A theoretical study on a permanent wavefilter

This paper investigates a wavefilter which could effectively eliminate re-reflected waves generated in a wave flume over a certain range of wave periods. The wavefilter is not required to change its thickness for different incoming wave periods and is, therefore, termed here a permanent wavefilter. The structure comprises of a set of multilayer porous media. Their related properties such as porosities, friction factors and thickness are sought with the hope of developing a structure which could serve the desired purpose.A theoretical study has been performed. It was found that a multilayer porous media could eliminate re-reflected waves without resorting to adjusting the medium property whenever wave period changes, only if the thickness of the media has to exceed a certain amount.Generally speaking, a thicker multilayer porous media is needed for longer-period waves, while a thinner one is suitable for shorter-period waves. The wavefilter considered here could serve both eliminating re-reflected waves and also providing a desired median wave height if all of those related factors of the media have been properly selected and if the generated waves are restricted to not too wide a range of wave periods. However, it is unlikely that a wavefilter with fixed thickness can remain effective over too wide a range of wave period.     

The elemination of re-reflected waves by a porous medium of finite thickness

The elimination of re-reflected waves in a wave channel by installing a porous medium in front of the wavemaker is investigated. The thickness of the porous wall required to eliminate the re-reflected waves is shown to be related to th porosity, friction coefficient, and wave period, as well as to both the positions of the porous medium and the test structure. However, this study indicates that the goal of eliminating re-reflected waves can be achieved by simply varying the thickness of the porous medium according to the wave period, with all the other factors arbitrarily selected.Assuming that the oscillation amplitude of the wavemaker board is constant, the primitive wave amplitude, before reaching the porous medium, becomes smaller as the wave period is increased. In addition, the study found that the required thickness of the porous medium for eliminating the re-reflected wave becomes larger as the wave period is increased. This results in a trend which further reduces the wave amplitude after the wave passes through the porous medium. In consequence, the oscillation amplitude of a wavemaker board has to be adjusted in a larger scale if the wave period is to be increased.     

Characteristics of turbulent boundary layers over a rough bed under saw-tooth waves and its application to sediment transport

A large number of studies have been done dealing with sinusoidal wave boundary layers in the past. However, ocean waves often have a strong asymmetric shape especially in shallow water, and net of sediment movement occurs. It is envisaged that bottom shear stress and sediment transport behaviors influenced by the effect of asymmetry are different from those in sinusoidal waves. Characteristics of the turbulent boundary layer under breaking waves (saw-tooth) are investigated and described through both laboratory and numerical experiments. A new calculation method for bottom shear stress based on velocity and acceleration terms, theoretical phase difference, φ and the acceleration coefficient, a_c expressing the wave skew-ness effect for saw-tooth waves is proposed. The acceleration coefficient was determined empirically from both experimental and baseline k–ω model results. The new calculation has shown better agreement with the experimental data along a wave cycle for all saw-tooth wave cases compared by other existing methods. It was further applied into sediment transport rate calculation induced by skew waves. Sediment transport rate was formulated by using the existing sheet flow sediment transport rate data under skew waves by Watanabe and Sato [Watanabe, A. and Sato, S., 2004. A sheet-flow transport rate formula for asymmetric, forward-leaning waves and currents. Proc. of 29th ICCE, ASCE, pp. 1703–1714.]. Moreover, the characteristics of the net sediment transport were also examined and a good agreement between the proposed method and experimental data has been found.     

