Numerical Solution of Boussinesq Equations to Simulate Dam-Break Flows

To investigate the effect of nonhydrostatic pressure distribution, dam-break flows are simulated by numerically solving the one-dimensional Boussinesq equations by using a fourth-order explicit finite-difference scheme. The computed water surface profiles for different depth ratios have undulations near the bore front for depth ratios greater than 0.4. The results obtained by using the Saint Venant equations and the Boussinesq equations are compared to determine the contribution of individual Boussinesq terms in the simulation of dam-break flow. It is found that, for typical engineering applications, the Saint Venant equations give sufficiently accurate results for the maximum flow depth and the time to reach this value at a location downstream of the dam.     

Lagrangian Modeling of Weakly Nonlinear Nonhydrostatic Shallow Water Waves in Open Channels

A Lagrangian, nonhydrostatic, Boussinesq model for weakly nonlinear and weakly dispersive flow is presented. The model is an extension of the hydrostatic model—dynamic river model. The model uses a second-order, staggered grid, predictor-corrector scheme with a fractional step method for the computation of the nonhydrostatic pressure. Numerical results for solitary waves and undular bores are compared with Korteweg-de Vries analytical solutions and published numerical, laboratory, and theoretical results. The model reproduced well known features of solitary waves, such as wave speed, wave height, balance between nonlinear steepening and wave dispersion, nonlinear interactions, and phase shifting when waves interact. It is shown that the Lagrangian moving grid is dynamically adaptive in that it ensures a compression of the grid size under the wave to provide higher resolution in this region. Also the model successfully reproduced a train of undular waves (short waves) from a long wave such that the predicted amplitude of the leading wave in the train agreed well with published numerical and experimental results. For prismatic channels, the method has no numerical diffusion and it is demonstrated that a simple second-order scheme suffices to provide an efficient and economical solution for predicting nonhydrostatic shallow water flows.     

Fully Nonhydrostatic Modeling of Surface Waves

A fully nonhydrostatic model is tested by simulating a range of surface-wave motions, including linear dispersive waves, nonlinear Stokes waves, wave propagation over bottom topographies, and wave–current interaction. The model uses an efficient implicit method to solve the unsteady, three-dimensional, Navier-Stokes equations and the fully nonlinear free-surface boundary conditions. A new top-layer pressure treatment is incorporated to fully include the nonhydrostatic pressure effect. The model results are verified against either analytical solutions or experimental data. It is found that the model using a small number of vertical layers is capable of accurately simulating both the free-surface elevation and vertical flow structure. By further examining the model’s performance of resolving wave dispersion and nonlinearity, the model’s efficiency and accuracy are demonstrated.     

Effect of Seepage-Induced Nonhydrostatic Pressure Distribution on Bed-Load Transport and Bed Morphodynamics

Bed-load transport is commonly evaluated in the condition of a hydrostatic pressure distribution of the flow field; while this condition is reasonable for quasi-steady, quasi-uniform rectilinear flows, it cannot be satisfied in a large variety of flow conditions, i.e., near an obstacle as in the case of a bridge pier. The dimensionless Shields number, which contains the assumption of a hydrostatic pressure distribution in its denominator, therefore cannot be strictly applied to evaluate bed-load transport in all the configurations where nonhydrostatic pressure distributions are observed. In the present work, a generalization of the Shields number is proposed for the case of nonhydrostatic pressure distribution produced by groundwater flow. Experiments showing the effects of vertical groundwater flow on the bed morphodynamics are presented. The comparison between the experimental observations and numerical results, obtained by means of a morphodynamic model which employs the new formulation of the Shields number, suggests that the proposed generalization of the Shields number is able to account the effect of the nonhydrostatic pressure distribution on the bed-load transport.     

Hydrostatic versus Nonhydrostatic Euler-Equation Modeling of Nonlinear Internal Waves

Basin-scale internal waves are inherently nonhydrostatic; however, they are frequently resolved features in three-dimensional hydrostatic lake and coastal ocean models. Comparison of hydrostatic and nonhydrostatic formulations of the Centre for Water Research Estuary and Lake Computer Model provides insight into the similarities and differences between these representations of internal waves. Comparisons to prior laboratory experiments are used to demonstrate the expected wave evolution. The hydrostatic model cannot replicate basin-scale wave degeneration into a solitary wave train, whereas a nonhydrostatic model does represent the downscaling of energy. However, the hydrostatic model produces a nonlinear traveling borelike feature that has similarities to the mean evolution of the nonhydrostatic wave.     

