Simulation Of Surface Waves Research Articles

A multiple-layer σ-coordinate model for simulation of wave–structure interaction

A 3D multiple-layer σ-coordinate model has been developed to simulate surface wave interaction with various types of structures including submerged structures, immersed structures, and floating structures. This model is the extension of the earlier model [Lin P, Li CW. A σ-coordinate three-dimensional numerical model for surface wave propagation. Int J Numer Methods Fluid 2002;38(11):1045–68] that solves Navier–Stokes equations in the transformed σ-coordinate, which is especially efficient for simulation of wave propagation over varying topography. By introducing the layered σ-coordinates, the present model overcomes the difficulty encountered by the earlier model in calculating waves past a depth discontinuity, e.g., a submerged rectangular breakwater. Furthermore, with the employment of 3-layer σ-coordinate the present model is able to simulate flow interaction with an immersed body or a floating body. The new model is validated against an established Volume-Of-Fluid (VOF) model [Lin P, Liu PL-F. A numerical study of breaking waves in the surf zone. J Fluid Mech 1998;359:239–64] for the 2D solitary wave interaction with a submerged, immersed, or floating rectangular obstacle. For the solitary wave interaction with a submerged breakwater, the numerical results are also compared to the experimental data by Zhuang and Lee [A viscous rotational model for wave overtopping over marine structure. In Proc 25th Int Conf Coast Eng, ASCE, 1996. p. 2178–91] and very good agreements have been obtained for velocities in the vortex behind the structure. Finally, the present model is used to simulate 3D wave interaction with a Very Large Floating Structure (VLFS) above a submerged shoal. It is proved that the model is an accurate and efficient numerical tool to investigate different wave–structure interactions problems.

An efficient model for three-dimensional surface wave simulations. Part II: Generation and absorption

Water wave generation procedures and efficient numerical beaches are crucial components of a fully non-linear numerical tank for water wave simulations. Linear formulae for pneumatic wave makers are optimized for efficient fully non-linear wave generation in three dimensions. Analytical integration of the (linear) applied free surface pressure provides formulae valid for all times of the simulation. The purely non-linear part of the wave making procedure becomes integrated in the fully non-linear formulation. Novel numerical beaches are introduced, damping the (scaled) tangential velocity at the free surface. More specifically, an additional term is introduced in the Bernoulli equation at the free surface, namely ∇ - 1 · ( γ ∇ ϕ ˜ ) , where γ is a non-zero (smooth) function in regions where damping is required and zero in the wave propagation domain, ∇ ϕ ˜ is the scaled tangential velocity at the free surface, and ∇ −1 the inverse horizontal gradient operator. The new term results in a modified dynamic free surface condition which is integrated in time in the fully non-linear formulation. Extensive numerical tests show that the energy of the outgoing waves is completely absorbed by the new damper. Neither wave reflection nor emission are observed. A steep solitary wave is completely absorbed at the numerical beach. Damping of waves due to advancing pressure distributions are efficient as well. The implementation of the absorber in any existing numerical tank is rather trivial.

An efficient model for three-dimensional surface wave simulations: Part I: Free space problems

An efficient numerical scheme for simulations of fully nonlinear non-breaking surface water waves in 3D is presented. The water depth is either shallow, finite or infinite. The method is based on a fast, rapidly converging, iterative algorithm to compute the Dirichlet to Neumann operator. This is evaluated by expanding the operator as a sum of global convolution terms and local integrals with kernels that decay quickly in space. The global terms are computed very quickly via FFT. The local terms are evaluated by numerical integration. Analytical integration of the linear part of the prognostic equations in Fourier space is obtained to machine precision. The remaining nonlinear components are integrated forward in time using an RK-scheme combined with a special step size control technique. This yields a very stable and accurate time marching procedure. Zeros-padding in the spectral space represents the anti-aliasing strategy. The method requires no smoothing. Illustration through examples show that the total energy is well conserved during the numerical simulations. The scheme is stable and accurate, even for very long time simulations of very steep wave events. The scheme is easily parallelizable. It propagates for example a Stokes wave of slope 0.2985 with a phase shift error of about 0.3° after 1000 periods of propagation.