Generation and propagation of transient nonlinear waves in a wave flume

A semi-analytical nonlinear wavemaker model is derived to predict the generation and propagation of transient nonlinear waves in a wave flume. The solution is very efficient and is achieved by applying eigenfunction expansions and FFT. The model is applied to study the effect of the wavemaker and its motion on the generation and propagation of nonlinear waves. The results indicate that the linear wavemaker theory may be applied to predict only the generation of waves of low steepness for which the nonlinear terms in the kinematic wavemaker boundary condition and free-surface boundary conditions are of secondary importance. For waves of moderate steepness and steep waves these nonlinear terms have substantial effects on wave profile and wave spectrum just after the wavemaker. A wave spectrum corresponding to a sinusoidally moving wavemaker possesses a multi-peak form with substantial nonlinear components, which disturbs or may even exclude physical modeling in wave flumes. The analysis shows that the widely recognized weakly nonlinear wavemaker theory may only be applied to describe the generation and propagation of waves of low steepness. This is subject to further restrictions in shallow and deep waters because the kinematic wavemaker boundary condition as well as the nonlinear interaction of wave components and the evolution of wave energy spectrum is not properly described by weakly nonlinear wavemaker theory. Laboratory experiments were conducted in a wave flume to verify the nonlinear wavemaker model. The comparisons show a reasonable agreement between predicted and measured free-surface elevation and the corresponding amplitudes of Fourier series. A reasonable agreement between theoretical results and experimental data is observed even for fairly steep waves.     

A polar method for obtaining wave resonating quadruplets in finite depths

A polar method for obtaining wave resonating quadruplets {K₁, K₂, K₃, K₄} in the computation of nonlinear wave–wave interaction source term of the wave model is presented with results for both deep and finite water depths. The method first determines the end radial points of the locus equation for K₂, for each set of input wave vectors (K₁, K₃) on the symmetry. The locus of K₂ (and hence K₄) is then traced in the anti-clockwise direction starting with the maximum radial point on the line of symmetry. It is shown that when k₃>k₁, the number of points on the locus varies when the orientations of the input wave vectors are changed and reduces when the difference in the magnitude of the input wave vectors is increased. A significant advantage in this method is that the angular increment on the locus for K₂ can be kept constant.     

A simple model of sediment initiation under waves

A simple model is developed to study the initial motion of sediment on a horizontal bed under non-breaking waves. The model is derived to be A=C(T−T₀) based on a wide range of experimental data collected in different flow regimes, where A is the nearbed semi-excursion of wave motion, T is the wave period, and C and T₀ are the coefficients dependent on sediment properties only. For a given sediment, the onset velocity of sediment motion derived from the model is shown to initially increase sharply with wave period T and then approach a constant. The flow Reynolds number Re corresponding to an initiated sediment is also calculated from the simple model and found to be a function of sediment properties and wave period. For the completeness of this study, the initial motion of light sediment under very short waves is also investigated. The present model agrees well with the available laboratory and field data.     

Wave reflection characteristics of plane, dentated and serrated seawalls

The hydrodynamic performance of vertical and sloped plane, dentated and serrated seawalls were investigated using physical model studies. Regular and random waves of wide range of heights and periods were used. Tests were carried out for different inclinations of the seawall (i.e. θ=30, 40, 50, 60 and 90°) and for a constant water depth of 0.7 m. The wave reflection was measured to assess the dissipation character of the seawalls. It was observed that the serrated seawall was superior to the plane and dentated seawall in reducing the wave reflection. Even for the vertical case, the coefficient of reflection due to regular waves for dentated seawall ranged from 0.6–0.99 and for the vertical serrated seawall it was 0.45–0.98, whereas for the vertical plane wall, it was almost 1.0. It was found that the characteristic dimension of the seawall (i.e. L/W) and the relative water depth (i.e. d/L) were better influencing parameters compared to the conventionally used surf similarity parameter ‘ξ’ (ξ=tan θ/(H_i/L)^0.5) in predicting the reflection from the dentated and serrated seawall, where L is the local wave length, W the width of the dent along the length of the seawall slope, d the water depth at the toe and H_i is the incident wave height. A similar trend was observed for the random waves too. The reduction in the wave reflection due to random waves for the dentated seawall as compared to the plane seawall was about 18% and for the serrated seawall, it was 20%. It was observed that the reflection due to random waves was lesser for all the three different walls than the regular waves, due to the mutual interaction of random waves. Multiple regression analysis on the measured data points was carried out and predictive equations for the reflection coefficient were obtained for both regular and random waves. This study will be useful in the design of energy dissipating type vertical quay walls in ports and harbours, sloped seawalls for shore protection from erosion and sloped caisson as breakwaters. Comparison of predictive formulae with the experimental results revealed that the prediction methods were good enough for practical purposes.     

