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.     

Simulation of a 3D Numerical Viscous Wave Tank

A numerical scheme was developed to solve the unsteady three-dimensional (3D) Navier–Stokes equations and the fully nonlinear free surface boundary conditions for simulating a 3D numerical viscous wave tank. The finite-analytic method was used to discretize the partial differential equations, and the marker-and-cell method was extended to treat the 3D free surfaces. A piston-type wave generator was incorporated in the computational domain to generate the desired incident waves. This wave tank model was applied to simulate the generation and propagation of a solitary wave in the wave tank and the diffraction of periodic waves by a semiinfinite breakwater. The computation was carried out by a PC cluster established by connecting several personal computers. The message passing interface (MPI) parallel language and MPICH software were used to write the computer code for parallel computing. High consistency between the numerical results and the theoretical solutions for the wave and velocity profiles confirms the accuracy of the proposed wave tank model.     

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.     

Classifying River Waves by the Saint Venant Equations Decoupled in the Laplacian Frequency Domain

The Saint Venant equations are often combined into a single equation for ease of solution. As a result however, this single equation gives rise to several redundant nonlinear terms that may impose significant limitations on model analyses. In order to avoid this, our paper employs a new procedure that separates, in the Laplace frequency domain, the governing equation of water depth from that of flow velocity and thus enables us to consider two independent equations rather than two coupled ones. The so-obtained analytical solutions are valid for prismatic channels of any shape. Solution validity is assured by repeated comparison with the corresponding numerical solutions based on Crump’s algorithm, which accelerates solution convergence. Utilizing this new procedure, this paper will construct a basic wave spectrum for classifying subcritical flow waves in a prismatic channel. The spectrum is basically a contour plot of the normalized specific energy loss for a small water wave moving in the channel for a finite distance of approximately 100?m. The distance is chosen so that four distinct regions with different contour patterns that represent kinematic, diffusion, gravity, and dynamic waves in a river are shown in the spectrum. By incorporating the spectrum with Ferrick’s criteria and Manning’s formula, a single contour line is also generated, which serves as the boundary of the four regions. Example computations show that the spectrum predicts a similar trend of wave attenuation for waves propagating in a trapezoidal channel. When the rising speed of a wave is of concern, the full Saint Venant equations are solved numerically to reconstruct a similar spectrum good for supercritical flow as well.     

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.     

Comparison between Computed and Experimentally Generated Impulse Waves

Large water waves caused by massive slide impacts are a potential hazard along waterways, coastal areas and Alpine regions. Experimental research has been conducted at the Swiss Laboratory of Hydraulics to assess the risk from landslide-generated impulse waves. Analogously, the Centro Elettrotecnico Sperimentale Italiano performed numerical simulations of initial landslide and consequent impulse wave propagation using two mathematical models based on the conservative shallow-water equations. This paper presents the experimental test results and numerical predictions of impulse waves in a flume for a range of stillwater depths, landslide volumes, and impact velocities at laboratory scale. The comparison between the measured and predicted wave free surface profiles generally produced corresponding wave heights, although the initial wave peak is too steep and arrives too early. Excluding spurious random effects, the relative differences between measured and numerically computed maximum wave heights ranged within ±20%, which can be considered satisfactory from the engineering point of view.     

Numerical investigation of the interface in a continuous steel casting mold water model

In this study, the steady-state Navier-Stokes equations are solved on a curvilinear nonorthogonal grid, following the finite volume approximation, with a pressure prediction-correction method, for the case of a flow in a model steel casting mold. The steel flow is simulated by water flow and the slag layer by an oil film, following conditions of previous experimental studies. The simulation aims at the understanding of the free wave and the interface surface wave behavior and the mechanism that leads to the breakup of the steel-slag interface, and thus induction of impurities inside the final steel product. Boundary conditions are set on the free and the interface surfaces, and an adaptive grid mechanism is used in order to update the grid’s shape so as to follow the wave formation. Several cases have been considered with the inlet velocity parameter, and results concerning the velocity field and the generated waves are reported. It is shown that a critical casting speed exists that leads to wave instability, which may be associated with emulsification phenomena.     

