Open-channel waves generated by propagation of a discontinuous wave over a bottom step

This paper presents the results of theoretical and experimental studies of open-channel waves generated by the propagation of a discontinuous dam-break wave over a bottom step. The cases where the initial tailwater level is higher than the step height (the step is under water) and where this value is smaller than the step height (at the initial time, water is absent on the step) are considered. Exact solutions are constructed using modified first-approximation equations of shallow-water theory, which admit the propagation of discontinuous waves in a dry channel. On the stationary hydraulic jump formed above the bottom step, the total free-stream energy is assumed to be conserved. These solutions agree with experimental data on various parameters (types of waves, wave propagation velocity, asymptotic depths behind the wave fronts). __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 31–44, January–February, 2008.     

A numerical scheme for strong blast wave driven by explosion

After the detonation of a solid high explosive, the material has extremely high pressure keeping the solid density and expands rapidly driving strong shock wave. In order to simulate this blast wave, a stable and accurate numerical scheme is required due to large density and pressure changes in time and space. The compressible fluid equations are solved by a fractional step procedure which consists of the advection phase and non‐advection phase. The former employs the Rational function CIP scheme in order to preserve monotone signals, and the latter is solved by interpolated differential operator scheme for achieving the accurate calculation. The procedure is categorized into the fractionally stepped semi‐Lagrangian. The accuracy of our scheme is confirmed by checking the one‐dimensional plane shock tube problem with 10³ times initial density and pressure jump in comparison with the analytic solution. The Sedov–Taylor blast wave problem is also examined in the two‐dimensional cylindrical coordinate in order to check the spherical symmetry and the convergence rates. Two‐ and three‐dimensional simulations for the blast waves from the explosion in the underground magazine are carried out. It is found that the numerical results show quantitatively good agreement with the experimental data. Copyright © 2006 John Wiley & Sons, Ltd.     

A numerical model for run‐up of breaking waves

In the present paper, the numerical method for the three‐dimensional run‐up, given in Johnsgard and Pedersen [‘A numerical model for three‐dimensional run‐up’, Int. J. Numer. Methods Fluids, 24 , 913–931 (1997)], is extended to include wave breaking. In the fundamental problem of run‐up of a uniform bore, the present model is compared with analytical solutions from the literature. The numerical solutions converge, but very slowly. This is not due to the numerical model, but rather to the structure of the solutions themselves. Numerical results for two realistic but simplified tsunami cases are also presented. In the first case, two‐dimensional simulations are performed concerning the run‐up of a tsunami in Portugal, in the second case, the three dimensional wave pattern generated after a slide in Tafjord, Norway in 1931, is studied. A discussion of different aspects of the model is summarized at the end of the paper. Copyright © 1999 John Wiley & Sons, Ltd.     

Surface characters of internal waves generated by Rankine ovoid

A linear theory on the internal waves generated in the stratified fluid with a pycnocline is presented in this paper. The internal wave fields such as the velocity fields in the stratified fluid and velocity gradient fields at the free surface are also investigated by means of the theoretical and numerical method. From the numerical results, it is shown that the internal wave generated by horizontally moving Rankine ovoid is a sort of trapped wave which propagates in a wave guide, and its waveform is a kind of Mach front-type internal wave in the pycnocline. Influence of the internal wave on the flow fields at the free surface is represented by the velocity gradient fields resulted from the internal waves generated by motion of the Rankine ovoid. At the same time, it is also shown that under the hypothesis of inviscid fluid, the synchronism between the surface velocity gradient fields at the free surface and the internal wave fields in the fluid is retained. This theory opens a possibility to study further the modulated spectrum of the Bragg waves at the free surface.The project supported by the National Natural Science Foundation of China (40576010). The English text was polished by Keren Wang.     

