Practical Aspects in Comparing Shock-Capturing Schemes for Dam Break Problems

The results of a survey aimed at comparing the performances of first-order and total variation diminishing (TVD) second-order upwind flux difference splitting schemes, first-order space-centered schemes, and second-order space-centered schemes with the TVD artificial viscosity term are reported here. The schemes were applied to the following dam-break wave cases: in a dry frictionless horizontal channel; in a dry, rough and sloping channel; and in a nonprismatic channel. Among first-order schemes, the diffusive scheme provides only slightly less accurate results than those obtained by the Roe scheme. For TVD second-order schemes, no significant difference between the upwind scheme and central schemes are reported. In the case of a dam break in a dry frictionless horizontal channel, the second-order schemes were two- to five-fold more accurate than the diffusive scheme and Roe’s scheme. These differences in scheme performances drastically reduce when the results obtained for the rough sloping channel test and for the nonprismatic channel test are analyzed. In particular, the accuracy of the diffusive and Roe’s schemes is similar to second-order schemes when such features of dam break wave, relevant from an engineering viewpoint, like wave peak arrival time and maximum water depths, are considered.     

Channel Routing in Open-Channel Flows with Surges

Using numerical models for the purpose of channel-routing calculation has been well accepted in engineering practice. However, most traditional models fail to predict the transcritical flows because of numerical instability. This paper presents two high-resolution, shock-capturing schemes for the simulation of 1D, rapidly varied open-channel flows. The present schemes incorporate the method of characteristics to deal with the unsteady boundary conditions. Also, the Strang-type splitting operator is used to include the effects of bottom slope and friction terms. To assess the performance of the proposed algorithms, several steady and unsteady problems are simulated to verify the accuracy and robustness in capturing strong shocks in open-channel flows. Furthermore, the results of dynamic flood routing and steady routing are compared to demonstrate the risk of using steady routing for flood mitigation.     

Godunov-Type Solution of Curvilinear Shallow-Water Equations

This paper presents details of a second-order accurate Godunov-type numerical model of the two-dimensional conservative hyperbolic shallow-water equations written in a nonorthogonal curvilinear matrix form and discretized using finite volumes. Roe's flux function is used for the convection terms, and a nonlinear limiter is applied to prevent spurious oscillations. Validation tests include frictionless rectangular and circular dam-breaks, an oblique hydraulic jump, jet-forced flow in a circular basin, and vortex shedding from a vertical surface-piercing cylinder. The results indicate that the model accurately simulates sharp fronts, a flow discontinuity between subcritical and supercritical conditions, recirculation in a basin, and unsteady wake flows.     

Numerical and Experimental Study on Two-Dimensional Flood Flows with and without Structures

The behavior of two-dimensional (2D) flood flows and the hydrodynamic force acting on structures are investigated numerically and experimentally. Numerical simulations are performed using a model based on the finite-volume method with an unstructured grid system and the flux-difference splitting technique. Experiments on flood propagation in a flood plain, with and without structures were conducted so as to obtain a comprehensive verification of the model. Front positions, depths, and surface velocities of flood flows as well as hydrodynamic forces on structures were observed. Comparisons of numerical results against these experimental data show that the model can predict 2D flood flows and the force on structures with reasonable accuracy.     

Finite Volume Model for Two-Dimensional Shallow Water Flows on Unstructured Grids

A numerical model based upon a second-order upwind finite volume method on unstructured triangular grids is developed for solving shallow water equations. The HLL approximate Riemann solver is used for the computation of inviscid flux functions, which makes it possible to handle discontinuous solutions. A multidimensional slope-limiting technique is employed to achieve second-order spatial accuracy and to prevent spurious oscillations. To alleviate the problems associated with numerical instabilities due to small water depths near a wet/dry boundary, the friction source terms are treated in a fully implicit way. A third-order total variation diminishing Runge–Kutta method is used for the time integration of semidiscrete equations. The developed numerical model has been applied to several test cases as well as to real flows. Numerical tests prove the robustness and accuracy of the model.     