The stability of the antifer units used on breakwaters in case of irregular placement

The primary aim of the study is to experimentally investigate the stability performance of antifer units on the trunk section of breakwaters under the effect of regular and irregular waves in case of irregular placement. The stability performance tests were conducted for different slopes, i.e. cot α=1.25, 1.5, 2.0, 2.5, under irregular waves and for cot α=2.5 under regular waves. Hudson’s formula was employed in order to characterize the stability performance of antifer units for the irregular placement technique. Different representative wave height parameters, i.e. H_s, H_1/10 and H_max, were examined to determine the one best characterizing breakwater stability. Furthermore, the effects of wave period and wave steepness on the stability of the breakwater were explored.     

Prediction of wave pressures on smooth impermeable seawalls

Simple prediction methods are proposed to estimate the wave induced pressures on smooth impermeable seawalls. Based on the physics of the wave structure interaction, the sloped seawall is divided into a total of five zones (zones 1, 2 and 3 during run-up (corresponding pressures are called as positive pressures) and zones 4 and 5 during run-down (corresponding pressures are called negative pressures)) (Fig. 1). Zone 1 (0<z<d−H_i/2), where the wave pressure is governed by the partial reflection and phase shift; Zone 2 (d−H_i/2<z<d), where the effect of wave breaking and turbulence is significant; Zone 3 (d<z<Run-up height), where the pressure is induced by the run-up water; Zone 4 (Run-down<z<d), where the wave pressure is caused by the run-down effect and Zone 5 (0<z<d-Run down), where the negative wave pressures are due to partial reflection and phase shift effects. Here d is the water depth at the toe of the seawall, H_i is the incident wave height and z is the vertical elevation with toe of the seawall as origin and z is positive upward. For wave pressure prediction in zones 1 and 5, the empirical formula proposed by Ahrens et al. (1993) to estimate the wave reflection and Sutherland and Donoghue's recommendations (1998) for the estimation of phase shift of the waves caused by the sloped structures are used. Multiple regression analysis is carried out on the measured pressure data and empirical formulas are proposed for zones 2, 3 and 4. The recommendations of Van der Meer and Breteler (1990) and Schüttrumpf et al. (1994) for the prediction of wave run-down are used for pressure prediction at zone 4. Comparison of the proposed prediction formulas with the experimental results reveal that the prediction methods are good enough for practical purposes. The present study also shows a strong relation between wave reflection, wave run-up, wave run-down and phase shift of waves on wave pressures on the seawalls.     

Eddy viscosities and velocities in combined wave-current flows

The eddy viscosities for the steady and the periodic components of combined wave-current flows have been studied quantitatively from the presently available experimental data. It has been found that inside the boundary interaction layer [z < δ] the eddy viscosity ε_c for the steady flow is increased in the presence of waves while outside the boundary interaction layer [z >δ] it is affected little by the wave motion, and that the eddy viscosity ε_w for the wave motion in the boundary layer is independent of the current strength U^*. On the other hand, a new eddy viscosity model is presented to give a good prediction of the velocity distributions of the waves and currents in comparison with experimental data.     

Numerical calculation of forces induced by short-crested waves on a vertical cylinder of arbitrary cross-section

Most off-shore oil platforms are supported by vertical cylinders extending to the ocean floor. An important problem in off-shore engineering is the calculation of the wave loading exerted on these vertical cylinders. Analytical solutions have been found for the case of plane incident waves incident on a circular cylinder by MacCamy and Fuchs [(1954), Wave forces on piles: a diffraction theory. U.S. Army Corps of Engineering, Beach Erosion Board, Technical Memorandum No. 69] and also for short-crested waves incident on a circular cylinder by Zhu [(1993), Diffraction of short-crested waves around a circular cylinder. Ocean Engng 20, 389–407]. However, for a cylinder of arbitrary cross-section, no analytic solutions currently exist. Au and Brebbia [(1983), Diffraction of water waves for vertical cylinders using boundary elements. Appl. Math. Modelling 7, 106–114] proposed an efficient numerical approach to calculate the wave loads induced by plane waves on vertical cylinders by using the boundary element method. However, wind-generated waves are better modelled by short-crested waves. Whether or not these short-crested waves can induce larger wave forces on a structure is of great concern to ocean engineers. In this paper wave loads, induced by short-crested incident waves, on a vertical cylinder of arbitrary cross-section are discussed. For a cylinder of certain cross-section, the wave loads induced by short-crested waves can be larger than those induced by plane waves with the same total wave number.     

