introduced for the shallow water equations. The algorithm is based on finite volume conservative discretisation method. It can deal with wet/dry and irregular beds. Second-order accuracy, in both time and space, is achieved using prediction and correction steps. A very simple and efficient flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purposes. In order to make sure about the balance between the flux gradient and the bed slope, treatment of the source term has been done using a new procedure inspired mainly by the physical rather than mathematical consideration. SEA has been applied to one-dimensional problems, although it can equally be applied to multi-dimensional problems. In order to assess the capability of proposed algorithm in dealing with practical applications, several test cases have been examined. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    Solution of the 2-D shallow water equations with source terms in surface elevation splitting form

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2007
    Dong-Jun Ma
    Abstract A vertex-centred finite-volume/finite-element method (FV/FEM) is developed for solving 2-D shallow water equations (SWEs) with source terms written in a surface elevation splitting form, which balances the flux gradients and source terms. The method is implemented on unstructured grids and the numerical scheme is based on a second-order MUSCL-like upwind Godunov FV discretization for inviscid fluxes and a classical Galerkin FE discretization for the viscous gradients and source terms. The main advantages are: (1) the discretization of SWE written in surface elevation splitting form satisfies the exact conservation property (,,-Property) naturally; (2) the simple centred-type discretization can be used for the source terms; (3) the method is suitable for both steady and unsteady shallow water problems; and (4) complex topography can be handled based on unstructured grids. The accuracy of the method was verified for both steady and unsteady problems, including discontinuous cases. The results indicate that the new method is accurate, simple, and robust. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    Baroclinic stability for a family of two-level, semi-implicit numerical methods for the 3D shallow water equations

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2007
    Francisco J. Rueda
    Abstract The baroclinic stability of a family of two time-level, semi-implicit schemes for the 3D hydrostatic, Boussinesq Navier,Stokes equations (i.e. the shallow water equations), which originate from the TRIM model of Casulli and Cheng (Int. J. Numer. Methods Fluids 1992; 15:629,648), is examined in a simple 2D horizontal,vertical domain. It is demonstrated that existing mass-conservative low-dissipation semi-implicit methods, which are unconditionally stable in the inviscid limit for barotropic flows, are unstable in the same limit for baroclinic flows. Such methods can be made baroclinically stable when the integrated continuity equation is discretized with a barotropically dissipative backwards Euler scheme. A general family of two-step predictor-corrector schemes is proposed that have better theoretical characteristics than existing single-step schemes. Copyright © 2006 John Wiley & Sons, Ltd. [source]

    Shallow flow simulation on dynamically adaptive cut cell quadtree grids

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2007
    Qiuhua Liang
    Abstract A computationally efficient, high-resolution numerical model of shallow flow hydrodynamics is described, based on dynamically adaptive quadtree grids. The numerical model solves the two-dimensional non-linear shallow water equations by means of an explicit second-order MUSCL-Hancock Godunov-type finite volume scheme. Interface fluxes are evaluated using an HLLC approximate Riemann solver. Cartesian cut cells are used to improve the fit to curved boundaries. A ghost-cell immersed boundary method is used to update flow information in the smallest cut cells and overcome the time step restriction that would otherwise apply. The numerical model is validated through simulations of reflection of a surge wave at a wall, a low Froude number potential flow past a circular cylinder, and the shock-like interaction between a bore and a circular cylinder. The computational efficiency is shown to be greatly improved compared with solutions on a uniform structured grid implemented with cut cells. Copyright © 2006 John Wiley & Sons, Ltd. [source]

    Well-balanced finite volume evolution Galerkin methods for the shallow water equations with source terms

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10-11 2005
    M. Luká, ová-Medvid'ová
    Abstract The goal of this paper is to present a new well-balanced genuinely multi-dimensional high-resolution finite volume evolution Galerkin method for systems of balance laws. The derivation of the method will be illustrated for the shallow water equation with geometrical source term modelling the bottom topography. The results can be generalized to more complex systems of balance laws. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    A semi-implicit method conserving mass and potential vorticity for the shallow water equations on the sphere

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8-9 2005
    Luca Bonaventura
    Abstract A semi-implicit discretization for the shallow water equations is discussed, which uses triangular Delaunay cells on the sphere as control volumes and conserves mass and potential vorticity. The geopotential gradient, the Coriolis force terms and the divergence of the velocity field are discretized implicitly, while an explicit time discretization is used for the non-linear advection terms. The results obtained with a preliminary implementation on some idealized test cases are presented, showing that the main features of large scale atmospheric flows are well represented by the proposed method. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    A stabilized SPH method for inviscid shallow water flows,