WAF Method and Splitting Procedure for Simulating Hydro- and Thermal-Peaking Waves in Open-Channel Flows

Hydro- and thermal-peaking waves, generated by hydroelectric power generation, have a strong impact on the ecological integrity of aquatic ecosystems. In order to reduce such effects, mitigation procedure must be studied and implemented. To this end a one-dimensional model which solves the coupling of hydrodynamics with heat transport is developed. The solution is obtained advancing simultaneously the hydrodynamic and thermal module with the same accuracy. For the numerical solution of the governing advection-reaction/diffusion problem a splitting procedure is adopted: the advection-reaction part is solved by means of the weight average flux (WAF) finite volume explicit method, while the diffusion part is solved using a nonlinear version of the implicit Crank-Nicolson method. The WAF method is extended to second-order in the presence of reaction terms. Numerical results are presented for different test examples, which demonstrate the accuracy and robustness of the scheme and its applicability in predicting temperature transport by shallow water flows. Application to the Adige River (Northern Italy) of this framework proves that the model is an effective tool for designing hydro- and thermal-peaking waves mitigation procedures.     

Finite Volume Model for Nonlevel Basin Irrigation

A finite volume model for unsteady, two-dimensional, shallow water flow is developed and applied to simulate the advance and infiltration of an irrigation wave in two-dimensional basins of complex topography. The fluxes are computed with Roe's approximate Riemann solver and the monotone upstream scheme for conservation laws is used in conjunction with predictor-corrector time-stepping to provide a second-order accurate solution. Flux-limiting is implemented to eliminate spurious oscillations and the model incorporates an efficient and robust scheme to capture the wetting and drying of the soil. Model predictions are compared with experimental data for one- and two-dimensional problems involving rough, impermeable, and permeable beds, including a poorly leveled basin.     

Godunov-Type Solutions for Water Hammer Flows

First- and second-order explicit finite volume (FV) Godunov-type schemes for water hammer problems are formulated, applied, and analyzed. The FV formulation ensures that both schemes conserve mass and momentum and produce physically realizable shock fronts. The exact solution of the Riemann problem provides the fluxes at the cell interfaces. It is through the exact Riemann solution that the physics of water hammer waves is incorporated into the proposed schemes. The implementation of boundary conditions, such as valves, pipe junctions, and reservoirs, within the Godunov approach is similar to that of the method of characteristics (MOC) approach. The schemes are applied to a system consisting of a reservoir, a pipe, and a valve and to a system consisting of a reservoir, two pipes in series, and a valve. The computations are carried out for various Courant numbers and the energy norm is used to evaluate the numerical accuracy of the schemes. Numerical tests and theoretical analysis show that the first-order Godunov scheme is identical to the MOC scheme with space-line interpolation. It is also found that, for a given level of accuracy and using the same computer, the second-order scheme requires much less memory storage and execution time than either the first-order scheme or the MOC scheme with space-line interpolation. Overall, the second-order Godunov scheme is simple to implement, accurate, efficient, conservative, and stable for Courant number less than or equal to one.     

Weighted Averaged Flux-Type Scheme for Shallow-Water Equations with Fractional Step Method

A numerical model describing two-dimensional fluid motions has been developed on an unstructured grid system. By using a fractional step method, a two-dimensional problem governed by the two-dimensional shallow-water equations is treated as two one-dimensional problems. Thus it is possible to simulate two-dimensional numerical problems with a higher computational efficiency. One-dimensional problems are solved by using an upwind total variation diminishing version of the second-order weighted averaged flux method with an approximate Riemann solver. Numerical oscillations commonly observed in second-order numerical schemes are controlled by exploiting a flux limiter. For the general purpose, the model can simulate on an arbitrary topography, treat a moving boundary, and resolve a shock. Five ideal and practical problems are tested. Very accurate results are observed.     

Role of Artificial Dissipation Scaling and Multigrid Acceleration in Numerical Solutions of the Depth-Averaged Free-Surface Flow Equations

A numerical model is developed for solving the depth-averaged, open-channel flow equations in generalized curvilinear coordinates. The equations are discretized in space in strong conservation form using a space-centered, second-order accurate finite-volume method. A nonlinear blend of first- and third-order accurate artificial dissipation terms is introduced into the discrete equations to accurately model all flow regimes. Scalar- and matrix-valued scaling of the artificial dissipation terms are considered and their effect on the accuracy of the solutions is evaluated. The discrete equations are integrated in time using a four-stage explicit Runge–Kutta method. For the steady-state computations, local time stepping, implicit residual smoothing, and multigrid acceleration are used to enhance the efficiency of the scheme. The numerical model is validated by applying it to calculate steady and unsteady open-channel flows. Extensive grid sensitivity studies are carried out and the potential of multigrid acceleration for steady depth-averaged computations is demonstrated.     

