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.     

Wave Propagation in a Pipe Pile for Low-Strain Integrity Testing

This paper presents an analytical solution methodology for a tubular structure subjected to a transient point loading in low-strain integrity testing. The three-dimensional effects on the pile head and the applicability of plane-section assumption are the main problems in low-strain integrity testing on a large-diameter tubular structure, such as a pipe pile. The propagation of stress waves in a tubular structure cannot be expressed by one-dimensional wave theory on the basis of plane-section assumption. This paper establishes the computational model of a large-diameter tubular structure with a variable wave impedance section, where the soil resistance is simulated by the Winkler model, and the exciting force is simulated with semisinusoidal impulse. The defects are classified into the change in the wall thickness and Young’s modulus. Combining the boundary and initial conditions, a frequency-domain analytical solution of a three-dimensional wave equation is deduced from the Fourier transform method and the separation of variables methods. On the basis of the frequency-domain analytic solution, the time-domain response is obtained from the inverse Fourier transform method. The three-dimensional finite-element models are used to verify the validity of analytical solutions for both an intact and a defective pipe pile. The analytical solutions obtained from frequency domain are compared with the finite-element method (FEM) results on both pipe piles in this paper, including the velocity time history, peak value, incident time arrival, and reflected wave crests. A case study is shown and the characteristics of velocity response time history on the top of an intact and a defective pile are investigated. The comparisons show that the analytical solution derived in this paper is reliable for application in the integrity testing on a tubular structure.     

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.     

Spectral Collocation Model for Solitary Wave Attenuation and Mass Transport over Viscous Mud

In this paper, wave attenuation and mass transport of a water-mud system due to a solitary wave on the free surface is modeled by using the Chebyshev-Chebyshev collocation spectral method for spatial discretization and a fourth-order multistage scheme for time integration. The governing equations are formulated in Lagrangian coordinates and perturbation equations for shallow water waves are derived. An iteration-by-subdomain technique is introduced to tackle the interface in the two-layer system. The numerical model is tested against available analytical solutions and good agreement has been found. Numerical simulations of the water-mud system with different layer thicknesses suggest that the accuracy of the existing boundary layer theory for fluid-mud interaction is limited when the mud layer is thin because the assumption of irrotational core may not be valid. Although the paper is focused on solitary waves and Newtonian fluid-mud, the methodology can be extended to oscillatory, nonlinear water waves over a non-Newtonian mud bottom.     

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.     

Wave Interaction with a Flexible Porous Breakwater in a Two-Layer Fluid

An analysis has been carried out to study the performance of a flexible porous plate breakwater in a two-layer fluid where each fluid is assumed to be of finite depth and the breakwater is extended over the entire water depth. The problem is analyzed in two dimensions with the assumption of small amplitude wave theory and plate response. The effects of both surface and internal waves are taken into account in the present study. The associated mixed boundary value problem is reduced to a linear system of equations by utilizing a more general orthogonal relation along with least squares approximation method. The reflection and transmission coefficients for the surface and internal modes, wave load, and breakwater response are computed for various physical parameters of interest to analyze the efficiency of the flexible porous plate as a breakwater in the two-layer fluid.     

Oblique Long Waves on Beach and Induced Longshore Current

This study considers the 3D runup of long waves on a uniform beach of constant or variable downward slope that is connected to an open ocean of uniform depth. An inviscid linear long-wave theory is applied to obtain the fundamental solution for a uniform train of sinusoidal waves obliquely incident upon a uniform beach of variable downward slope without wave breaking. For waves at nearly grazing incidence, runup is significant only for the waves in a set of eigenmodes being trapped within the beach at resonance with the exterior ocean waves. Fourier synthesis is employed to analyze a solitary wave and a train of cnoidal waves obliquely incident upon a sloping beach, with the nonlinear and dispersive effects neglected at this stage. Comparison is made between the present theory and the ray theory to ascertain a criterion of validity. The wave-induced longshore current is evaluated by finding the Stokes drift of the fluid particles carried by the momentum of the waves obliquely incident upon a sloping beach. Currents of significant velocities are produced by waves at incidence angels about 45° and by grazing waves trapped on the beach. Also explored are the effects of the variable downward slope and curvature of a uniform beach on 3D runup and reflection of long waves.     

Time-Domain Boundary Element Method for Underground Structures in Orthotropic Media

Based on the fundamental solutions due to Dirac’s δ function for transversely isotropic media, a numerical scheme for wave propagation in two-dimensional orthotropic media is developed within the boundary element method in the time domain. Criteria for selection of the time and space discretizations for convergent and accurate results are established. The applicability and accuracy of the scheme are verified by a number of benchmark problems including dynamic responses of an orthotropic half plane under uniform harmonic surface loadings and a circular cavern in slightly orthotropic media under the incidence of pseudo-P and -SV waves. The procedure is then applied to examine wave propagation for an underground tunnel with different ratios of orthotropy in the media. The effects of different wave parameters and ratios of orthotropy on dynamic response of the tunnel are presented.     

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.     

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.     