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 2 2005
    Riadh Ata
    Abstract In this paper, the smoothed particle hydrodynamics (SPH) method is applied to the solution of shallow water equations. A brief review of the method in its standard form is first described then a variational formulation using SPH interpolation is discussed. A new technique based on the Riemann solver is introduced to improve the stability of the method. This technique leads to better results. The treatment of solid boundary conditions is discussed but remains an open problem for general geometries. The dam-break problem with a flat bed is used as a benchmark test. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    Experimental and numerical analysis of solitary waves generated by bed and boundary movements

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2004
    L. Cea
    Abstract This paper is an experimental and numerical study about propagation and reflection of waves originated by natural hazards such as sea bottom movements, hill slope sliding and avalanches. One-dimensional flume experiments were conducted to study the characteristics of such waves. The results of the experimental study can be used by other researchers to verify their numerical models. A finite volume numerical model, which solves the shallow water equations, was also verified using our own experimental results. In order to deal with reflection on sloping surfaces and overtopping walls, a new condition for the treatment of the coastline is suggested. The numerical simulation of wave generation is also studied considering the bed movement. A boundary condition is proposed for this case. Those situations when the shallow water equations are valid to simulate this type of phenomena have been studied, as well as their limitations. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    Improved non-staggered central NT schemes for balance laws with geometrical source terms

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2004
    rnjari
    Abstract In this paper we extend the non-staggered version of the central NT (Nessyahu,Tadmor) scheme to the balance laws with geometrical source term. This extension is based on the source term evaluation that includes balancing between the flux gradient and the source term with an additional reformulation that depends on the source term discretization. The main property of the scheme obtained by the proposed reformulation is preservation of the particular set of the steady-state solutions. We verify the improved scheme on two types of balance laws with geometrical source term: the shallow water equations and the non-homogeneous Burger's equation. The presented results show good behaviour of the considered scheme when compared with the analytical or numerical results obtained by using other numerical schemes. Furthermore, comparison with the numerical results obtained by the classical central NT scheme where the source term is simply pointwise evaluated shows that the proposed reformulations are essential. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    High-order boundary conditions for linearized shallow water equations with stratification, dispersion and advection,

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 4 2004
    Vince J. van Joolen
    Abstract The two-dimensional linearized shallow water equations are considered in unbounded domains with density stratification. Wave dispersion and advection effects are also taken into account. The infinite domain is truncated via a rectangular artificial boundary ,, and a high-order open boundary condition (OBC) is imposed on ,. Then the problem is solved numerically in the finite domain bounded by ,. A recently developed boundary scheme is employed, which is based on a reformulation of the sequence of OBCs originally proposed by Higdon. The OBCs can easily be used up to any desired order. They are incorporated here in a finite difference scheme. Numerical examples are used to demonstrate the performance and advantages of the computational method, with an emphasis is on the effect of stratification. Published in 2004 by John Wiley & Sons, Ltd. [source]

    A staggered conservative scheme for every Froude number in rapidly varied shallow water flows

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2003
    G. S. Stelling Professor
    Abstract This paper proposes a numerical technique that in essence is based upon the classical staggered grids and implicit numerical integration schemes, but that can be applied to problems that include rapidly varied flows as well. Rapidly varied flows occur, for instance, in hydraulic jumps and bores. Inundation of dry land implies sudden flow transitions due to obstacles such as road banks. Near such transitions the grid resolution is often low compared to the gradients of the bathymetry. In combination with the local invalidity of the hydrostatic pressure assumption, conservation properties become crucial. The scheme described here, combines the efficiency of staggered grids with conservation properties so as to ensure accurate results for rapidly varied flows, as well as in expansions as in contractions. In flow expansions, a numerical approximation is applied that is consistent with the momentum principle. In flow contractions, a numerical approximation is applied that is consistent with the Bernoulli equation. Both approximations are consistent with the shallow water equations, so under sufficiently smooth conditions they converge to the same solution. The resulting method is very efficient for the simulation of large-scale inundations. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    Dispersion and stability analyses of the linearized two-dimensional shallow water equations in boundary-fitted co-ordinates