Hydrodynamic Investigation of Coastal Bridge Collapse during Hurricane Katrina

A number of U.S. coastal bridges have been destroyed by hurricanes, including three highway bridges in Mississippi and Louisiana during Hurricane Katrina (2005). This paper addresses three fundamental questions on the coastal bridge failures: (1) what were the hydrodynamic conditions near the failed bridge during the hurricane; (2) what was the cause of the bridge collapse; and (3) what was the magnitude of the hydrodynamic loading on the bridge under the extreme hurricane conditions. Guided by field observations of winds, waves, and water levels, two numerical models for storm surges and water waves are coupled to hindcast the hydrodynamic conditions. Fairly good agreement between the modeled and measured high watermarks and offshore wave heights is found, allowing an estimate of the surge and wave conditions near the bridges in nested domains with higher resolutions. The output of the coupled wave-surge models is utilized to determine the static buoyant force and wave forces on the bridge superstructure based on empirical equations derived from small-scale hydraulic tests for elevated decks used in the coastal and offshore industry. It is inferred that the bridge failure was caused by the wind waves accompanied by the storm surge, which raised the water level to an elevation where surface waves generated by strong winds over a relatively short fetch were able to strike the bridge superstructure. The storm waves produced both an uplift force and a horizontal force on the bridge decks. The magnitude of wave uplift force from individual waves exceeded the weight of the simple span bridge decks and the horizontal force overcame the resistance provided by the connections of the bridge decks to the pilings. The methodology for determining the hydrodynamic forcing on bridge decks can be used to produce a preliminary assessment of the vulnerability of existing coastal bridges in hurricane-prone areas.     

Management of Fluid Mud in Estuaries, Bays, and Lakes. I: Present State of Understanding on Character and Behavior

Fluid mud is a high concentration aqueous suspension of fine-grained sediment in which settling is substantially hindered. It constitutes a significant management problem in rivers, lakes, estuaries, and shelves by impeding navigation, reducing water quality and damaging equipment. Fluid mud accumulations have been observed in numerous locations worldwide, including Savannah Harbor, U.S., the Severn Estuary, U.K., and the Amazon River Delta, Brazil. This paper describes the present state of knowledge on fluid mud characteristics, processes, and modeling. Fluid mud consists of water, clay-sized particles, and organic material and displays a variety of rheological behaviors ranging from elastic to pseudo-plastic. It forms by three principle mechanisms: (1) the rate of sediment aggregation and settling into the near-bottom layer exceeds the dewatering rate of the suspension; (2) soft sediment beds fluidized by wave agitation; and (3) convergence of horizontally advected suspensions. Once formed, fluid mud is transported vertically by entrainment and horizontally by shear flows, gravity, and streaming. If not resuspended, it slowly consolidates to form bed material. Quantitative relationships have been formulated for key fluid mud formation and movement mechanisms, but they rely on empirical coefficients that are often site- or situation-specific and are not generally transferable. Research to define general relationships is needed.     

Extended Fourth-Order Depth-Integrated Model for Water Waves and Currents Generated by Submarine Landslides

In this paper, a preexisting higher-order depth-integrated wave propagation model is extended to include a moving seabed. As a result, the extended model can be applied to both wave propagation and the dynamic process of wave generation by a seabed disturbance such as a submarine landslide. The model has the linear dispersion relation in a form of (4,4) Padè approximant, and approximates the water velocity profiles along the water depth with a fourth-order polynomial of the vertical coordinates. The fourth-order model is aimed at extending the validity of the lower-order depth-integrated models from long waves to both long and shorter waves, as well as improving the approximation of the velocity field from the second order to the fourth order. Laboratory experiments are carried out in a wave flume to study wave generation by a submerged landslide model. Both water waves and water velocities are measured by using resistance-type wave gauges and a particle image velocimetry. The experimental data are then compared with the predicted wave height and water current based on the new model and two existing lower-order Boussinesq-type models. The results clearly show that the new model predicts the fluid velocity more accurately and is also able to predict the shorter trailing waves very well where the traditional Boussinesq model may be inadequate, thus validating the improvement provided by the fourth-order model.     