Mathematical simulation of surface wave created in a mold due to submerged entry nozzle

Two‐dimensional numerical simulations have been performed using a finite volume method that employs unstructured grids with cell‐wise local refinement and an interface‐capturing scheme to predict the shape of the free surface, thus simulating the surface wave that is created in a mold due to the flow from the submerged entry nozzle (SEN). Simulation has been done for 1:6.25 aspect ratio of the mold having a height of 2 m with parallel rectangular ports as well as 15° upward and downward ports. It has been found that for low inlet velocity of the SEN (<1 m/s) the maximum wave amplitude of the surface remains below 12 mm and no outside air is entrapped by the wave to form a bubble. However, for high inlet velocity (2 m/s or more) there is considerable fluctuations on the free surface and the maximum wave amplitude shoot up beyond 70 mm at the start up and slowly falls to about 30 mm entrapping air bubbles from the surroundings. The movement of the air bubble within the mold and its rise to the free surface where it subsequently collapses has been captured well in the numerical simulation. The overall shape of the free surface matches well, excepting the initial transience, with that of the experimentally observed free surface, although the free surface never attains a perfect steady shape neither in the experiment nor in the numerical simulation due to its continuous oscillation and breaking.

Boussinesq modeling of longshore currents

A time domain Boussinesq model for nearshore hydrodynamics is improved to obtain the conservation of vertical vorticity correct to second order and extended for use on an open coast using longshore periodic boundary conditions. The model is utilized to simulate surface waves and longshore currents under laboratory and field conditions. Satisfactory agreement is found between numerical results and measurements, including root mean square wave height, mean water level, and longshore current. One striking result of the simulations is the prediction of the strong longshore current in the trough shoreward of the bar as observed during the Duck Experiment on Low‐frequency and Incident‐band Longshore and Across‐shore Hydrodynamics field campaign. The model results give insight into the spatial and temporal variability of wave‐driven longshore currents and the associated vertical vorticity field under the phase‐resolving, random wave forcing with wave/current interaction. Numerical experiments are carried out to examine the response of the modeled longshore currents to the randomness of surface waves and the cross‐shore distributions of bed shear stress coefficient. We find that both regular and irregular waves lead to very similar mean longshore currents, while the input of monochromatic, unidirectional waves results in much more energetic shear waves than does the input of random waves. The model results favor Whitford and Thornton's [1996] finding that the bed shear stress coefficient for the area offshore the bar is larger than that in the trough, as better agreement with the field data for both regular and irregular waves is found if such coefficients are used in the Boussinesq model.

Parasitic modes on printed circuit boards and their effects on EMC and signal integrity

In this paper, parasitic modes, such as slotline, parallel plane, and surface wave (SW) modes, commonly found on printed circuit boards (PCBs) are analyzed and their effects on electromagnetic compatibility (EMC) and signal integrity are discussed. The analysis is based on numerical simulations using the finite difference time domain (FDTD) method which is shown to be very well suited for rigorous modeling of parasitic mode effects. The EMC and signal integrity problems discussed include power loss, crosstalk, ground bounce, and free space radiation. Design guidelines for improved EMC and signal integrity are derived from the results obtained. Comprehensive simulation and characterization of SWs using FDTD is presented for the first time.

Measurements and numerical simulations of surface waves using a plane model

A falling-weight impactor with a controlled force energy spectrum was used to generate surface waves in soft clay soil. Field measurements of both load and response in terms of vertical acceleration were made. The acceleration-type frequency response function (FRF) between the point of source and the point of response was established. The Rayleigh wave velocity of the soil was estimated from the measured acceleration time histories and from the established FRFs. Simulations of the field measurements were conducted and the dynamic response of the soil was studied in terms of the vertical acceleration and displacement at the soil surface. Two-dimensional numerical results were scaled to three dimensions. The effect of incorporating contributions from shear waves and compression waves at the surface was demonstrated. FRFs based on simulated data were established. The scaled numerical results and measured values are in good agreement.

Numerical simulation of nonlinear radiation water surface wave caused by large-amplitude oscillation of a cylinder

The BEM combined with the time-stepping scheme is employed in the nonlinear numerical simulation of the development process of radiation wave profile and hydrodynamic force caused by large amplitude oscillating cylinder. To ensure stability and accuracy of the simulation, special consideration is paid on a series of difficulties in numerical modeling, such as the compatibility of free surface and body surface conditions at the intersection point and the open boundary condition problem. Numerical calculation is carried out for some study cases of different shaped cylinders undergoing large amplitude oscillating motions with distinct modes, the results show some nonlinear phenomena and indicate the feasibility of the numerical approach.

