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 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.     

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.     

Analytical Solution of the Burgers Equation for Simulating Translatory Waves in Conveyance Channels

A Burgers equation model (BEM) for simulating translatory waves in conveyance channels is extracted from the Saint-Venant equations for small perturbations in initial uniform flow. The present study improves upon the previous model and presents analytical solutions for the simulation of translatory waves occurring in conveyance channels. The BEM is reduced to the linear diffusion equation using the Cole–Hopf transformation and then solved by means of the Green’s function assuming an infinite domain. The simulation studies performed show that the BEM results are comparable to those of the Saint-Venant equations for small perturbations in the initial uniform flow conditions and for Froude numbers within the subcritical region. The BEM could be useful for flood routing and for simulating release of water from a reservoir into a conveyance channel when the flow perturbation is small.     

Depth-Averaged Two-Dimensional Numerical Modeling of Unsteady Flow and Nonuniform Sediment Transport in Open Channels

A depth-averaged two-dimensional (2D) numerical model for unsteady flow and nonuniform sediment transport in open channels is established using the finite volume method on a nonstaggered, curvilinear grid. The 2D shallow water equations are solved by the SIMPLE(C) algorithms with the Rhie and Chow’s momentum interpolation technique. The proposed sediment transport model adopts a nonequilibrium approach for nonuniform total-load sediment transport. The bed load and suspended load are calculated separately or jointly according to sediment transport mode. The sediment transport capacity is determined by four formulas which are capable of accounting for the hiding and exposure effects among different size classes. An empirical formula is proposed to consider the effects of the gravity on the sediment transport capacity and the bed-load movement direction in channels with steep slopes. Flow and sediment transport are simulated in a decoupled manner, but the sediment module adopts a coupling procedure for the computations of sediment transport, bed change, and bed material sorting. The model has been tested against several experimental and field cases, showing good agreement between the simulated results and measured data.     

One-Dimensional Model of Shallow Water Surface Waves Generated by Landslides

A numerical model is presented as the basis for the study of surface waves generated by bank and bottom landslides in rivers. The flow is assumed to be properly modeled by the shallow water equations. The solid movement is introduced in the model as an external action, and assumed rigid and impervious. Two situations are identified in the flow subsequent to a solid movement: longitudinal and transversal sliding. A discussion on the modeling difficulties associated with the latter is included. The flow equations are solved by means of an upwind scheme specially adapted to advancing fronts over dry irregular beds.     

Effect of Long Internal Waves on the Quality of Water Withdrawn from a Stratified Reservoir

The properties of water withdrawn from a stratified reservoir are investigated in a field study conducted in Lake Burragorang, Australia. It is shown that temperature and turbidity fluctuations of the extracted water are directly correlated to the vertical displacement of the thermal structure of the reservoir immediately in front of the offtake and the thickness of the selective withdrawal layer. Scaling of the unsteady withdrawal revealed that the timescale associated with the formation of selective withdrawal is an order of magnitude smaller than the typical period of the internal wave. This means the withdrawal layer is acting as a filter, extracting water of a particular quality as it is swept past the outlet by the internal seiches; the steady-state theory of the selective withdrawal can be used to predict outflow temperature fluctuations in reservoirs where long internal waves are present. To correctly interpret other outflow water parameters, such as turbidity or dissolved oxygen, it is important not only to know the stratification conditions in front of the offtake, but also to understand the local flow dynamics in the lower reaches of the reservoir.     

One-Dimensional Study on Propagation of Tsunami Wave in River Channels

A study of one-dimensional tsunami propagation up river channels is presented. Laboratory experiments were conducted to examine the wave propagation characteristics and provide data for validating a numerical model. The validated numerical model, employing a Boussinesq-type equation was applied to the Tokachi-oki Earthquake tsunami which occurred on September 2003 in Hokkaido, Japan. The computational results of arrival time and water level at each wave gauge agree well with the observed data.     

Channel Bed Slope Effect on the Height of Gravity Waves Produced by a Sudden Downstream Discharge Stoppage

We derive a simple analytical correction of a well-known standard formulation of the gravity wave height produced in a prismatic channel due to a sudden discharge stoppage at the downstream end of the channel. The proposed analytical correction considers the vertical growth of the wave and, as a result, takes into account the effect of the channel bed slope on the wave height. This simple correction is useful to be considered in preliminary designs of relatively long channels subject to unsteady flow conditions.     