Explicit Schemes for Dam-Break Simulations

Dam-break problems involve the formation of shocks and rarefaction fans. The performance of 20 explicit numerical schemes used to solve the shallow water wave equations for simulating the dam-break problem is examined. Results from these schemes have been compared with analytical solutions to the dam-break problem with finite water depth and dry bed downstream of the dam. Most of the numerical schemes produce reasonable results for subcritical flows. Their performance for problems where there is a transition between subcritical and supercritical flows is mixed. Although many numerical schemes satisfy the Rankine-Hugoniot condition, some produce solutions which do not satisfy the entropy condition, producing nonphysical solutions. This was the case for the majority of first-order schemes examined. Numerical schemes which consider critical flow in the solution are guaranteed to produce entropy satisfying solutions. Second-order schemes avoid the generation of expansive shocks; however, some form of flux or slope limiter must be used to eliminate oscillations that are associated with these schemes. These limiters increase the complexity and the computational effort required, but they are generally more accurate than their first-order counterparts. The limiters employed by these second-order schemes will produce monotone or total variation diminishing solutions for scalar equations. Some limiters do not exhibit these properties when they are applied to the nonlinear shallow water wave equations. This comparative study shows that there are a variety of shock-capturing numerical schemes that are efficient, accurate, robust, and are suitable for solving the shallow water wave equations when discontinuities are encountered in the problem.     

Accurate Two-Dimensional Simulation of Advective-Diffusive-Reactive Transport

The present paper presents an accurate numerical algorithm for the simulation of 2D solute∕heat transport by unsteady advection-diffusion-reaction. The model was specifically developed for the study of convective exchange processes in a cross section of lakes and ponds, when the currents are predominantly driven by density (temperature) gradients. The numerical scheme is based on the split-operator approach, in which advection and diffusion with chemical∕biological kinetic processes are calculated separately at each time step. Special attention is given to the advection operator in order to avoid excessive numerical damping or oscillations, as well as to the source∕sink term, which may cause numerical instability and inaccuracy if improperly treated. The model has been verified on standard test problems for a wide range of Courant, Fourier, Péclet, and Thiele numbers, and found to produce stable results of high accuracy.     

Gradient-Adaptive-Sigma (GAS) Grid for 3D Mass-Transport Modeling

The accuracy of three-dimensional (3D) mass transport modeling strongly relies on how well the vertical structures of the transport processes are resolved with the model resolution. The widely used vertical “stair-stepped” and topography following sigma (σ)-coordinate systems are fixed in time and space. Hence, these techniques often fail to resolve the details of the time varying, and highly nonlinear, activities in the water column. To better capture the geometrical details of these activities, a new vertical solution-adaptive grid method is introduced in this paper. The method takes into account the gradient variation of a selected variable in the mass transport field as an additional controlling factor in σ-coordinate transformation and so the new transformed grid is called a Gradient-Adaptive-Sigma (GAS) grid. The transformed grid spacing automatically adjusts in time and space according to the local solution gradient of the selected variable and converges in the high gradient regions for better resolution. The solution-adaptive method is implemented in a surface water numerical model to transform Cartesian coordinates into GAS coordinates. The model is established on the basis of an operator splitting scheme coupled with a Eulerian-Lagrangian method. Four numerical experiments describing the sediment transport activities of net entrainment and net deposition, and also pollutant dispersion, are performed with the transformed model. The computed results agree with the laboratory measurements. A comparison between the results computed by the GAS-gridded model and a σ-gridded model show that using the GAS-grid arrangement can improve the solution accuracy by as much as one-fold in regions of high solution gradients.     

Flood Simulation Using a Well-Balanced Shallow Flow Model

This work extends and improves a one-dimensional shallow flow model to two-dimensional (2D) for real-world flood simulations. The model solves a prebalanced formulation of the fully 2D shallow water equations, including friction source terms using a finite volume Godunov-type numerical scheme. A reconstruction method ensuring nonnegative depth is used along with a Harten, Lax, and van Leer approximate Riemann solver with the contact wave restored for calculation of interface fluxes. A local bed modification method is proposed to maintain the well-balanced property of the algorithm for simulations involving wetting and drying. Second-order accurate scheme is achieved by using the slope limited linear reconstruction together with a Runge-Kutta time integration method. The model is applicable to calculate different types of flood wave ranging from slow-varying inundations to extreme and violent floods, propagating over complex domains including natural terrains and dense urban areas. After validating against an analytical case of flow sloshing in a domain with a parabolic bed profile, the model is applied to simulate an inundation event in a 36?km2 floodplain in Thamesmead near London. The numerical predictions are compared with analytical solutions and alternative numerical results.     