Model equations for nonlinear surface waves

The theoretical framework developed originally for modeling nonlinear Rayleigh waves in isotropic elastic half-spaces [Zabolotskaya, J. Acoust. Soc. Am. 91, 2569 (1992); see also Knight et al., ibid. 102, 1402 (1997)] has been extended to encompass a broad class of nonlinear surface wave problems. The theory is based on Hamiltonian mechanics, in which spatial Fourier amplitudes are used as generalized coordinates, and coupled spectral equations are obtained for the harmonic amplitudes in the wave. The nonlinearity coefficient matrices are expressed explicitly in terms of the second- and third-order elastic moduli of the materials. Solutions of the equations describe harmonic generation, waveform distortion, and shock formation not only at the interface but also in the interior of the solid. Cylindrical spreading and beam diffraction may be taken into account, as can transient effects in pulses. Model equations, and simulations of nonlinear surface waves in real materials, shall be presented for Rayleigh, Stoneley, Scholte, and Lamb waves, and surface waves in crystals and piezoelectric materials. The validity of the theory has been demonstrated via quantitative agreement with measurements of laser-generated Rayleigh wave pulses with shocks in fused quartz [Lomonosov et al., J. Acoust. Soc. Am. 101, 3080(A) (1997)]. [Work supported by NSF and ONR.]

Modeling water for computer graphics

Water motion is a phenomenon the full complexity of which has yet to be realized in simulation. Although a good deal of work has been done in modeling water waves, there remain interesting behaviours which have not been captured in any existing model. The variety of phenomena realizable by water is extensive, including waves, foam, bubbles, and spray. The present work is motivated by the complexity and range of phenomena possible as well as by the frequency with which such phenomena are encountered. Water motion has been extensively studied from a physical perspective. A computational model should take advantage of past work in this area. Computer scientists have incorporated some physics into models of water; however, currently existing models are incomplete. This paper considers the strengths and shortcomings of previous attempts to model water. The work involves the development of an extension and refinement of an existing microscopic model of fluid, and a formalization of some of the results of using this model. Some results of the model have been compared with the predictions due to hydrodynamic theory. Other results have been presented in the context of the situation being simulated, and the discussion hinges on observations of the real situation rather than on theory. In brief, the model displays great potential for simulation of surface waves, spray, and interactions between solid objects and fluid. The experimental work is largely able to substantiate claims as to the accuracy of the model at simulating the underlying physics.

Dependence of the emission characteristics of magnetized surface-wave plasmas on the azimuthal configuration of the wave field

A 1D plasma-wave model initially developed for describing a plasma sustained by a surface wave in the azimuthally symmetric configuration (m=0) was modified for simulating surface waves in the dipolar mode (m=1). Using this model, it is shown that the maintenance limit for unmagnetized plasmas in the symmetric mode, which, as was previously determined, requires that the product of the wave frequency and the plasma radius be smaller than 2 GHz cm, is no longer valid for magnetized plasmas. Comparison of calculated and measured radial distributions of argon atoms in a radiative excited state shows that changes observed in this distribution when varying some of the operating conditions can be explained by a switching of the discharge from one mode to the other. Conditions for selection of the mode are derived from considerations on wave propagation and power dissipation along the plasma column.

Investigation of a lattice gas model for surface gravity waves

A recently proposed lattice gas model for simulating surface waves at a free boundary is described in detail. Simulated waves are compared to linear theory and are seen to compare well, provided an additional parameter ε is introduced. This parameter, ε, is investigated and found to be required due to the density gradient produced across the fluid by the gravitational interaction. Its value and the values of the other model parameters are found for a range of gravitational strengths.

A Seismic Refraction Profile Acress the Longitudinal Valley Near Hualien Taiwan

The tectonic plate movement in eastern Taiwan is very active, especially along the Longitudinal Valley, which is believed to be a suture zone between two plates. Except for the high seismicity and rapid ground deformation, the geological literatures would normally suggest two high angle faults on two sides of the Valley, i.e., the Central Range fault on the west and the Coastal Range fault on the east. The structure of the Longitudinal Valley itself must represent a certain degree of complexity due to the tectonic compression. In January, 1994, we conducted a 7.5 km long refraction survey at 0.5 km intervals across the Valley with two 50 kg dynamite explosions at both ends. The purpose was to explore the detailed structure under the Longitudinal Valley and to detect two faults near the Hualien area. Very clear first arrivals were obtained, from which we are able to draw a concave shaped valley profile (2.0 km in axial depth and 5.0 km in width) of the valley basement. The eastern flank of the basement is steeper than the western flank with good indications of the Coastal Range fault located just below the axial place, but the Central Range fault itself is not so obvious. A transition zone, with Central Range materials under the west half of the Valley, is suggested. The axis of the Valley would be thus the suture line. This fault, along the axial suture line, should be named the 'Longitudinal Valley fault' instead of using an ambiguous name such as the Coastal Range fault. Except for the refraction measurement, we also used surface wave simulations to derive an S-wave velocity constraint for the Valley. It is found that the P-wave and S-wave velocities for most alluvia layers are 3.2 km and 0.90 km/sec, respectively. The basement under the valley has P and S velocities of 5.0 km/sec and 2.0 km/sec, respectively. A low velocity near-surface layer with Vp=2.1 km/sec, Vs=0.4 km/sec, and thickness=0.3 km is also found, which would adequately explain a large amplitude surface wave group with a 0.54 km/sec apparent velocity. The result of this study is consistent with the refraction survey carried out in 1973 by Tsai et al (1974). Although our velocity values are lower by 10%, the dimensions for the concave shaped basement of the Longitudinal Valley are in good agreement.