Computing Nonhydrostatic Shallow-Water Flow over Steep Terrain

Flood and dambreak hazards are not limited to moderate terrain, yet most shallow-water models assume that flow occurs over gentle slopes. Shallow-water flow over rugged or steep terrain often generates significant nonhydrostatic pressures, violating the assumption of hydrostatic pressure made in most shallow-water codes. In this paper, we adapt a previously published nonhydrostatic granular flow model to simulate shallow-water flow, and we solve conservation equations using a finite volume approach and an Harten, Lax, Van Leer, and Einfeldt approximate Riemann solver that is modified for a sloping bed and transient wetting and drying conditions. To simulate bed friction, we use the law of the wall. We test the model by comparison with an analytical solution and with results of experiments in flumes that have steep (31°) or shallow (0.3°) slopes. The law of the wall provides an accurate prediction of the effect of bed roughness on mean flow velocity over two orders of magnitude of bed roughness. Our nonhydrostatic, law-of-the-wall flow simulation accurately reproduces flume measurements of front propagation speed, flow depth, and bed-shear stress for conditions of large bed roughness.     

Effect of Applying Different Distribution Shapes for Velocities and Pressure on Simulation of Curved Open Channels

Most of the computational models of curved open channel flows use the conventional depth averaged De St. Venant equations. De St. Venant equations assume uniform velocity and hydrostatic pressure distributions. They are thus applicable only to cases of meandering rivers and curved open channels where vertical details are not of importance. The two-dimensional vertically averaged and moment equations model, developed by the writers, is used to study the effect of applying different distribution shapes for velocities and pressure on the simulation of curved open channels. Linear and quadratic distribution shapes are proposed for the horizontal velocity components, while a quadratic distribution shape is considered for the vertical velocity. Linear hydrostatic and quadratic nonhydrostatic distribution shapes are proposed for the pressure. The proposed model is applied to problems involved in curved open channels with different degrees of curvature. The implicit Petrov–Galerkin finite element scheme is applied in this study. Computed values for depth averaged longitudinal and transverse velocities across the channel width and vertical profiles of longitudinal and transverse velocities are compared to the observed experimental data. A fairly good agreement is attained. Predictions of overall flow characteristics suggest that the results are not very sensitive to different approximations of the preassumed applied velocity and pressure distribution shapes.     

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.     

Multilayer Averaged and Moment Equations for One-Dimensional Open-Channel Flows

A model is developed to account for the vertical distribution of velocity and nonhydrostatic pressure in one-dimensional open-channel flows. The model is based on both classical multilayer models and depth-averaged and moment equations. The establishment of its governing equations and the flow simulation are performed over a number of flow layers as in classical multilayer models. However, the model also allows for vertical distributions within a flow layer by including both Boussinesq terms and effective stress terms due to depth-averaging operations. These terms are evaluated on the basis of vertically linearly approximated profiles of velocity and pressure. The resulting additional coefficients can be solved by the moment equations for the relevant layers. Three verifications demonstrate satisfactory simulations for water surface profile, as well as vertical distributions for horizontal velocity, vertical velocity, and nonhydrostatic pressure. Sensitivity analysis shows that the model can be applied with fewer flow layers, more flexibility of layer division, and less computational cost than classical multilayer models, without a remarkable compromise in accuracy.     

Flux-Difference Splitting Schemes for 2D Flood Flows

Two numerical models for 2D flood flows are presented. One model is first-order accurate and another is second-order accurate. Roe's numerical flux is used to develop the first-order accurate model, while second-order accuracy, in space and time, is obtained by using the Lax-Wendroff numerical flux. A simple operator splitting is found to yield the same results as that obtained by using more complicated, and thus, time consuming, operator splitting. Roe's approximate Jacobian is used for conservative properties and Harten and Hyman's procedure is followed for the entropy inequality condition. Flux limiter is used in the second-order accurate model that removes oscillations while maintaining the order of accuracy. The models are verified against available experimental data of a 2D flood wave due to partial dam-break. Numerical experiments are conducted to verify the models' ability to correctly predict behavior of the free surface, in addition to prediction of depth and velocity.     

