Validation of a new generation system for bottom boundary layer beneath solitary wave

Understanding of sea bottom boundary layer characteristics, especially bottom shear stress acting on the sea bed, is an important step needed in sediment transport modeling for practical application purposes. In the present study, a new generation system for bottom boundary layer under solitary wave is proposed. Applicability of this system is examined by comparing measured and numerical solution velocities. Moreover, transitional behavior from laminar to turbulence was investigated. It is concluded that the critical Reynolds number in the experiments shows good agreement with DNS result of Vittori and Blondeaux (2008) and laboratory data of Sumer et al. (2010), indicating validity of the generation system. Since the present generation system enables continuous measurement to obtain ensemble averaged quantities, it can be effectively utilized for future experimental studies on solitary wave boundary layers, including sediment transport experiments with movable bed.     

Numerical simulation of interaction between internal solitary waves and submerged ridges

Direct numerical simulations are performed to study the transformation of internal solitary waves (ISWs) of depression type propagating over an underwater ridge in a two-layer fluid system. Bottom ridges with relatively smooth vertex are employed to represent sills in natural lakes and oceans. Consistent with previous experiments, three interaction types (weak, moderate and strong) are observed to be based on the energy loss. In addition, the moderate interaction are found to be categorized into transmitted and reflected type according to their distinct transformation process. General flow characteristics for ISW–ridge interaction in the benthic boundary layer and in the pycnocline is monitored and analysed. A modified degree of blocking B_m considering both the nonlinear effect of incident ISWs and the blockage effect of the submerged ridge is proposed. Different ISW–ridge interactions are discovered to be linked with B_m. Maximum wave-induced velocities, wave energy losses, reflected and transmitted wave amplitudes are found to have a self-similar feature with B_m. The maximum energy loss is up to 35% and the maximum wave-induced velocity can reach 1.8 times of the phase speed of the incident ISW. Empirical equations are obtained based on the data fitting to predict some useful physical parameters during ISW–ridge interaction.     

Computation of solitary waves during propagation and runup on a slope

A numerical time-simulation algorithm for analysing highly nonlinear solitary waves interacting with plane gentle and steep slopes is described by employing a mixed Eulerian–Lagrangian method. The full nonlinear free surface conditions are considered here in a Lagrangian frame of reference without any analytical approximations, and thus the method is valid for very steep waves including overturning. It is found that the runup height is crucially dependent on the wave steepness and the slope of the plane. Pressures and forces exerted on impermeable walls of different inclinations (slopes) by progressive shallow water solitary waves are studied. Strong nonlinear features in the form of pronounced double peaks are visible in the time history of pressure and force signals with increasing heights of the oncoming solitary waves. The effect of nonlinearity is less pronounced as the inclination of the wall decreases with respect to the bottom surface.     

Breaking solitary wave evolution over a porous underwater step

Solitary wave evolution over a shelf including porous damping is investigated using Volume-Averaged Reynolds Averaged Navier–Stokes equations. Porous media induced damping is determined based on empirical formulations for relevant parameters, and numerical results are compared with experimental information available in the literature. The aim of this work is to investigate the effect of wave damping on soliton disintegration and evolution along the step for both breaking and non-breaking solitary waves. The influence of several parameters such as geometrical configuration (step height and still water level), porous media properties (porosity and nominal diameter) or solitary wave characteristics (wave height) is analyzed. Numerical simulations show the porous bed induced wave damping is able to modify wave evolution along the step. Step height is observed as a relevant parameter to influence wave evolution. Depth ratio upstream and downstream of the edge appears to be the more relevant parameter in the transmission and reflection coefficients than porosity or the ratio of wave height–water depth. Porous step also modifies the fission and the solitary wave disintegration process although the number of solitons is observed to be the same in both porous and impermeable steps. In the absence of breaking, porous bed triggers a faster fission of the incident wave into a second and a third soliton, and the leading and the second soliton reduces their amplitude while propagating. This decrement is observed to increase with porosity. Moreover, the second soliton is released before on an impermeable step. Breaking process is observed to dominate over the wave dissipation at the porous bottom. Fission is first produced on a porous bed revealing a clear influence of the bottom characteristics on the soliton generation. The amplitude of the second and third solitons is very similar in both impermeable and porous steps but they evolved differently due to the effect of bed damping.     

