Form Of Shallow Water Equations Research Articles

Application of Large Time Step TVD High Order Scheme to Shallow Water Equations

The numerical modeling of actual river floods faces three challenges related to computational efficiency, accuracy, and proper balancing of terms in the governing equations, all of which are discussed in this paper. Herein, a large time step (LTS) scheme is used to improve efficiency, a high order scheme is used to enhance accuracy, and specific treatment of the bed slope term achieves a well-balanced form of shallow water equations. The LTS scheme, originally proposed by LeVeque in 1998, has led to the development of highly efficient computational solvers of the shallow water equations (SWEs). This paper examines use of a total variation diminishing (TVD) high order scheme in conjunction with LTS. We first applied the scheme to the solution of the homogeneous 1D SWEs and obtained satisfactory results for three cases, even though small oscillations nevertheless occur when the CFL number is very large. The additional source term makes the issue more complicated and can introduce a spurious flow when the term is not correctly handled. Many methods have been developed in traditional differential schemes, but not all are fit for the TVD-LTS scheme; for example, the method of decomposing the source term into simple characteristic waves has proved infeasible. In this paper the TVD-LTS scheme was implemented for the first time for well-balanced SWEs containing bed slope source terms. We found that oscillations were not as suppressed as for the homogeneous SWEs when the TVD-LTS scheme was applied to the three cases of step Riemann problems (SRP) tested for CFL numbers 1 to 10. For free surface flow over a bed hump, the TVD-LTS scheme can only reach CFL number 4 before the solution breaks down.

Urban Stormwater Modeling with Local Inertial Approximation Form of Shallow Water Equations: A Comparative Study

This study focused on the performance and limitations of the local inertial approximation form model (LIM) of the shallow water equations (SWEs) when applied in urban flood modeling. A numerical scheme of the LIM equations was created using finite volume method with a first-order spatiotemporal Roe Riemann solver. A simplified urban stormwater model (SUSM) considering surface and underground dual drainage system was constructed based on LIM and the US Environmental Protection Agency Storm Water Management Model. Moreover, a complete urban stormwater model (USM) based on the SWEs with the same solution algorithm was used as the evaluation benchmark. Numerical results of the SUSM and USM in a highly urbanized area under four rainfall return periods were analyzed and compared. The results reveal that the performance of the SUSM is highly consistent with that of the USM but with an improvement in computational efficiency of approximately 140%. In terms of the accuracy of the model, the SUSM slightly underestimates the water depth and velocity and is less accurate when dealing with supercritical flow in urban stormwater flood modeling. Overall, the SUSM can produce comparable results to USM with higher computational efficiency, which provides a simplified and alternative method for urban flood modeling.

A meshless generalized finite difference method for solving shallow water equations with the flux limiter technique

In this study, a novel meshless stable numerical solver is proposed to solve the non-conservative form of shallow water equations. Since they form a hyperbolic system of equations, discontinuous solutions are allowed to transmit during the simulation. The generalized finite difference-split coefficient matrix method, recently proposed, is applied and improved using the flux limiter to eliminate the possible-appearing numerical oscillations. In the proposed scheme, the split-coefficient matrix method is adopted to convert the shallow water equations to the characteristic form. Then, the generalized finite difference method and the second-order Runge-Kutta method are employed for spatial and temporal discretization, respectively. The upwinding spatial derivatives can be approximated at every node using the half-disk shape of the star and generalized finite difference method. Applying the flux limiter technique, the expressions can automatically switch the proper discrete order when facing discontinuous solutions. Although the limiter function required the derivatives of different orders, the generalized finite difference method can solve these necessary expressions using the first- and second-order Tayler series. Several numerical examples are provided to demonstrate the capability of the proposed scheme, and the results are compared with other numerical schemes to show the effectiveness of the proposed generalized finite difference-flux limiter method.

Comparison of depth-averaged concentration and bed load flux sediment transport models of dam-break flow

This paper presents numerical simulations of dam-break flow over a movable bed. Two different mathematical models were compared: a fully coupled formulation of shallow water equations with erosion and deposition terms (a depth-averaged concentration flux model), and shallow water equations with a fully coupled Exner equation (a bed load flux model). Both models were discretized using the cell-centered finite volume method, and a second-order Godunov-type scheme was used to solve the equations. The numerical flux was calculated using a Harten, Lax, and van Leer approximate Riemann solver with the contact wave restored (HLLC). A novel slope source term treatment that considers the density change was introduced to the depth-averaged concentration flux model to obtain higher-order accuracy. A source term that accounts for the sediment flux was added to the bed load flux model to reflect the influence of sediment movement on the momentum of the water. In a one-dimensional test case, a sensitivity study on different model parameters was carried out. For the depth-averaged concentration flux model, Manning's coefficient and sediment porosity values showed an almost linear relationship with the bottom change, and for the bed load flux model, the sediment porosity was identified as the most sensitive parameter. The capabilities and limitations of both model concepts are demonstrated in a benchmark experimental test case dealing with dam-break flow over variable bed topography.