Fully Nonlinear Boussinesq-Type Equations for Waves and Currents over Porous Beds

The paper introduces a complete set of Boussinesq-type equations suitable for water waves and wave-induced nearshore circulation over an inhomogeneous, permeable bottom. The derivation starts with the conventional expansion of the fluid particle velocity as a polynomial of the vertical coordinate z followed by the depth integration of the vertical components of the Euler equations for the fluid layer and the volume-averaged equations for the porous layer to obtain the pressure field. Inserting the kinematics and pressure field into the Euler and volume-averaged equations on the horizontal plane results in a set of Boussinesq-type momentum equations with vertical vorticity and z-dependent terms. A new approach to eliminating the z dependency in the Boussinesq-type equations is introduced. It allows for the existence and advection of the vertical vorticity in the flow field with the accuracy consistent with the level of approximation in the Boussinesq-type equations for the pure wave motion. Examination of the scaling of the resistance force reveals the significance of the vertical velocity to the pressure field in the porous layer and leads to the retention of higher-order terms associated with the resistance force. The equations are truncated at O(μ4), where μ = measure of frequency dispersion. An analysis of the vortical property of the resultant equations indicates that the energy dissipation in the porous layer can serve as a source of vertical vorticity up to the leading order. In comparison with the existing Boussinesq-type equations for both permeable and impermeable bottoms, the complete set of equations improve the accuracy of potential vorticity as well as the damping rate. The new equations retain the conservation of potential vorticity up to O(μ2). Such a property is desirable for modeling wave-induced nearshore circulation but is absent in existing Boussinesq-type equations.     

Nonhydrostatic Three-Dimensional Method for Hydraulic Flow Simulation. II: Validation and Application

A three-dimensional computational method, without the use of hydrostatic assumption, is developed to solve fluid flows for hydraulic applications. Numerical algorithms and verification of the nonhydrostatic model are described in our companion paper. The model employs unstructured grid technology with arbitrarily shaped cells, offering the potential to unify many grid topologies into a single formulation. Herein, the model is applied to two practical steady hydraulic flows to provide further validation of the model and demonstrate its use in practical flows. The flows in a hydroturbine draft tube and in the forebay of Rocky Reach Dam for the fish passage facility design are simulated. Comparisons with experimental data in the former and physical and field measurements in the latter establish the scope of the model.     

Lagrangian Simulation of One-Dimensional Dam-Break Flow

The note demonstrates the application of a pure Lagrangian numerical method to dam-break flows by solving the St. Venant equations. The method is developed based on the smoothed particle hydrodynamics. It is easy to apply and is shown to be capable of providing accurate simulations for mixed flow regimes with strong shocks.     

Extended Boussinesq Equations for Water-Wave Propagation in Porous Media

This paper presents a new Boussinesq-type model equations for describing nonlinear surface wave motions in porous media. The mathematical model based on perturbation approach reported by Hsiao et al. is derived. The drag force and turbulence effect suggested by Sollitt and Cross are incorporated for observing the flow behaviors within porous media. Additionally, the approach of Chen for eliminating the depth-dependent terms in the momentum equations is also adopted. The model capability on an applicable water depth range is satisfactorily validated against the linear wave theory. The nonlinear properties of model equations are numerically confirmed by the weakly nonlinear theory of Liu and Wen. Numerical experiments of regular waves propagating in porous media over an impermeable submerged breakwater are performed and the nonlinear behaviors of wave energy transfer between different harmonics are also examined.     

Upwind Conservative Scheme for the Saint Venant Equations

An upwind conservative scheme with a weighted average water-surface-gradient approach is proposed to compute one-dimensional open channel flows. The numerical scheme is based on the control volume method. The intercell flux is computed by the one-sided upwind method. The water surface gradient is evaluated by the weighted average of both upwind and downwind gradients. The scheme is tested with various examples, including dam-break problems in channels with rectangular and triangular cross-sections, hydraulic jump, partial dam-break problem, overtopping flow, a steady flow over bump with hydraulic jump, and a dam-break flood case in a natural river valley. Comparisons between numerical and exact solutions or experimental data demonstrated that the proposed scheme is capable of accurately reproducing various open channel flows, including subcritical, supercritical, and transcritical flows. The scheme is inherently robust, stable, and monotone. The scheme does not require any special treatment, such as artificial viscosity or front tracking technique, to capture steep gradients or discontinuities in the solution.     

Modeling 3D Supercritical Flow with Extended Shallow-Water Approach

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

Simulation of the St. Francis Dam-Break Flood

A two-dimensional (2D) simulation of flooding from the 1928 failure of St. Francis Dam in southern California is presented. The simulation algorithm solves shallow-water equations using a robust unstructured grid Godunov-type scheme designed for wetting and drying and achieves good results. Flood extent and flood travel time are predicted within 4 and 10% of observations, respectively. Representation of terrain by the mesh is identified as the dominant factor affecting accuracy, and an iterative process of mesh refinement and convergence checks is implemented to minimize errors. The most accurate predictions are achieved with a uniformly distributed Manning n = 0.02. A 50% increase in n increases travel time errors to 25% but has little effect on flood extent predictions. This highlights the challenge of a priori travel time prediction but robustness in flood extent prediction when topography is well resolved. Predictions show a combination of subcritical and supercritical flow regimes. The leading edge of the flood was supercritical in San Francisquito Canyon, but due to channel tortuosity, the wetting front reflected off canyon walls causing a transition to subcritical flow, considerably larger depths, and a standing wave in one particular reach that accounts for a 30% fluctuation in discharge. Elsewhere, oblique shocks locally increased flood depths. The 2D dam-break model is validated by its stability and accuracy, conservation properties, ability to calibrate with a physically realistic and simple resistance parametrization, and modest computational cost. Further, this study highlights the importance of a dynamic momentum balance for dam-break flood simulation.     