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2003
    S. Sankaranarayanan
    Abstract In the present investigation, a Fourier analysis is used to study the phase and group speeds of a linearized, two-dimensional shallow water equations, in a non-orthogonal boundary-fitted co-ordinate system. The phase and group speeds for the spatially discretized equations, using the second-order scheme in an Arakawa C grid, are calculated for grids with varying degrees of non-orthogonality and compared with those obtained from the continuous case. The spatially discrete system is seen to be slightly dispersive, with the degree of dispersivity increasing with an decrease in the grid non-orthogonality angle or decrease in grid resolution and this is in agreement with the conclusions reached by Sankaranarayanan and Spaulding (J. Comput. Phys., 2003; 184: 299,320). The stability condition for the non-orthogonal case is satisfied even when the grid non-orthogonality angle, is as low as 30° for the Crank Nicolson and three-time level schemes. A two-dimensional wave deformation analysis, based on complex propagation factor developed by Leendertse (Report RM-5294-PR, The Rand Corp., Santa Monica, CA, 1967), is used to estimate the amplitude and phase errors of the two-time level Crank,Nicolson scheme. There is no dissipation in the amplitude of the solution. However, the phase error is found to increase, as the grid angle decreases for a constant Courant number, and increases as Courant number increases. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    Relaxation schemes for the shallow water equations

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2003
    A. I. Delis
    Abstract We present a class of first and second order in space and time relaxation schemes for the shallow water (SW) equations. A new approach of incorporating the geometrical source term in the relaxation model is also presented. The schemes are based on classical relaxation models combined with Runge,Kutta time stepping mechanisms. Numerical results are presented for several benchmark test problems with or without the source term present. Copyright © 2003 John Wiley & Sons, Ltd. [source]

    Performance of finite volume solutions to the shallow water equations with shock-capturing schemes

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2002
    K. S. Erduran
    Abstract Numerical methods have become well established as tools for solving problems in hydraulic engineering. In recent years the finite volume method (FVM) with shock capturing capabilities has come to the fore because of its suitability for modelling a variety of types of flow; subcritical and supercritical; steady and unsteady; continuous and discontinuous and its ability to handle complex topography easily. This paper is an assessment and comparison of the performance of finite volume solutions to the shallow water equations with the Riemann solvers; the Osher, HLL, HLLC, flux difference splitting (Roe) and flux vector splitting. In this paper implementation of the FVM including the Riemann solvers, slope limiters and methods used for achieving second order accuracy are described explicitly step by step. The performance of the numerical methods has been investigated by applying them to a number of examples from the literature, providing both comparison of the schemes with each other and with published results. The assessment of each method is based on five criteria; ease of implementation, accuracy, applicability, numerical stability and simulation time. Finally, results, discussion, conclusions and recommendations for further work are presented. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    A cascadic conjugate gradient algorithm for mass conservative, semi-implicit discretization of the shallow water equations on locally refined structured grids

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 1-2 2002
    Luca Bonaventura
    Abstract A semi-implicit, mass conservative discretization scheme is applied to the two-dimensional shallow water equations on a hierarchy of structured, locally refined Cartesian grids. Different resolution grids are fully interacting and the discrete Helmholtz equation obtained from the semi-implicit discretization is solved by the cascadic conjugate gradient method. A flux correction is applied at the interface between the coarser and finer discretization grids, so as to ensure discrete mass conservation, along with symmetry and diagonal dominance of the resulting matrix. Two-dimensional idealized simulations are presented, showing the accuracy and the efficiency of the resulting method. Copyright © 2002 John Wiley & Sons, Ltd. [source]

    The shallow flow equations solved on adaptive quadtree grids

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2001
    A. G. L. Borthwick
    Abstract This paper describes an adaptive quadtree grid-based solver of the depth-averaged shallow water equations. The model is designed to approximate flows in complicated large-scale shallow domains while focusing on important smaller-scale localized flow features. Quadtree grids are created automatically by recursive subdivision of a rectangle about discretized boundary, bathymetric or flow-related seeding points. It can be fitted in a fractal-like sense by local grid refinement to any boundary, however distorted, provided absolute convergence to the boundary is not required and a low level of stepped boundary can be tolerated. Grid information is stored as a tree data structure, with a novel indexing system used to link information on the quadtree to a finite volume discretization of the governing equations. As the flow field develops, the grids may be adapted using a parameter based on vorticity and grid cell size. The numerical model is validated using standard benchmark tests, including seiches, Coriolis-induced set-up, jet-forced flow in a circular reservoir, and wetting and drying. Wind-induced flow in the Nichupté Lagoon, México, provides an illustrative example of an application to flow in extremely complicated multi-connected regions. Copyright © 2001 John Wiley & Sons, Ltd. [source]