Analytical and numerical studies for Seiches in a closed basin with bottom friction

A standing wave oscillation in a closed basin, known as a seiche, could cause destruction when its period matches the period of another wave generated by external forces such as wind, quakes, or abrupt changes in atmospheric pressure. It is due to the resonance phenomena that allow waves to have higher amplitude and greater energy, resulting in damages around the area. One condition that might restrict the resonance from occurring is when the bottom friction is present. Therefore, a modified mathematical model based on the shallow water equations will be used in this paper to investigate resonance phenomena in closed basins and to analyze the effects of bottom friction on the phenomena. The study will be conducted for several closed basin shapes. The model will be solved analytically and numerically in order to determine the natural resonant period of the basin, which is the period that can generate a resonance. The computational scheme proposed to solve the model is developed using the staggered grid finite volume method. The numerical scheme will be validated by comparing its results with the analytical solutions. As a result of the comparison, a rather excellent compatibility between the two results is achieved. Furthermore, the impacts that the friction coefficient has on the resonance phenomena are evaluated. It is observed that in the prevention of resonances, the bottom friction provides the best performance in the rectangular type while functioning the least efficient in the triangular basin. In addition, non-linearity effect as one of other factors that provide wave restriction is also considered and studied to compare its effect with the bottom friction effect on preventing resonance.     

A fluctuating lattice-Boltzmann model for direct numerical simulation of particle Brownian motion

A single-relaxation-time fluctuating lattice-Boltzmann （LB） model for direct numerical simulation （DNS） of particle Brownian motion is established by adding a fluctuating component to the lattice-Boltzmann equations （LBEs）. The fluctuating term is proved to be the random stress tensor in fluctuating hydrodynamics by recovering Navier-Stokes equations from LBEs through a Chapman-Enskog expansion. A three-dimensional implementation of the model is also presented, along with simulations of a single spherical particle and 125 spherical particles at short times. Numerical results including the meansquare displacement, velocity autocorrelation function and self-diffusion coefficient of particles compare favorably with theoretical results and previous numerical results.     

Focusing shock waves generated by an accelerating projectile

Unsteady flow fields past a projectile either in parabolic motion or in accelerated motion are investigated numerically, with special attention to shock wave focusing phenomena induced by the motions. The two-dimensional Euler equations are solved by a moving-cell, MUSCL TVD finite-volume method with Steger-Warming flux-vector splitting. The pattern of shock wave focusing induced by the projectile is found to be similar to that of shock waves reflected from a concave wall observed in a laboratory experiment using a shock tube. Effects of projectile shapes and trajectories on focusing are also examined.     

A meshless numerical wave tank for simulation of nonlinear irregular waves in shallow water

Time domain simulation of the interaction between offshore structures and irregular waves in shallow water becomes a focus due to significant increase of liquefied natural gas (LNG) terminals. To obtain the time series of irregular waves in shallow water, a numerical wave tank is developed by using the meshless method for simulation of 2D nonlinear irregular waves propagating from deep water to shallow water. Using the fundamental solution of Laplace equation as the radial basis function (RBF) and locating the source points outside the computational domain, the problem of water wave propagation is solved by collocation of boundary points. In order to improve the computation stability, both the incident wave elevation and velocity potential are applied to the wave generation. A sponge damping layer combined with the Sommerfeld radiation condition is used on the radiation boundary. The present model is applied to simulate the propagation of regular and irregular waves. The numerical results are validated by analytical solutions and experimental data and good agreements are observed. Copyright © 2008 John Wiley & Sons, Ltd.     

Large-scale edge waves generated by a moving atmospheric pressure

Long waves generated by a moving atmospheric pressure distribution, associated with a storm, in coastal region are investigated numerically. For simplicity the moving atmospheric pressure is assumed to be moving only in the alongshore direction and the beach slope is assumed to be a constant in the on-offshore direction. By solving the linear shallow water equations we obtain numerical solutions for a wide range of physical parameters, including storm size (2a), storm speed (U), and beach slope (α). Based on the numerical results, it is determined that edge wave packets are generated if the storm speed is equal to or greater than the critical velocity, U_cr, which is defined as the phase speed of the fundamental edge wave mode whose wavelength is scaled by the width of the storm size. The length and the location of the positively moving edge wave packet is roughly Ut/2 ≤ y ≤ Ut, where y is in the alongshore direction and t is the time. Once the edge wave packet is generated, the wavelength is the same as that of the fundamental edge wave mode corresponding to the storm speed and is independent of the storm size, which can, however, affect the wave amplitude. When the storm speed is less than the critical velocity, the primary surface signature is a depression directly correlated to the atmospheric pressure distribution.     