Interaction of solitary waves with emergent, rigid vegetation

In this study, solitary wave interaction with emergent, rigid vegetation was studied numerically and experimentally. Laboratory experiments were carried out in a wave flume with vegetation models of different lengths and porosities; the Boussinesq equations with the effects of the vegetation being modeled by a quadratic drag law are used to simulate the wave scattering by and the wave propagation through vegetation. Effects of incident wave height, vegetation density, and vegetation length are discussed. An empirical expression for the mean drag coefficient of emergent, rigid vegetation is presented and compared with other available data. The results are useful for studying tsunami hazard mitigation by coastal forests.     

Characteristics of the boundary layer at the bottom of a solitary wave

The results of direct numerical simulations of the boundary layer generated at the bottom of a solitary wave are described. The numerical results, which agree with the laboratory measurements of Sumer et al. (2010) show that the flow regime in the boundary layer can be laminar, laminar with coherent vortices and turbulent. The average velocity and bottom shear stress are computed and the results obtained show that the logarithmic law can approximate the velocity profile only in a restricted range of the parameters and at particular phases of the wave cycle. Moreover, the maximum value of the bottom shear stress is found to depend on the dimensionless wave height only, while the minimum (negative) value depends also on the dimensionless boundary layer thickness. Diagrams and simple formulae are proposed to evaluate the minimum and maximum bottom shear stresses and their phase shift with respect to the wave crest.     

孤立波与升沉水平板相互作用数值模拟研究

合理的刚度和潜深设计可以使升沉水平板获得优异的消浪性能。基于考虑流体黏性的二维不可压缩Navier-Stokes方程,以高阶紧致插值CIP(constrained interpolation profile)方法求解方程对流项,采用VOF(volume of fluid)方法重构自由液面,构建二维数值波浪水槽。采用试验数据验证模型后,研究孤立波与升沉水平板相互作用,分析相对刚度K*、相对潜深d/h、相对波高H/h对于升沉板的消浪性能和运动响应的影响,揭示升沉板对孤立波的消浪机理。研究表明:在孤立波通过时,升沉板会经历一个先上升后下降的运动,随后非线性自由振动,板下方水体近似均匀流动,且水流的垂向流动与板的垂荡方向一致;升沉板主要通过不对称涡旋脱落、浅水变形、波浪反射与辐射波转化等方式消耗孤立波能量;一定条件下,采用最优相对刚度K*=4.0和最优相对潜深d/h=0.52可以取得良好的消浪效果,此时透射系数最小,同时升沉板的运动响应在合理的范围内。     

Analytic solution of long wave propagation over a submerged hump

A new analytical solution of the long wave refraction by a submerged circular hump is presented. The geometry of the hump is assumed to be axisymmetric and be described by a power function in the radial direction with arbitrary values of both the exponent and the scaling factor. The submergence of the hump is also variable. The water surface elevation governed by the long wave version of the mild slope wave equation is solved by separation of variables, and a series solution of the Frobenius type is obtained. The solution is shown to be valid when the hump is sufficiently submerged or is of a relatively small height. Matching method is employed to illustrate the refraction of long waves under given conditions of incidence. Effects of the shape, the scale, and the submergence of the hump on wave refraction are discussed.     

Fully nonlinear 3D interaction of bubble dynamics and a submerged or floating structure

This paper is concerned with the interaction of bubbles, a submerged or floating structure, and free surface waves. A three-dimensional fully nonlinear model has been developed based on the coupling of the boundary integral method (BIM) for bubble dynamics and free surface waves and the finite element method for structure deformation. The present method is well validated by comparing the numerical results with the experimental data. Three structure characteristics, including fixed, rigidly moving and flexible, are investigated separately to determine their influence on bubble dynamics. For a free-floating structure, the free surface causes not only a larger reduction in peak pressure for a rigid structure compared with a fixed body but also the modification of the bubble period and structural response. The interaction between a bubble and a flexible structure, in the absence of a free surface, is simulated. Both the rigid motion and the deformation at the local structure appear in the simulation. The effect of the structural thickness on the reduction in peak pressure is also considered.     