Laboratory Study of Unidirectional Focusing Waves in Intermediate Depth Water

The results of laboratory measurements of large focusing wave groups, which were generated using the New Wave theory, are presented. The influences of both the steepness and frequency bandwidth on focused wave characteristics were examined. The influence of frequency bandwidth on focused wave groups with small and moderate steepness was very small. However, for cases with the large steepness, the nonlinearity increased with increasing bandwidth frequency and widened free-wave regimes are identified for those cases with large steepness at the focal location. The underlying nonlinear phase coupling of focused waves was examined using wavelet-based bicoherence and biphase, which can detect nonlinear phase coupling in a short time series. For wave groups with large initial steepness, as wave groups approached the focal location, the values of bicoherence between primary waves and its higher harmonics progressively increased to 1 and the corresponding biphase was gradually close to zero, suggesting that an extreme wave event can be produced by considering Stokes-like nonlinearity to very high-order. Furthermore, the fast change of bicoherence of focused wave groups indicates that the nonlinear energy transfer within focusing waves is faster than that of nonfocusing wave trains.     

Interaction of Nonlinear Progressive Waves with Two Serially Arranged Submerged Obstacles

The purpose of the present study is to develop a numerical model for the investigation of water waves propagating over a pair of impermeable submerged obstacles. The mathematic model is formulated by coupling solutions of the Navier–Stokes equations and transport equations for the surface elevation using the volume of fluid method. Based on a staggered computational mesh, an explicit numerical algorithm is employed with a predictor–corrector procedure of pressure and velocity field. The proposed model provides good agreement with other experimental results and validates its good performance. Regarding the spatial harmonic evolutions of various cases, it is noted that the present fluctuating mode of harmonic amplitudes exists upstream and at the gap between obstacles. The results show that the nonlinearity of propagating waves becomes stronger than the initial wave in such areas, and reveals much steeper wave profiles compared to the initial ones. The fluctuating harmonic amplitudes vary with the gap width and form two hydrodynamic cycles. The vortices play an important role in the wave reflection as they form a water column wall to reflect the incoming waves. The reflection ratio depends on the extent of vortex development near the upstream obstacle. The maximum wave reflection occurs in cases with dimensionless gap width S/L equal to 3/8 and 7/8 in this study.     

Flow Conditions of Undular Hydraulic Jumps in Horizontal Rectangular Channels

The flow conditions of undular jumps for fully developed inflow condition have been investigated systematically. If the inflow Froude number is larger than 1.2, an undular jump has lateral shock waves near the toe of the jump. For a narrow channel, the shock waves cross upstream of the first wave crest, and the flow conditions of undular jumps depend on the aspect ratio and the inflow Froude number. In contrast, for a wide channel the shock waves do not cross upstream of the first wave crest, and the flow conditions of undular jumps are independent of the aspect ratio. The flow conditions of undular jumps are classified by considering the cross position of the lateral shock waves and the inflow Froude number. Also, the hydraulic conditions for the formation of nonbreaking and breaking undular jumps are determined. The effect of the Reynolds number on undular-jump formations is discussed, and changes of the flow conditions with the Reynolds number are described.     

Analysis of Seepage from Polygon Channels

An exact analytical solution for the quantity of seepage from a trapezoidal channel underlain by a drainage layer at a shallow depth has been obtained using an inverse hodograph and a Schwarz-Christoffel transformation. The symmetry about the vertical axis has been utilized in obtaining the solution for half of the seepage domain only. The solution also includes relations for variation in seepage velocity along the channel perimeter and a set of parametric equations for the location of phreatic line. From this generalized case, particular solutions have also been deduced for rectangular and triangular channels with a drainage layer at finite depth and trapezoidal, rectangular, and triangular channels with a drainage layer and water table at infinite depth. Moreover, the analysis includes solutions for a slit, which is also a special case of polygon channels, for both cases of the drainage layer. These solutions are useful in quantifying seepage loss and/or artificial recharge of groundwater through polygon channels.     

Two-Dimensional SPH Simulations of Landslide-Generated Water Waves

Water waves generated by landslides have been of interest to ocean and coastal engineers, as well as to dam engineers. The present study uses a meshless and pure Lagrangian method known as smoothed particle hydrodynamics (SPH) to simulate nonlinear water waves due to landslides, with the aim of an accurate numerical prediction of the generation and propagation of such water waves. Validation is carried out by comparison between the computed prediction and experimental data of water waves generated by a two-dimensional triangular rigid body sliding into water. The calculated results show that the simulated water waves agree well with those observed in the experiment confirming the effectiveness and the accuracy of the proposed scheme.     