Wave interaction with ‘’-type breakwaters

The wave transmission, reflection and energy dissipation characteristics of ‘’-type breakwaters were studied using physical models. Regular and random waves in a wide range of wave heights and periods and a constant water depth were used. Five different depths of immersion (two emerged, one surface flushing and two submerged conditions) of this breakwater were selected. The coefficient of transmission, K_t, and coefficient of reflection, K_r, were obtained from the measurements, and the coefficient of energy loss, K_l was calculated using the law of balance of energy. It was found that the wave transmission is significantly reduced with increased relative water depth, d/L, whether the vertical barrier of the breakwater is surface piercing or submerged, where ‘d’ is the water depth and ‘L’ is the wave length. The wave reflection decreases and energy loss increases with increased wave steepness, especially when the top tip of the vertical barrier of this breakwater is kept at still water level (SWL). For any incident wave climate (moderate or storm waves), the wave transmission consistently decreases and the reflection increases with increased relative depth of immersion, Δ/d from −0.142 to 0.142. K_t values less than 0.3 can be easily obtained for the case of Δ/d=+0.071 and 0.142, where Δ is the height of exposure (+ve) or depth of immersion (−ve) of the top tip of the vertical barrier. This breakwater is capable of dissipating wave energy to an extent of 50–80%. The overall performance of this breakwater was found to be better in the random wave fields than in the regular waves. A comparison of the hydrodynamic performance of ‘’-type and ‘T’-type shows that ‘T’-type breakwater is better than ‘’-type by about 20–30% under identical conditions.     

The effects of a porous-elastic seabed on interfacial wave propagation

A theoretical model for the decay of progressive interfacial gravity waves propagating above a porous bed is developed assuming potential flow in a two-layer system with a free surface and a sharp interface. A new wave dispersion relation for two-layer flow above a quasi-static porous seabed is derived and investigated. The solutions for the nonlinear wave profile are derived using a perturbation method and the effects of geometric and flow parameters including bed characteristics, depth ratios and the densities of the two fluids are studied and discussed. Comparisons with existing analytical solutions for viscous interfacial wave attenuation over a rigid bed demonstrate the relative importance of the porous bed as a mechanism for wave decay. It is shown that the influence of a porous seabed on wave propagation is significant when the depth of the lower layer, normalised by the wavenumber, is less than π.     

Geometrical properties of perfect breaking waves on composite breakwaters

The experimental results have so far shown that when a wave breaks on a vertical wall with an almost vertical front face at the instant of impact that is called perfect breaking or perfect impact, the greatest impact forces are produced on the wall. Therefore, the configuration of breaking waves is important in the design considerations of coastal structures. The present study is concerned with determining the geometrical properties of oscillatory waves that break perfectly on the vertical wall of composite-type breakwaters. The laboratory tests for perfect breaking waves on composite breakwaters are conducted with base slopes of 1/2, 1/4 and 1/6, and with berm widths of 0.00, 0.10, 0.20, 0.30 and 0.40 m. The shape and the dimensions of waves at the instant of perfect breaking on the wall are determined using a video camera. The experimental results for the geometrical properties of the breakers are presented non-dimensionally. Within the range of present experimental conditions, it is found that the dimensionless breaker crest height, h_b/d_w, and dimensionless breaker height, H_b/d_w, decrease; and, dimensionless breaker depth, d_w/H₀, increases with increasing relative berm width, B/D. The breaker height index, H_b/H₀, is almost unaffected by B/D. The deep-water wave steepness and the base slope of the breakwater do not seem to influence the geometrical properties of the breakers at wall systematically.     