内孤立波与海脊相互作用的模拟研究

A nonhydrostatic numerical model was developed and numerical experiments performed on the interaction of an internal solitary wave(ISW) with a sill, for a two-layer fluid with a diffusive interface. Based on the blocking parameter(Br), the flow was classified into three cases:(1) when bottom topography has little influence on the propagation and spatial structure of the ISW(Br0.5),(2) where the ISW is distorted significantly by the blocking effect of the topography(though no wave breaking occurs,(0.5Br0.7), and(3) where the ISW is broken as it encounters and passes over the bottom topography(0.7Br). The numerical results obtained here are consistent with those obtained in laboratory experiments. The breaking process of the incident ISW when Br≈0.7 was completely reproduced. Dissipation rate was linearly related to the blocking parameter when Br0.7, and the maximum dissipation rate could reach about 34% as Br raised to about 1.0. After that, instead of breaking, more reflection happened. Similarly, breaking induced mixing was also most effective during Br around 1.0, and can be up to 0.16.     

Dynamic loads on submerged bodies in a viscous numerical wave tank at small KC numbers

In this paper, an improved version of the MAC method (SIMAC), recently developed at the University of Trieste, is employed for the study of the wave generation and propagation into a numerical wave tank and for the evaluation of dynamic loads over submerged fixed bodies.In the first part of the work, a numerical wave tank was developed. A pneumatic wave-maker at the left-hand side of the tank was implemented by the use of a pressure perturbation at the free surface. The pressure varies in time with a sinusoidal law. Grid sensitivity tests, checks on mass conservation and the Fourier analysis of the waves which propagate in the tank showed the effectiveness of SIMAC in treating such problems. The wave-maker was then calibrated.In the second part of the work, the dynamic loads over submerged square and rectangular cylinders were evaluated. The time records of the horizontal and vertical forces which act over the body were then treated using the Morison equation in order to derive the inertial and damping coefficients. The analysis was carried out for KC numbers ranging between 0.447 and 3.58. Numerical results satisfactorily tallied with experimental data. The analysis of the velocity field near the body evidenced the influence of vortex generation and vortex shedding on the coefficients of inertial forces.     

Numerical simulation of solitary wave scattering by a circular cylinder array

The interaction of a solitary wave with an array of surface-piercing vertical circular cylinders is investigated numerically. The wave motion is modeled by a set of generalized Boussinesq equations. The governing equations are discretized using a finite element method. The numerical model is validated against the experimental data of solitary wave reflection from a vertical wall and solitary wave scattering by a vertical circular cylinder respectively. The predicted wave surface elevation and the wave forces on the cylinder agree well with the experimental data. The numerical model is then employed to study solitary wave scattering by arrays of two circular cylinders and four circular cylinders respectively. The effect of wave direction on the wave forces and the wave runup on the cylinders is quantified.     

On the evolution and run-up of breaking solitary waves on a mild sloping beach

This paper presents new laboratory experiments carried out in a supertank (300 m × 5 m × 5.2 m) of breaking solitary waves evolution on a 1:60 plane beach. The measured data are employed to re-examine existing formulae that include breaking criterion, amplitude evolution and run-up height. The properties of shoreline motion, underwater particle velocity and scale effect on run-up height are briefly discussed. Based on our analyses, it is evidently found that there exist five zones during a wave amplitude evolution course on the present mild slope. A simple formula which is capable of predicting maximum run-up height for a breaking solitary wave on a uniform beach with a wide range of beach slope (1:15–1:60) is also proposed. The calculated results from the present model agree favorably with available laboratory data, indicating that our method is compatible with other predictive models.     

Numerical investigation of tsunami-like wave hydrodynamic characteristics and its comparison with solitary wave

Solitary waves have been commonly used as an initial condition in the experimental and numerical modelling of tsunamis for decades. However, the main component of a tsunami waves acts at completely different spatial and temporal scales than solitary waves. Thus, use of solitary waves as approximation of a tsunami wave may not yield realistic model results, especially in the coastal region where the shoaling effect restrains the development of the tsunami wave. Alternatively, N-shaped waves may be used to give a more realistic approximation of the tsunami wave profile. Based on the superposition of the sech²(*) waves, the observed tsunami wave profile could be approximated with the N-shaped wave method, and this paper presents numerical simulation results based on the tsunami-like wave generated based on the observed tsunami wave profile measured in the Tohoku tsunami. This tsunami-like wave was numerically generated with an internal wave source method based on the two-phase incompressible flow model with a Volume of Fluid (VOF) method to capture the free surface, and a finite volume scheme was used to solve all the governing equations. The model is first validated for the case of a solitary wave propagating within a straight channel, by comparing its analytical solutions to model results. Further, model comparisons between the solitary and tsunami-like wave are then made for (a) the simulation of wave run-up on shore and (b) wave transport over breakwater. Comparisons show that use of these largely different waveform shapes as inputs produces significant differences in overall wave evolution, hydrodynamic load characteristics as well as velocity and vortex fields. Further, it was found that the solitary wave uses underestimated the total energy and hence underestimated the run-up distance.     

