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.     

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.     

Laminar and Turbulent Bottom Boundary Layer Induced by Nonlinear Water Waves

Results of a numerical study to investigate wave-induced boundary layer flows are reported. In this study, the writers consider a coupled viscous-inviscid approach, in which the fully nonlinear free surface boundary conditions are satisfied in the inviscid flow calculation, while the viscous flow near the seabed is solved via the Reynolds-averaged Navier-Stokes equations, instead of the thin boundary layer equation. To simulate the turbulent flow, a two-layer k-ε model is applied. Coupling of the viscous and inviscid computations is accomplished by the direct matching of the velocity and pressure distributions on the matching boundaries. Validation of the numerical model is carried out separately for the inviscid and viscous models, and the coupling approach as a whole. The numerical results are compared with theoretical solutions and available experimental data. A parametric study of the laminar and turbulent boundary layers for highly and weakly nonlinear waves is performed using the coupled viscous-inviscid approach. The results are compared with corresponding U-tube simulations, and the discrepancy is highlighted and discussed.     

Generation of Three-Dimensional Fully Nonlinear Water Waves by a Submerged Moving Object

This paper describes a numerical investigation on the generation of three-dimensional (3D) fully nonlinear water waves by a submerged object moving at speeds varied from subcritical to supercritical conditions in an unbounded fluid domain. Considering a semispheroid as the moving object, simulations of the time evolutions of 3D free-surface elevation and flow field are performed. The present 3D model results are found to agree reasonably well with other published vertical two-dimensional (2D) and quasi-3D numerical solutions using Boussinesq-type models. Different from the 2D cases with near critical moving speeds, the 3D long-term wave pattern suggests that in addition to the circularly expanded upstream advancing solitonlike waves, a sequence of divergent and transverse waves are also developed behind the moving object. The velocity distributions and associated fluid-particle trajectories at the free-surface and middle layers are presented to show the 3D feature of the motion. The results under various vertical positions (referred as gap) of a moving object are also compared. It is found the gap has shown a substantial influence on the generated waves, especially in the wake region, when an object moves at a near critical or subcritical speed. However, the results under the case with a high supercritical moving speed indicate the gap has a negligible effect on the generated upstream and downstream waves.     

Effects of Weakly Nonlinear Water Waves on Soft Poroelastic Bed with Finite Thickness

Since porous material is usually of a finite thickness in nature, the effects of periodically nonlinear water waves propagating over a soft poroelastic bed with finite thickness are hence noticed and studied in this work. The water waves are simulated by potential theory while the porous bed is governed by Biot’s theory of poroelasticity herein. The conventional Stokes expansion of water waves based on a one-parameter perturbation expansion fails to solve the soft poroelastic bed problem; therefore, the boundary layer correction approach combined with a two-parameter perturbation expansion is proposed, which enables us to solve the problem of soft poroelastic bed with finite thickness. The results are compared to the similar problem with an infinite-thickness porous bed. The boundary effects of the impervious rock are significant on wave-induced pore water pressure and effective stresses, but are of very little significance on wave profiles at the free surface and the porous bed surface. However, the rigid boundary is insignificant to the pore water pressure and effective stresses when the thickness of porous bed is larger than about one wavelength.     

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.     

Modeling Soil Water Uptake by Plants Using Nonlinear Dynamic Root Density Distribution Function

Water uptake by plants is one of the major components of water balance of the vadose zone that greatly influences the contaminant and moisture movement in variably saturated soils. In this study, a nonlinear macroscopic root water uptake model that includes the impact of soil moisture stress is developed. The model incorporates the spatial and temporal variation of root density in addition to the dynamic root depth considerations. The governing moisture flow equation coupled with the water extraction by plants term is solved numerically by an implicit finite-difference method. The simulation is performed for various physical scenarios subjected to different boundary conditions. The model is tested first without considering the water uptake and results are compared with observed data available in the literature for two cases. A nonlinear water uptake term is subsequently incorporated in the model which is then simulated for corn crop for constant root depth under various characteristic moisture availability environments. Results show that the water extraction rate is closely related to the soil moisture availability in addition to the root density. The plants are observed to extract moisture mainly from the upper root dense soil profile when water content is in an optimal range, otherwise, the peak of the uptake moves to other soil layers where the moisture is easily available. Finally, the model is applied to a corn field and simulated results are validated with field data. The simulated moisture content for 2 months of crop growing season shows a reasonably good agreement with the observed data.     