In-plane wave motion in finite element model

The analysis method of lattice dynamics in classical physics is extended to study the properties of in-plane wave motion in the hybrid-mass finite element model in this paper. The dispersion equations of P and SV waves in the discrete model are first obtained by means of separating the characteristic equation of the motion equation, and then used to analyse the properties of P-and SV-homogeneous, inhomogeneous waves and other types of motion in the model. The dispersion characters, cut-off frequencies of P and SV waves, the polarization drift and appendent anisotropic property of wave motion caused by the discretization are finally discussed. The project sponsored by the Earthquake Science Foundation under Contract No. 90141     

Experimental and numerical analysis of solitary waves generated by bed and boundary movements

This paper is an experimental and numerical study about propagation and reflection of waves originated by natural hazards such as sea bottom movements, hill slope sliding and avalanches. One‐dimensional flume experiments were conducted to study the characteristics of such waves. The results of the experimental study can be used by other researchers to verify their numerical models. A finite volume numerical model, which solves the shallow water equations, was also verified using our own experimental results. In order to deal with reflection on sloping surfaces and overtopping walls, a new condition for the treatment of the coastline is suggested. The numerical simulation of wave generation is also studied considering the bed movement. A boundary condition is proposed for this case. Those situations when the shallow water equations are valid to simulate this type of phenomena have been studied, as well as their limitations. Copyright © 2004 John Wiley & Sons, Ltd.     

A numerical study of the load on cylindrical piles exerted by internal solitary waves

A continuously stratified nonlinear model is employed to simulate the generation of internal solitary waves (ISWs) over a sill by tidal flows, and it is shown that the simulated ISW-induced current field basically agrees with that observed. Then the force and torque on a supposed small-diameter vertical cylindrical pile exerted by the simulated ISW packet are calculated. According to the calculation, it is found that, no matter whether the direction of the ISW-induced current is the same as that of the tidal current or not, the force exerted by the ISW would be much larger than that by only the tidal current; if the direction of the ISW-induced current is the same as that of the tidal current, then the torque exerted by the ISW would also be much larger than that by only the tidal current; whilst if the direction of the ISW-induced current is against that of the tidal current, then the torque exerted by the ISW has the same order as that exerted by only the tidal current. It is shown that, under the same conditions, the maximum force on the cylindrical pile is 6.58×10² kN, which is larger than that by the modal separation method of Cai et al., whilst the maximum torque is 2.46×10⁵ kN m, which is less than that given by Cai et al. During the passage of the ISW, the time series of the force and torque on the cylindrical pile can also be shown. Finally, the effect of the characteristics of the Gaussian sill on the force is studied, and the resulted empirical formulas on the force with the wave amplitude and the non-dimensional variable of the sill parameters are put forward.     

A three-dimensional numerical study on instability of sinusoidal flame induced by multiple shock waves

The instabilities of a three-dimensional sinusoidally premixed flame induced by an incident shock wave with Mach = 1.7 and its reshock waves were studied by using the Navier–Stokes(NS) equations with a single-step chemical reaction and a high resolution, 9th-order weighted essentially non-oscillatory scheme. The computational results were validated by the grid independence test and the experimental results in the literature. The computational results show that after the passage of incident shock wave the flame interface develops in symmetric structure accompanied by large-scale transverse vortex structures. After the interactions by successive reshock waves, the flame interface is gradually destabilized and broken up, and the large-scale vortex structures are gradually transformed into small-scale vortex structures. The small-scale vortices tend to be isotropic later.The results also reveal that the evolution of the flame interface is affected by both mixing process and chemical reaction. In order to identify the relationship between the mixing and the chemical reaction, a dimensionless parameter, η, that is defined as the ratio of mixing time scale to chemical reaction time scale, is introduced. It is found that at each interaction stage the effect of chemical reaction is enhanced with time.The enhanced effect of chemical reaction at the interaction stage by incident shock wave is greater than that at the interaction stages by reshock waves. The result suggests that the parameter η can reasonably character the features of flame interface development induced by the multiple shock waves.     