Field Measurement of Ship Waves in Victoria Harbor

Field measurement of ship waves has been carried out in Victoria Harbor of Hong Kong. The wave data were collected at 18 monitoring stations by the underwater pressure/current instrument S4ADW and a floating wave rider TYPE 1156. It is found that the wave conditions in the study area are predominantly affected by the ship-generated waves under normal weather conditions. By comparing with the wind-generated waves in the open region which are irregular but with dominant directional characteristics, the existing harbor–wave field is much more complex. The irregular ship–wave field is not statistically stationary. It is also found that the ship–wave spectra are either of multiple peaks or have wide crests without distinct peaks.     

Estimation of Laboratory Wave Reflection by a Transfer Function Method

A theory of transfer function method for separating two-dimensional wave data obtained in laboratory experiments into incident and reflected waves is presented in this paper. Based on the linear wave assumption, specific transfer functions are derived from mathematical manipulations of the composite wave field, and the corresponding impulse response functions are obtained by implementing the inverse Fourier transform of transfer functions. These response functions are used to perform convolution integrals with time series data measured by fixed wave gauges at different locations in a wave flume and then to separate the incident and reflected waves. Compared with other available methods, the phase difference between two wave signals is considered in the transfer functions. Thus, the separation of waves does not involve the phase calculation and the corresponding error is avoided. The validity of the present method is examined through numerical examples and laboratory experiments of physical models carried out in a wave flume. A comparison of results from physical experiments shows that the present method gives much better estimates of incident and reflected waves than other methods available in the literature.     

Using Finite-Element Method to Simulate Wave Transformations in Surf Zone

In this study, a finite element method proposed by Hsu et al. in 2003 is extended to develop a numerical model for the simulation of wave transformation in the surf zone. The governing equation is the elliptic mild-slope equation including the energy dissipation of wave breaking. At the open boundaries with varying depth, the reflected waves caused by shoaling are adopted to the radiation boundary conditions. The rationality of the present numerical model is examined through the cases of offshore parallel breakwater problems. The results of calculation are in good agreement with experimental results.     

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.     

Simulation of Free Surface Flow over Spillway

A numerical approach is proposed to simulate and study the effect of geometry on the free surface flow over a tunnel spillway. A three-step solution procedure is proposed to speed up the solution. The first step is to obtain an approximate free surface profile and mean velocity distribution, assuming 1D steady flow. Next, the 3D turbulent flow field is computed while the water surface profile is kept fixed. Finally, the water surface is set free to move and generate waves. The governing equations for weakly compressible flow (compressible hydrodynamic flow) are solved with an explicit finite volume method. A boundary fitted grid system is used to accurately resolve the flow near the free surface with steep waves. A mixed Lagrangian-Eulerian approach is proposed to calculate the new free surface position. The numerical results of a time-averaged free surface profile as well as pressure and velocity distribution have been compared with some experimental data.     

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.     

Hydrodynamic Behavior of Asymmetric Oscillatory Boundary Layers at Low Reynolds Numbers

An experimental and numerical study has been carried out to study the wave boundary layers under asymmetric waves. The experiments were conducted in an oscillating tunnel using a simple mechanical system to generate an asymmetric oscillatory motion similar to cnoidal waves. The velocities were measured by laser Doppler velocimetry and the bottom shear stress was calculated from the cross-stream velocity profile. A low Reynolds number k–ε model was used to predict the hydrodynamic properties of the cnoidal wave boundary layers. After validating the model with the experimental data, a series of numerical experiments were carried out to study the transitional behavior of these boundary layers by virtue of friction factor and phase difference between mean free-stream velocity and bottom shear stress. Finally a stability diagram was drawn to demarcate the laminar, transition, and fully turbulent regimes using the numerical results. The present study would be useful for the hydraulic and coastal engineers interested in calculating bottom shear stress in order to compute the sediment transport in coastal environments.     

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.     

金属材料内部非金属夹杂超声检测的数值模拟

在金属材料内部夹杂物的超声检测中,如何通过检测获得的回波信号辨识夹杂物的属性和位置,一直是其重点和难点问题.通过建立包含夹杂物缺陷的二维金属板模型,采用有限元数值模拟的方法,对材料内部超声波场进行计算,获得了两种最典型的夹杂物Al₂O₃和TiN,以及二者在材料内部不同深度时的超声回波信号.研究了夹杂物类型和夹杂物深度对超声回波时域波形以及对界面波、夹杂物缺陷回波和底面回波频谱分布的影响规律.      

Scattering of Harmonic Waves by a Circular Cavity in a Porous Medium: Complex Functions Theory Approach

An analytical solution for the evaluation of scattering of waves by a circular cavity in infinite isotropic elastic porous media is presented. Two groups of complex functions for solid skeleton and pore fluid in a two-dimensional complex plane are introduced in order to solve the Biot equations. Stress, displacement, and pore pressure fields induced by incident and scattered waves in the medium and especially in the vicinity of the cavity are evaluated in this complex plane. The validation of the proposed solution is shown by various numerical examples. A parametric study including the effects of fluid compressibility changes, shear modulus, and permeability variations, several wave numbers, and wave types (fast, slow, and shear waves) is performed.