Discrete mixed subdomain least squares (DMSLS) meshless method with collocation points for modeling dam-break induced flows

This paper presents a new meshless numerical scheme to overcome the problem of shock waves and to apply boundary conditions in cases of dam-break flows in channels with constant and variable widths. The numerical program solves shallow water equations based on the discrete mixed subdomain least squares (DMSLS) meshless method with collocation points. The DMSLS meshless method is based on the minimization of a least squares functional defined as the weighted summation of the squared residuals of the governing equations over the entire domain and requiring the summation of residual function to be zero at collocation points in boundary subdomains. The collocated discrete subdomain meshless method is applied on the boundary, whereas the collocated discrete least squares meshless technique is applied to the interior domain. The meshless scheme extends for dam-break formulation of shallow water equations. The model is verified by comparing computed results with analytical and experimental data for constant and varying width channels. The developed model is also used to study one-dimensional dam-break problems involving different flow situations by considering changes to the channel width, a bumpy channel with various downstream boundary conditions, and the effects of bed friction and bed slope as source terms on wave propagation. The accuracy of the results is acceptable.

A new well-posed vorticity divergence formulation of the shallow water equations

A new vorticity–divergence formulation of the two-dimensional shallow water equations including boundary conditions is derived. The new formulation is necessary since the conventional one does not lead to a well-posed initial boundary value problem for limited-area modelling.The new vorticity–divergence formulation includes four dependent variables instead of three and requires more equations and boundary conditions than the conventional formulation. On the other hand, it forms a hyperbolic set of equations with well-defined boundary conditions that leads to a well-posed problem with bounded energy.

Validation of depth-averaged flow model using flat-bottomed benchmark problems.

In this study, a shallow water flow code was developed and tested against four benchmark problems of practical relevance. The results demonstrated that as the eddy viscosity increased, the velocity slope along the spanwise direction decreased, and the larger roughness coefficient induced a higher flow depth over the channel width. The mass conservation rate was determined to be 99.2%. This value was measured by the variation of the total volume of the fluid after a cylinder break. As the Re increased to 10,000 in the internal recirculating flow problem, the intensity of the primary vortex had a clear trend toward the theoretically infinite Re value of −1.886. The computed values of the supercritical flow evolved by the oblique hydraulic jump agreed well with the analytic solutions within an error bound of 0.2%. The present model adopts the nonconservative form of shallow water equations. These equations are weighted by the SU/PG scheme and integrated by a fully implicit method, which can reproduce physical problems with various properties. The model provides excellent results under various flow conditions, and the solutions of benchmark tests can present criteria for the evaluation of various algorithmic approaches.

Numerical simulation of wave-induced currents combined with parabolic mild-slope equation in curvilinear coordinates

Researches on breaking-induced currents by waves are summarized firstly in this paper. Then, a combined numerical model in orthogonal curvilinear coordinates is presented to simulate wave-induced current in areas with curved boundary or irregular coastline. The proposed wave-induced current model includes a nearshore current module established through orthogonal curvilinear transformation form of shallow water equations and a wave module based on the curvilinear parabolic approximation wave equation. The wave module actually serves as the driving force to provide the current module with required radiation stresses. The Crank-Nicolson finite difference scheme and the alternating directions implicit method are used to solve the wave and current module, respectively. The established surf zone currents model is validated by two numerical experiments about longshore currents and rip currents in basins with rip channel and breakwater. The numerical results are compared with the measured data and published numerical results.

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

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

A Discontinuous Galerkin Method for Two-Dimensional Shock Wave Modeling

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

Integration of the boussinesq equations for steady channel and jet flows

The equations of long nonlinear waves in round jets and channels of arbitrary cross section are considered with account for the transverse acceleration of the fluid particles (Boussinesq approximation). In the general case of steady flows, the equations in the form of shallow water equations with the pressure expressed in terms of the variational derivative of the kinetic energy of a thin transverse fluid layer, have three first integrals with three arbitrary constants. Examples of solutions of the equations for solitary capillary-gravitation waves in rectangular and triangular channels are presented and compared with the higher approximations. The shape of the free boundary of the round jet is determined. In the case of outflow from a conical nozzle an analytical dependence of the jet contraction ratio on the conicity angle is obtained. The dependence is in agreement with the experimental data for angles of less than 45°.

A unified treatment of steady-state shallow water and two-dimensional navier-stokes equations — finite element penalty function approach

Abstract The objectives of this paper are threefold: 1. (a) To introduce new generalized formulation of shallow water equations which permits taking into account the effect of return current distribution in the depth direction. 2. (b) To introduce a penalty formulation in the steady state solution of the shallow water equations. 3. (c) To test the validity of various arbitrary approximations introduced in simplified shallow water equations by different investigators. In the development of the first objective we start, as is common practice, introducing the only approximation necessary in shallow water formulations, i.e. the neglect of vertical acceleration terms. The essential variables remaining are the water surface elevation and the two horizontal velocities. The problem thus is still three dimensional albeit with two velocity components to be represented by an expansion in the depth coordinate direction. The problem is reduced to a set of two equations in two space dimensions but with a finite set k of parameters defining the vertical distributions. This permits, when k ⩾ 2, to obtain return currents which have been included in general formulation given by other authors at the expense of omitting horizontal viscosity and convective terms. The second objective, i.e. that of developing a “penalty” form for the steady state solution of the resulting equations is accomplished in the manner previously used by authors for Navier-Stokes equations and results in the elimination of the surface elevation variable from the final discrete form by imposing the approximate simultaneous satisfaction of the continuity constraint with a penalty number. In the last section of the paper the third objective is addressed by the study of several examples in which the full solution as well as its various approximations are used. One of the examples concerns lake circulation driven by wind forces and another wave generated currents near a shore obstacle.