Modeling Acoustic Waves in Saturated Poroelastic Media

In this paper we present a comparison of the linear wave analysis for four models of poroelastic materials. A nonlinear thermodynamical construction of a two-component model of such materials requires a dependence on the porosity gradient. In the linear version this dependence may or may not be present. Consequently, we may work with the model without a dependence on this gradient which is identical to Biot’s model or we can use the so-called full model. In both cases we can construct simplified models without a coupling between partial stresses introduced by Biot. These simplified models have the advantage that their application to, for instance, surface wave analysis yields much simpler mathematical problems. In the present work we show that such a simplification for granular materials leads to a good qualitative agreement of all four models in ranges of porosity and Poisson’s ratio commonly appearing in geotechnical applications. Quantitative differences depend on the mode of propagation and vary between 10 and 20%. We illustrate the analysis with a numerical example corresponding to data for sands. Simultaneously we demonstrate severe limitations of the applicability of Gassmann relations which yield an instability of models in a wide range of practically important values of parameters.     

Nonlinear Response of Fluid-Filled Membrane in Gravity Waves

This paper describes a time-domain model for the nonlinear response of fluid-filled membranes in gravity waves. A formulation based on the principle of virtual work provides an integral governing equation for membrane deformation that fully accounts for geometric nonlinearity, which is known to be important even for relatively small deformation. The incident wave amplitude and membrane deformation are considered to be small, to allow linearization of the hydrodynamic problems. The potential flows inside and outside the membrane are solved by two boundary element models, which are coupled to the finite element model of the membrane. An iterative scheme based on Newmark’s method integrates the resulting nonlinear equations of motion in time. The computed results for a bottom-mounted fluid-membrane system show favorable agreement with available experimental and numerical data. Membrane geometric nonlinearity increases the system stiffness due to strain-stiffening and gives rise to hysteresis response at some frequencies.     

Modeling the Behavior of Flow Regulating Devices in Water Distribution Systems Using Constrained Nonlinear Programming

Currently the modeling of check valves and flow control valves in water distribution systems is based on heuristics intermixed with solving the set of nonlinear equations governing flow in the network. At the beginning of a simulation, the operating status of these valves is not known and must be assumed. The system is then solved. The status of the check valves and flow control valves are then changed to try to determine their correct operating status, at times leading to incorrect solutions even for simple systems. This paper proposes an entirely different approach. Content and co-content theory is used to define conditions that guarantee the existence and uniqueness of the solution. The work here focuses solely on flow control devices with a defined head discharge versus head loss relationship. A new modeling approach for water distribution systems based on subdifferential analysis that deals with the nondifferentiable flow versus head relationships is proposed in this paper. The water distribution equations are solved as a constrained nonlinear programming problem based on the content model where the Lagrangian multipliers have important physical meanings. This new method gives correct solutions by dealing appropriately with inequality and equality constraints imposed by the presence of the flow regulating devices (check valves, flow control valves, and temporarily closed isolating valves). An example network is used to illustrate the concepts.     

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 Simulations of Nonlinear Short Waves Using a Multilayer Model

The multilayer model developed by Lynett and Liu is used for simulating the evolution of deep-water waves in a constant depth. The computational model is tested with experimental data for nonlinear monochromatic and biharmonic waves with kh values as high as 8.3. The experiments were conducted in a super wave tank with dimensions of 300?m×5?m×5.2?m located at Tainan Hydraulics Laboratory of National Cheng-Kung University. The nonlinearity of the waves tested, ka, range from 0.0627 to 0.1577. The overall comparisons between the multilayer model and the experiments are quite good, indicating that the multilayer model is adequate for both linear and nonlinear deep-water waves.     

Modeling Nonlinear Systems by Volterra Series

The Volterra-series expansion is widely employed to represent the input-output relationship of nonlinear dynamical systems. This representation is based on the Volterra frequency-response functions (VFRFs), which can either be estimated from observed data or through a nonlinear governing equation, when the Volterra series is used to approximate an analytical model. In the latter case, the VFRFs are usually evaluated by the so-called harmonic probing method. This operation is quite straightforward for simple systems but may reach a level of such complexity, especially when dealing with high-order nonlinear systems or calculating high-order VFRFs, that it may loose its attractiveness. An alternative technique for the evaluation of VFRFs is presented here with the goal of simplifying and possibly automating the evaluation process. This scheme is based on first representing the given system by an assemblage of simple operators for which VFRFs are readily available, and subsequently constructing VFRFs of the target composite system by using appropriate assemblage rules. Examples of wind and wave-excited structures are employed to demonstrate the effectiveness of the proposed technique.     