Accuracy Order of Crank–Nicolson Discretization for Hydrostatic Free-Surface Flow

Application of Crank–Nicolson (CN) discretization to the hydrostatic (or shallow-water) free-surface equation in two-dimensional or three-dimensional Reynolds-averaged Navier–Stokes models neglects a second order term. The neglected term is zero at steady state, so it does not appear in steady-state accuracy analyses. A new correction term is derived that restores second-order accuracy. The correction is significant when the amplitude of the surface oscillation is within two orders of magnitude of the water depth and the barotropic Courant–Friedrichs–Lewy (CFL) stability condition is less than unity. Analysis shows that the CN accuracy for an unforced free-surface oscillation is degraded to first order when the barotropic CFL stability condition is greater than unity, independent of whether or not the new correction term is applied. The results indicate that the semi-implicit Crank–Nicolson method, applied to the hydrostatic free-surface evolution equation, is only first-order accurate for the time and space scales typically used in lake, estuarine, and coastal ocean studies.     

Coupled 1D–Quasi-2D Flood Inundation Model with Unstructured Grids

A simplified numerical model for simulation of floodplain inundation resulting from naturally occurring floods in rivers is presented. Flow through the river is computed by solving the de Saint Venant equations with a one-dimensional (1D) finite volume approach. Spread of excess flood water spilling overbank from the river onto the floodplains is computed using a storage cell model discretized into an unstructured triangular grid. Flow exchange between the one-dimensional river cells and the adjacent floodplain cells or that between adjoining floodplain cells is represented by diffusive-wave approximated equation. A common problem related to the stability of such coupled models is discussed and a solution by way of linearization offered. The accuracy of the computed flow depths by the proposed model is estimated with respect to those predicted by a two-dimensional (2D) finite volume model on hypothetical river-floodplain domains. Finally, the predicted extent of inundation for a flood event on a stretch of River Severn, United Kingdom, by the model is compared to those of two proven two-dimensional flow simulation models and with observed imagery of the flood extents.     

3D Analytic Model for Testing Numerical Tidal Models

A 3D analytic solution is presented for tides in channels with arbitrary lateral depth variation. The solution is valid for narrow channels in which the lateral variation of the amplitude of tidal elevation is small. The error introduced by the solution is on the order of a few percent in a tidal channel of a few kilometers in width. The solution allows an arbitrary lateral depth variation and thus provides a wide choice of depth functions, especially those with large bottom slopes. The largest amplitude of the along-estuary velocity appears on the surface in the deepest water. The depth-averaged velocity is the largest in the deepest water. The time of flood (ebb) in deep water lags that in shallow water. The time of flood (ebb) on the surface lags that at the bottom. Since this solution is simple and allows arbitrary lateral depth variations, it can be used to demonstrate the first-order tidal flow in narrow tidal channels of variable depth, and to test high-resolution numerical models with large depth gradients.     

Non-Gaussian Air Gap Response Models for Floating Structures

The air gap response of a specific semisubmersible platform subjected to irregular waves is considered. Statistical analyses are performed on model test data for the absolute near-structure wave elevation, and these measured data are compared with predictions resulting from probabilistic models. Models applied are first-order and second-order diffraction models typical of standard practice, and two new hybrid models that include second-order effects in the incident wave, but not in the diffracted wave. The first of these hybrid models is moment based, while the second relies on narrow-band random process theory. Either of these new models can be implemented in place of the standard linear-only model with little additional computational effort, as only linear diffraction analysis is required. Both new models are found to better predict the air gap demand than standard linear diffraction analysis.     

Observation and Simulation of Floodwater Intrusion and Sedimentation in the Shichikashuku Reservoir

Wash load sedimentation during flood events is studied through comprehensive field measurements at the Shichikashuku Reservoir, Japan and numerical simulation. Field data during a flooding event caused by a typhoon in 1996 are used to set the boundary conditions for the model and to verify the simulation. In the field experiments, continuous monitoring of inflow turbidity is shown to be an effective and reliable means of estimating sediment influx. In the simulations, the proposed model is shown to provide an accurate reproduction of floodwater currents and sedimentation in the reservoir. The model achieves good accuracy and resolution by adopting an orthogonal curvilinear grid to discretize the reservoir and by solving the flow field using a three-dimensional standard k–ε turbulence model. The simulation reproduces the observed velocity, turbidity, and distribution of sedimentation well, demonstrating the potential utility of such models in the management of reservoir water quality and sedimentation.