A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves

Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed. Results of the Boussinesq model with and without strong nonlinearity are compared in detail to those of a boundary element solution of the fully nonlinear potential flow problem developed by Grilli et al. (1989). The fully nonlinear variant of the Boussinesq model is found to predict wave heights, phase speeds and particle kinematics more accurately than the standard approximation.

A three-dimensional panel method for nonlinear free surface waves on vector computers

A number of recent improvements in a higher-order three-dimensional (3D) panel method for highly nonlinear free surface wave simulations is discussed. Special attention is paid to grid evolution techniques, and stability of the time-dependent problem. Due to the improvements, stable and accurate results can be obtained for linear and highly nonlinear wave problems. Although no artificial smoothing is applied, even extreme problems like the development of breaking waves in a 3D configuration can be simulated. The computer code has been specially developed for implementation on a vector computer. The program is highly vectorized, and use is made of mathematical libraries for acceptable CPU-times.

A numerical procedure for time domain nonlinear surface wave calculations

A numerical procedure based upon Boundary Integral Method for time domain wave-making problems is developed. The free-surface boundary condition is expressed in a Lagrangian notation to follow the free-surface's motion in time. The corresponding material derivative term is approximated by a finite difference expression, and the velocity terms are extrapolated in time for the completion of the formulations. Solutions with a linear free-surface condition and a non-linear free-surface condition are obtained at small time increment values. Modelling of surface wave problems are studied in two dimensions as well as in three dimensions, and comparisons are made to results based on linear theory as well as to published experimental results. Good agreements between the numerical solutions and measured values are found. Applications are on the simulations of surface waves in a towing tank, ship's bow wave calculation, and wave diffraction problem for a vertical, surface piercing circular cylinder. For the modelling of the three-dimensional wave diffraction problem, results at high wave amplitude are restricted because of the use of quadrilateral elements. The near cylinder region of the free-surface is not considered to be well represented because of the coarse element size. Wave forces calculated on the vertical cylinder are found to be affected by the modeled tank length. When the simulated wave length is comparable to the wave tank's length, numerical results are found to be less than the experimental measurements. However, when the wave length is much shorter than the wave tank's length, solutions are obtained with very good precision.

Absorbing boundary conditions for free surface wave simulations with a panel method

The numerical implementation and stability of first- and second-order absorbing boundary conditions for simulating free surface gravity waves are considered. The free surface waves are solved with a panel method. The stability of the boundary conditions is proved, assuming certain properties of the integral operator. Arguments are given that support these assumptions. The theoretical results are confirmed in the test cases and the boundary conditions give low reflections for plane waves.

Absorbing boundary conditions for free surface wave simulations with a panel method

Absorbing boundary conditions for free surface wave simulations with a panel method

Numerical simulation of solitary surface waves in a discrete model of an incompressible liquid

Numerical simulation of solitary surface waves in a discrete model of an incompressible liquid

A new interpretation of the generators of BAEP waves I–V: Results of a spatio-temporal dipole model

Brain-stem auditory evoked potential (BAEP) scalp distribution, recorded in 10 normal subjects, was analysed using a spatio-temporal dipole model. This model can simulate surface wave forms due to overlapping activity from multiple dipolar sources within a 3-shell head model. The conventional 5-peak (I–V) hypothesis could not fully account for the experimental BAEP distribution, particularly around waves I- and III-. When the temporal course of dipole strength was modelled according to a triphasic compound action potential, 6 dipolar sources were sufficient to fit all BAEP wave forms. Location, orientation and latency of the dipoles indicated generation of dipole I at the distal end of the auditory nerve (AN), of dipole III in, or near to, the cochlear nucleus (CN), of dipole III- in the trapezoid body and of dipoles IV and V in the superior olivary complexes and in the lateral lemnisci. Dipole I-, peaking only 0.65 msec after wave I, is suggested to result from the electric inhomogeneity at the porus acusticus internus. Wave II required no extra model dipole, but was attributable, in part, to the second peak of dipole I. Estimation of latencies based on AN conduction velocity, synaptic delay and differences in AN length amongst species confirmed that second order neuronal activity cannot arise before wave III. Second order axons, spreading widely through the brain-stem, apparently cause major contributions also to waves III-, IV and V. The new source hypothesis must leave open questions concerning the amount of contribution from third order neurones or from ipsi- versus contralateral structures to the IV/V wave complex.