Dynamic Response of Soft Poroelastic Bed to Nonlinear Water Wave—Boundary Layer Correction Approach

When an oscillatory water wave propagates over a soft poroelastic bed, a boundary layer exists within the porous bed and near the homogeneous water∕porous bed interface. Owing to the effect of the boundary layer, the conventional evaluation of the second kind of longitudinal wave inside the soft poroelastic bed by one parameter, ε1 = k0a, is very inaccurate so that a boundary layer correction approach for a soft poroelastic bed is proposed to solve the nonlinear water wave problem. Hence a perturbation expansion for the boundary layer correction approach based on two small parameters, ε1 and ε2 = k0∕k2, is proposed and then solved. The solutions carried out to the first three terms are valid for the first kind and the third kind of waves throughout the whole domain. The second kind of wave is solved systematically inside the boundary layer, whereas it disappears outside the boundary layer. The result is compared with the linear wave solution of Huang and Song in order to show the nonlinearity effect. The present study is very helpful to formulate a simplified boundary-value problem in numerical computation for soft poroelastic medium with irregular geometry.     

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.     

Time Domain Analysis on the Dynamic Response of a Flexible Floating Structure to Waves

A time-domain numerical method is developed to analyze the hydroelastic responses of flexible floating structures to waves; in which, the boundary element method is applied to evaluate the fluid motion and the finite-element method to analyze the elastic deformation of structure. The dynamic wave-structure interaction is simulated by prescribing the conditions on a wave generation boundary for each time step and by satisfying the continuity of the pressure and displacement on the fluid-structure interface. A time-domain solution is obtained in a predictor-corrector scheme and through a time-stepping computation. The effect of space and time discretizations on the convergence and stability of solution for regular, random and solitary waves is discussed by comparing among numerical solutions. The validity of the present method is verified by comparing it with the experimental results for the three kinds of waves mentioned. Further, the fission of a solitary wave under a flexible floating structure is observed both in numerical analysis and experiments.     

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.     

Laminar Poroelastic Media Flow

The continuity and momentum equations of laminar flow through poroelastic media and the sufficient boundary conditions are found in the present paper. The land subsidence equation, Brinkman equation, Darcy's law, and so forth, can be recovered by the simplified governing equations of the present model. A water wave passing over a poroelastic bed is taken as an example of application of this laminar poroelastic media flow model. It is found from the example that, besides the two kinds of longitudinal waves and one kind of transverse wave inside the porous media that the conventional poroelastic model can provide, a second kind of transverse wave is obtained by this model. It is also found that the tangential stress and flow velocity inside the porous bed, according to the laminar poroelastic media flow model, are very different from those obtained by the conventional potential poroelastic media flow model. Finally, the limiting rigid bed solution of the application example, which is useful for the experimental verifications, is also given in this study.     

Analysis of Dynamic Wave Model for Unsteady Flow in an Open Channel

The models for flood propagation in an open channel are governed by Saint-Venant’s equations or by their simplified forms. Assuming the full form of hyperbolic type nonlinear expressions, the complete or dynamic wave model is obtained. Hence, after first-order linearization procedure, the dispersion relation is obtained by using the classical Fourier analysis. From this expression, the phase and group speed and the variations of the amplitude of the waves are defined and investigated. Adopting Manning’s resistance formula, the effects of the variations of the Froude number, Courant number, and friction parameter are examined in the wave number domain for progressive and regressive waves. For small and high wave numbers, the simplified kinematic and gravity wave models are recovered, respectively. Moreover, the analysis confirms, according to the Vedernikov criterion, the Froude number value corresponding to the stability condition to contrast the development of roll waves. In addition, for stable flow on the group speed versus wave number curves, the results show critical points, maximum and minimum for progressive and regressive waves, respectively.     