Fully dispersive dynamic models for surface water waves above varying bottom, Part 1: Model equations

In this paper we formulate relatively simple models to describe the propagation of coastal waves from deep parts in the ocean to shallow parts near the coast. The models have good dispersive properties that are based on smooth quasi-homogeneous interpolation of the exact dispersion above flat bottom. This dispersive quality is then maintained in the second order nonlinear terms of uni-directional equations as known from the AB-equation. A linear coupling is employed to obtain bi-directional propagation which includes (interactions with) reflected waves.The derivation of the models is consistent with the basic variational formulation of surface waves without rotation. A subsequent spatial discretization that takes this variational structure into account leads to efficient and accurate codes, as will be shown in Part 2.     

Higher order Boussinesq-type equations for water waves on uneven bottom

Higher order Boussinesq-type equations for wave propagation over variable bathymetry were derived. The time dependent free surface boundary conditions were used to compute the change of the free surface in time domain. The free surface velocities and the bottom velocities were connected by the exact solution of the Laplace equation. Taking the velocities on half relative water depth as the fundamental unknowns, terms relating to the gradient of the water depth were retained in the inverse series expansion of the exact solution, with which the problem was closed. With enhancements of the finite order Taylor expansion for the velocity field, the application range of the present model was extended to the slope bottom which is not so mild. For linear properties, some validation computations of linear shoaling and Booij' s tests were carried out. The problems of wave-current interactions were also studied numerically to test the performance of the enhanced Boussinesq equations associated with the effect of currents. All these computational results confirm perfectly to the theoretical solution as well as other numerical solutions of the full potential problem available.     

金属圆管内爆轰波相互作用效应的数值模拟研究

采用动力学有限元程序,针对大变形带来的网格扭曲造成计算的锁死现象进行了ALE技术处理,结合高速摄影和X光照相实验结果,对金属圆管内爆轰波的相互作用效应进行数值模拟,给出了爆轰波对碰作用下金属圆管运动计算结果,复现了金属圆管对碰区鼓包现象,计算结果与实验吻合较好。可以预见,基于该方法能够初步分析影响金属圆管运动规律的诸多因素。     

Fully dispersive dynamic models for surface water waves above varying bottom, Part 2: Hybrid spatial-spectral implementations

In Part 1 (van Groesen and Andonowati [1]), we derived models for the propagation of coastal waves from deep parts in the ocean to shallow parts near the coast. In this paper, we will describe hybrid spatial-spectral implementations of the models that retain the basic variational formulation of irrotational surface waves that underlays the derivation of the continuous models. It will be shown that the numerical codes are robust and efficient from results of simulations of two test cases of waves above a 1:20 sloping bottom from 30 m to 15 m depth: one simulation of a bichromatic wave train, and one of irregular waves of JONSWAP type. Measurements of scaled experiments at MARIN hydrodynamic laboratory and simulations with two other numerical codes will be used to test the performance. At the end of the full time trace of 3.5 h details of the irregular waves that travelled over more than 5000 m are clearly resolved with a correlation of more than 90%, in calculation times of less than 5% of the physical time. Also freak-like waves that appear in the irregular wave are shown to be modelled to a high degree of accuracy.     

Two notes on numerical simulation for the properties of solitary waves

In order to suppress the error induced by the modified differential equations, a compensative procedure is designed to desoribe the higher order effect of the non-linear interaction of the solitary waves. The project supported by National Natural Science Foundation of China     

Quantitative guided wave testing by applying the time reversal principle on dispersive waves in beams

A method for the localization and characterization of defects in waveguide-like structures is presented in this paper. In contrast to traditional ultrasound and guided wave techniques, a broadband signal is used to enforce strong dispersion of the flexural wave mode. Since dispersion is well compensated in a time reversal experiment we use a time reversal numerical simulation to identify the origin and the original shape of the flexural wave excited at a local non-uniformity due to mode conversion. Limitations of the time reversal process for broadband signals due to multimode and evanescent behavior of guided waves are discussed and eliminated using a Timoshenko beam model. The resulting novel process which uses both flexural and longitudinal wave information allows detection, localization and size estimation of several defects in a beam with only a single measurement. The method proposed is experimentally validated on rectangular solid beams and cylindrical hollow beams with notches of different sizes and positions. Up to three notches could be localized from one measurement, with a maximum error of 3% with respect to the propagation distance. The size was accurately predicted for notches as small as 0.5 mm depth or 8.3% of the cross section, using a generalized spring model of a crack.