Higher harmonics induced by a submerged horizontal plate and a submerged rectangular step in a wave flume

The decomposition of a monochromatic wave over a submerged plate is investigated experimentally in a wave flume. Bound and free higher harmonic modes propagating upstream and downstream the structure are discriminated by means of moving resistive probes. The first-order analysis shows a resonant behaviour linked to the ratio of the plate's width and the fundamental mode wavelength over the plate. The second-order analysis shows an energy transfer from the fundamental mode towards free harmonics propagating downstream the structure. This transfer is linked to the ratio of the width of the plate and the bound harmonic wavelength over the plate. We also performed experiments with a submerged step to compare the efficiency of both structures. The submerged plate is shown to be a more efficient breakwater than the step, at the first as well as the second-order.     

Comparative study on the accuracy of solitary wave generations in an ISPH-based numerical wave flume

In the present study, six solitary wave generations by different mathematical approximations are investigated using a piston type wave maker at dimensionless amplitudes ranging from 0.1 to 0.6 and two water depths. Incompressible smoothed particle hydrodynamics is used to simulate solitary wave propagation along the fixed depth channel. The present numerical results are compared with analytical results and experimental data in terms of free surface displacements, fluid particle velocity, phase speed, paddle motion, etc. The present mesh-free numerical results of wave profile variations over time proved that “Rayleigh” has the lowest relative wave height variation. However, its solitary wave has notable phase lead, while “Third order” and “Ninth order” have the least wave lags. Furthermore, the record of present numerical free surface elevation at different distances and the loss of amplitude of the main pulse showed that regarding both of them, “Ninth order” has supremacy over five others. Considering the numerical velocity components of generated solitary wave, “Third order” and “Ninth order” trace analytical results more accurately than other four ones, whereas “Rayleigh” is the most accurate one in predicting the maximum runup. Finally, the paddle motion, its velocity, and displacement, as well as phase speed and outskirts decay coefficient are also compared and discussed intensely.     

An experimental study on the flow characteristics of a standing wave: application of FLDV measurements

The determination of the characteristics of a standing wave boundary layer, such as water particle velocities and shear stresses, is a vital issue in the prediction of sediment transport rate. Thus, an accurate measure of boundary layer characteristics is a key factor in the study of the wave boundary layer. In this study, a fiber-optic laser Doppler velocimeter (FLDV) is applied to directly measure the velocity profile in the standing wave boundary layer. As the experimental data presented in this paper, the antinode points of standing waves are found to move in the temporal domain. The previous second-order solution for a standing wave boundary layer is insufficient for the prediction of larger Ursell numbers.     

孤立波与直墙式多孔介质结构相互作用数值分析

以连续性方程和雷诺平均方程为控制方程,用k-ε模型封闭方程组,采用VOF方法追踪自由面,建立了具有造波和消波功能的二维孤立波数值水槽。并基于改进的Darcy多孔介质模型研究了二维孤立波与直墙式多孔介质结构的相互作用问题。数值分析了孤立波与直墙式多孔结构相互作用引起的流场和波形变化特性,并与文献中的实验结果进行了比较,数值结果与实验结果吻合良好。     

Propagation of a solitary wave over rigid porous beds

The unsteady two-dimensional Navier–Stokes equations and Navier–Stokes type model equations for porous flows were solved numerically to simulate the propagation of a solitary wave over porous beds. The free surface boundary conditions and the interfacial boundary conditions between the water region and the porous bed are in complete form. The incoming waves were generated using a piston type wavemaker set up in the computational domain. Accuracy of the numerical model was verified by comparing the numerical results with the theoretical solutions. The main characteristics of the flow fields in both the water region and the porous bed were discussed by specifying the velocity fields. Behaviors of boundary layer flows in both fluid and porous bed regions were also revealed. Effects of different parameters on the wave height attenuation were studied and discussed. The results of this numerical model indicate that for the investigated incident wave as the ratio of the porous bed depth to the fluid depth exceeds 10, any further increase of the porous bed depth has no effect on wave height attenuation.