Laminar Water Wave and Current Passing Over Porous Bed

The problem of the dynamic interaction of water waves, current, and a hard poroelastic bed is dealt with in this study. Finite-depth homogeneous water with harmonic linear water waves passing over a semi-infinite poroelastic bed is investigated. In order to reveal the importance of viscous effect for different bed forms, viscosity of water is considered herein. In a boundary layer correction approach, the governing equations of the poroelastic material are decoupled without losing physical generality. The contribution of pressure effect and shear effect to the hard poroelastic bed, which is a valuable indication to the mechanism of ripple formation, is clarified in the present study. This approach will be helpful in saving time and storage capacity when it is applied to numerical computation.     

Modeling Nonlinear Dispersive Water Waves

An expository review is given on various theories of modeling weakly to strongly nonlinear, dispersive, time-evolving, three-dimensional gravity-capillary waves on a layer of water. It is based on a new model that allows the nonlinear and dispersive effects to operate to the same full extent as in the Euler equations. Its relationships with some existing models are discussed. Various interesting phenomena will be illustrated with applications of these models and with an exposition on the salient features of nonlinear waves in wave-wave interactions and the related processes of transport of mass and energy.     

Hydrodynamic Performance of Wave-Driven Artificial Upwelling Device

Artificial upwelling and mixing is a new technology for enhancing open ocean mariculture using nutrient-rich deep ocean water. A wave-driven artificial upwelling device was developed based on mathematical modeling and hydraulic experiments. Results of the numerical modeling are in good agreement with hydraulic experiments. The mathematical model was applied to simulate the operation of a prototype device in typical Hawaiian waves. The prototype device consists of a buoy with a water chamber 4.0 m in diameter, a flow-controlling valve, and a long tail pipe 300 m in length and 1.2 m in diameter. This device can produce an upwelling flow of 0.62 m3∕s in regular Hawaiian waves with a period of 12 s and a wave height of 1.9 m. Two analytical solutions to the simplified governing equations were also derived that provide a basis for the derivation of predictive equations, useful for preliminary engineering design and analysis.     

Kinematic and Diffusion Waves: Analytical and Numerical Solutions to Overland and Channel Flow

Flood wave propagation is the unifying concept in representing open channel and overland flow. Therefore, understanding flood wave routing theory and solving the governing equations accurately is an important issue in hydrology and hydraulics. In an attempt to contribute to the understanding of this subject, in this study: (1) an analytical solution is derived for diffusion waves with constant wave celerity and hydraulic diffusivity applied to overland flow problems; and (2) an algorithm is developed using the MacCormack explicit finite difference method to solve the kinematic and diffusion wave governing equations for both overland and open channel flow. The MacCormack method is particularly well suited to approximate nonlinear differential equations. The analytical solutions provide the practicing engineer with computational speed in obtaining results for overland flow problems, and a means to check the validity of the numerical models. On the other hand, for larger scale catchment-stream problems, the verified numerical methods provide efficient and accurate algorithms to obtain solutions. Both the analytical approaches and the MacCormack algorithm are used to solve the same synthetic examples. Comparison of results shows that the numerical and analytical solutions are in close agreement. Furthermore, the MacCormack algorithm is applied to a real catchment: a segment of the Duke University West Campus storm water drainage system. In order to check the accuracy of the results obtained by the MacCormack method, the results are compared to predictions of the Environmental Protection Agency storm water management model (SWMM) as calibrated with measured rainfall and surface runoff flow data. The results obtained from SWMM are in good agreement with the results obtained from applying the MacCormack algorithm.