Shallow Water Wave Propagation in Convectively Accelerating Open-Channel Flow Induced by the Tailwater Effect

Propagation of shallow water waves in viscous open-channel flows that are convectively accelerating or decelerating under gradually varying water surface profiles is theoretically investigated. Issues related to the hydrodynamics of wave propagation in a rectangular open channel are studied: the effect of viscosity in terms of the Manning coefficient; the effect of gravity in terms of the Froude number; wave translation and attenuation characteristics; nonlinearity and wave shock; the role of tailwater in wave propagation; and free surface instability. A uniformly valid nonlinear solution to describe the unsteady gradually varying flow throughout the complete wave propagation domain at and away from the kinematic wave shock as well as near the downstream boundary that exhibits the tailwater effect is derived by employing the matched asymptotic method. Different scenarios of hydraulically spatially varying surface profiles such as M1, M2, and S1 type profiles are discussed. Results from the nonlinear wave analysis are further interpreted and the influence of the tailwater effect is identified. In addition to the nonlinear wave analysis, a linear stability analysis is introduced to quantify the impact from such water surface profiles on the free surface instability. It is shown that the asymptotic flow structure is composed of three distinct regions: an outer region that is driven by gravity and channel resistance; a near wave shock region dominated by the convective inertia, pressure gradient, gravity and channel resistance; and a downstream boundary impact region where the convective inertia, pressure gradient, gravity and channel resistance terms are of importance. The tailwater effect is demonstrated influential to the flow structure, free surface stability, wave transmission mechanism, and hydrostatic pressure gradient in flow.     

Numerical Solution of the Dam-Break Problem with a Discontinuous Galerkin Method

A discontinuous Galerkin method for the solution of the dam-break problem is presented. The scheme solves the shallow water equations with spectral elements, utilizing an efficient Roe approximate Riemann solver in order to capture bore waves. The solution is enhanced by a projection limiter that eliminates spurious oscillations near discontinuities. The main advantage of the model is the flexibility in approximating smooth solutions with high-order polynomials and resolving at the same time discontinuous shock waves. Furthermore, the finite element discretization is capable of handling complex geometries and producing correct results near the boundaries. Both the h- and p-type extensions are investigated for the one-dimensional dam break, and the results are verified by comparison with analytical solutions. The application to a two-dimensional dam-break problem shows the efficiency and stability of the method.     

SWAN-Mud: Engineering Model for Mud-Induced Wave Damping

This paper describes the implementation of a new dispersion relation and energy-dissipation equation obtained from a viscous two-layer model schematization in the state-of-the-art wave forecasting model SWAN to simulate wave damping in coastal areas by fluid mud deposits. This new dispersion relation is derived for a nonviscous, nonhydrostatic upper layer and a viscous, hydrostatic lower layer, covering most conditions encountered in nature. An algorithm is developed for a robust numerical solution of this new implicit dispersion relation through proper starting values in the iteration procedure. The implementation is tested against a series of analytical solutions and three schematic test cases. Next, four dispersion relations published in the literature are evaluated and compared with the new dispersion relation. The solution of the dispersion relations forms a multidimensional space. Comparison of the various model solutions through one-dimensional graphs can therefore become quite misleading, as shown in the discussion of a two-dimensional representation of the solution space, explaining for instance the variation in ambient conditions at which maximum wave damping is to be expected. The various models have been developed for a variety of conditions, such as shallow and deep water and shallow and thick mud layers; the various models agree well in their domain of applicability, but they show significant deviations when used outside their domain. Because the ambient and mud conditions may vary considerable in space and time at a particular site, the use of the new model is advocated because it covers most water depths and fluid mud thicknesses encountered in nature. The strength of the new SWAN-mud model lies in its large-scale applicability, assessing the two-dimensional evolution of wave fields in coastal areas. Therefore, the new implementation is evaluated with respect to the behavior of waves on a sloping seabed, representing real-world coasts. In all cases, the new SWAN-mud model behaves satisfactorily; a critical remaining issue, though, is the assessment of the relevant fluid mud parameters.