RANS∕Laplace Calculations of Nonlinear Waves Induced by Surface-Piercing Bodies

An interactive zonal numerical method has been developed for the prediction of free surface flows around surface-piercing bodies, including both viscous and nonlinear wave effects. In this study, a Laplace solver for potential flow body-wave problems is used in conjunction with a Reynolds-averaged Navier-Stokes (RANS) method for accurate resolution of viscous, nonlinear free surface flows around a vertical strut and a series 60 ship hull. The Laplace equation for potential flow is solved in the far field to provide the nonlinear waves generated by the body. The RANS method is used in the near field to resolve the turbulent boundary layers, wakes, and nonlinear waves around the body. Both the kinematic and dynamic boundary conditions are satisfied on the exact free surface to ensure accurate resolution of the divergent and transverse waves. The viscous-inviscid interaction between the potential flow and viscous flow regions is captured through a direct matching of the velocity and pressure fields in an overlapping RANS and potential flow computational region. The numerical results demonstrate the capability of an interactive RANS∕Laplace coupling method for accurate and efficient resolution of the body boundary layer, the viscous wake, and the nonlinear waves induced by surface-piercing bodies.     

Propagation of Long Water Waves through Branching Channels

This paper investigates the fundamental behavior of long water waves propagating through branching channels of uniform depth and width. Numerical simulations based on the Boussinesq long wave model were carried out to investigate the effects of channel width b, effective wavelength λe, wave amplitude α, and angle θ between channel branches on wave transmission and reflection. Our results showed that the transmission and reflection of long waves through branching channels are dominantly governed by a single dimensionless parameter b/λe, whereas other parameters such as the angle between channel branches are less important. Detailed quantitative results for predicting long wave transmission and reflection in different branching channels based on the similarity parameter b/λe were determined. From these results, we have discovered some very interesting and distinct long wave behaviors in narrow and wide branching channels. To verify the numerical results, experiments were conducted in a right-angled branching channel, and the experimental results showed good agreement with the numerical predictions.     

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.     

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.     

Modeling Controlled Water Systems

The particular challenges of modeling controlled water systems are discussed. The high degree of freedom due to the control structures increases the risk of producing the right output for the wrong reasons. On the other hand, many controlled water systems are (partly) manually operated or at least supervised by an operational water manager. The decisions of these managers are not as rigid as a computer simulated control strategy. Therefore, getting a very close fit with a water-system control model is mostly not possible. A modeling framework is proposed that takes advantage of the vast availability of measurement data in controlled water systems. The water level and flow data at control structures allow for intensive validation and subsystem calibration to reduce the degree of modeling freedom and to model separately the natural rainfall-runoff and hydrodynamic processes. The framework is successfully applied to improve a simulation model of the controlled water system of Rijnland, The Netherlands. The yearly volume error was reduced from 11% to less than 1% and as a consequence, the short-term peak events were modeled more accurately as well. The resulting water-system control model is more reliable for both design studies and operational decision support. The framework will contribute to prepare more reliable simulation models of controlled water systems.     

Nonhydrostatic Modeling of Nonlinear Deep-Water Wave Groups

A nonhydrostatic model with a higher-order top-layer pressure treatment is developed. Accuracy with respect to linear wave dispersion and wave nonlinearity is carefully examined. The model is thereafter applied to simulate nonlinear deep-water wave groups. For slowly modulated wave groups, the model well predicts the characteristics of bichromatic waves better than those obtained by the fourth-order nonlinear Schr?dinger equation and the multilayer Boussinesq model. For fast evolution of focusing wave groups, the model accurately captures the limiting extreme wave conditions. Particularly the predicted local wave steepness of a narrow-banded wave group is higher than that of a broad-banded wave group, supporting the importance of spectral bandwidth in determining the limiting wave condition in the previous study. Overall, the agreement between the model’s results and experimental data are excellent, demonstrating the capability of the model on resolving wave-wave interactions in the nonlinear deep-water wave groups.     

General Third-Order Solutions for Irregular Waves in Deep Water

The third-order analytical solutions for the strong interactions among three free-wave components are derived using two well-known perturbation methods, namely, the mode-coupling method and the Zakharov equation method. A term-by-term comparison of their solutions shows that the two methods render virtually identical results. The validity and convergence range of the solution are investigated for the interacting free-wave components of various frequencies but within the frequency range of storm seas. It is found that the solution may encounter two types of convergence difficulties, which occur, respectively, in the cases (1) when two of the interacting free-wave components are of quite different frequencies; and (2) when two of the interacting free-wave components are very close in frequency but the frequency of the third free-wave component is relatively larger or smaller.