Numerical Investigation into Secondary Currents and Wall Shear in Trapezoidal Channels

This paper presents the use of computational fluid dynamics (CFD) to determine the distribution of the bed and sidewall shear stresses in trapezoidal channels. The impact of the variation of the slant angle of the side walls, aspect ratio, and composite roughness on the shear stress distribution is analyzed. The shear stress data can be directly output from the CFD models at the boundaries, but they can also be derived using the Guo and Julien equations for the average bed and side wall shear stresses. These equations compute the shear stress as a function of three components; gravitational, secondary flows, and interfacial shear stress, and are hence used to gauge the respective merits of the different components of wall shear. The results show a significant contribution from the secondary currents and internal shear stresses on the overall shear stress at the boundaries. This work also extends previous work of the authors on rectangular channels.     

Characterization of Heterogeneous Soils Using Surface Waves: Homogenization and Numerical Modeling

Seismic surface waves are well-adapted to study the elastic parameters, and hence the mechanical properties, of soils. The aim herein is to evaluate whether Rayleigh waves in heterogeneous soils may be used to estimate average elastic parameters and to determine how these parameters are influenced by heterogeneities. The heterogeneous medium, underlain by a homogeneous half-space, is considered as a homogeneous matrix with one or several types of randomly distributed inclusions (with a normal distribution) in the matrix. Seismic waves generated by surface loads and propagating in this medium were calculated using the finite element method (FEM) and then compared with single- and multiple-scattering homogenization methods. For the FEM calculation, special care was taken to reduce numerical dispersion through the use of elements smaller than 1/20 of the dominant wavelength. The group and phase velocity dispersion curves were measured and inverted in order to obtain the effective shear wave velocity of the heterogeneous medium. The results show a clear dependence of the wave velocity with respect to the nature, concentration, and size of the inclusions. The dependence with respect to the nature and concentration of inclusions coincides with that obtained from a multiple-scattering homogenization method up to an inclusion concentration of approximately 50%.     

Dynamic Analysis of Multilayered Soils to Water Waves and Flow

In nature, a soil profile generally consists of several heterogeneous layers. This study is aimed at discussing the interactive problem of oscillatory water waves and flow passing over multilayered soils. The soil behavior is considered as viscoelastic in the present mathematical model modified from Biot’s poroelastic theory. Employing this model, the dynamic response including the profiles of pore water pressure and effective stress in the multilayered soils is discussed. The results reveal that the perturbed pore pressure is different from that inside a single-layered soil where the thickness of the first soil layer is less than the water wavelength. The discrepancy of the vertical effective stresses between multilayered and single-layered soils is even much more apparent under the same conditions. Moreover, seepage force is examined and is found to be larger near the bed surface and the bottom of the first soil layer where soils are easily disturbed by external disturbance. The locations where soil failure might happen are found near the troughs of surface water waves.     

Transient Hydrodynamic Dispersion in Rough Open Channels: Theoretical Analysis of Bed-Form Effects

A stochastic Lagrangian approach is proposed to describe dispersion in a two-dimensional low-regime river flow past random bed undulations characterized by the superposition of periodical and stationary exponential correlations, through a suitable time-dependent coefficient. The resulting dimensionless expression depends on the overall resistance factor, which is proportional to the average Peclet number of the process. A graphical analysis shows that the oscillatory transient originating from the periodic component of the bottom elevation pattern is enhanced by reduced global flow resistance, while relatively more intense tracer spreading is associated with wavelike profiles affected by persistent trendless random noise, which also determines the characteristic time needed by the plume to achieve the asymptotic domain. Numerical simulations, validating the first-order analytical approach in a wide range of heterogeneous bed geometries, are also discussed. A semianalytical procedure is finally suggested for the study of depth-averaged transport processes in real three-dimensional streams, based on the use of the derived dispersion coefficient.     

Analysis of Water Waves Passing Over a Submerged Rectangular Dike

The analytic solutions of inviscid and viscous water waves passing over a submerged rectangular dike are investigated. Owing to the fact that the orthogonality of eigenfunctions is invalid for viscous wave problem, two newly developed orthogonal inner products are applied to reduce the mathematical difficulty of viscous wave problem. Both inviscid and viscous water wave solutions are obtained under the assumption of linear water wave without separation. It shows that two solutions have no significant kinematic difference but the viscous contribution of dynamic effect is not negligible. Beside giving a better theoretical approach, which reduces the error of the conventional minimal squares method, the result of the present analytical solution can be used to quantitatively evaluate the correctness of experiments and also provides helpful information such as near wall boundary layer thickness and oscillating free surface for computational use.