A coupled numerical model for simulation of wave breaking and hydraulic performances of a composite seawall

A numerical model is developed by combining a porous flow model and a two-phase flow model to simulate wave transformation in porous structure and hydraulic performances of a composite type low-crest seawall. The structure consists of a wide submerged reef, a porous terrace at the top and an impermeable rear wall. The porous flow model is based on the extended Navier-Stokes equations for wave motion in porous media and k−ε turbulence equations. The two-phase flow model combines the water domain with the air zone of finite thickness above water surface. A unique solution domain is established by satisfying kinematic boundary condition at the interface of air and water. The free surface advection of water wave is modeled by the volume of fluid method with newly developed fluid advection algorithm. Comparison of computed and measured wave properties shows reasonably good agreement. The influence of terrace width and structure porosity is investigated based on numerical results. It is concluded that there exist optimum value of terrace width and porosity that can maximize hydraulic performances. The velocity distributions inside and in front of the structure are also investigated.     

JONSWAP's parameters: Sorting out the inconsistencies

The original prognostic equations for the JONSWAP-spectrum contained inconsistencies. A subsequent paper (Hasselmann et al., 1976, J. phys. Oceanogr.6, 200–208) attempted to regularise the situation. This paper shows that there were still inconsistencies in the prognostic equations giving overestimations of the first moment of the spectrum and consequently the significant wave height. The prognostic equations are reworked systematically and results presented. It is shown that variable σ_a, σ_b and γ must be used to achieve consistent results     

A comparison of two spectral wave models in the Southern California Bight

Two models, a spectral refraction model (Longuet-Higgins) and a parabolic equation method (PEM) refraction-diffraction model (Kirby), are used to simulate the propagation of surface gravity waves across the Southern California Bight. The Bight contains numerous offshore islands and shoals and is significantly larger (≈ 300 km by 300 km) than regions typically studied with these models. The effects of complex bathymetry on the transformation of incident wave directional spectra, S₀(f,θ₀), which are very narrow in both frequency and direction are difficult to model accurately. As S₀(f,θ₀) becomes broader in both dimensions, agreement between the models improves and the spectra predicted at coastal sites become less sensitive to errors in the bathymetry grid, to tidal changes in the mean water depth, and to uncertainty in S₀(f,θ₀) itself. The smoothing associated with even relatively narrow (0.01 Hz-5° bandwidth) S₀(f,θ₀) is usually sufficient to bring the model predictions of shallow water energy into at least qualitative agreement. However, neither model is accurate at highly sheltered sites. The importance of diffraction degrades the predictions of the refraction model, and a positive bias [O (10%) of the deep ocean energy] in the refraction-diffraction model estimates, believed to stem from numerical “noise” (Kirby), may be comparable to the low wave energy. The best agreement between the predicted spectra generally occurs at moderately exposed locations in deeper waters within the Bight, away from shallow water diffractive effects and in the far-field of the islands. In these cases, the differences between the models are small, comparable to the errors caused by tidal fluctuations in water depth as waves propagate across the Bight. The accuracy of predicted energies at these sites is likely to be limited by the uncertainty in specifying S₀(f,θ₀).     

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.     

Topocaustics

Deep (2000 m) observations near the Sigsbee escarpment in the Gulf of Mexico show short-period (approximately 5–12 days) energetic currents due to topographic Rossby waves (TRW’s). We suggest that the phenomenon is due to the focusing and accumulation of TRW energy by the slopes coupled with a bend in isobaths, in a topographic caustic (topocaustic). The idea draws on a simple mathematical equivalence between the propagation of internal waves and of TRW’s. Topocaustics occur near regions of maximum N_T = N|h| (N = Brunt–Väisälä frequency; h = water depth). Because of the one-sided propagation property of TRW’s, energy also tends to accumulate at the “western” end of closed contours of N_T. The process is demonstrated here using a nonlinear primitive-equation numerical model with idealized bathymetry and forcing. A Gulf of Mexico simulation initialized with a data-assimilated analysis covering the period of the Sigsbee observation is then conducted. The mooring is near a localized maximum N_T, and Intrinsic Mode Functions confirm the existence of energetic bursts of short-period deep-current events. The strong currents are locally forced from above, either by an extended Loop Current or a warm ring.     

双消浪室局部开孔沉箱防波堤反射特性的理论与试验研究

双消浪室局部开孔沉箱防波堤具有低反射、结构受力小、适宜较大水深和工程造价低等优点。为明确双消浪室局部开孔沉箱水动力特性的主要影响因素,采用理论分析和物理模型试验相结合的方法,对规则波和不规则波作用下双消浪室局部开孔沉箱防波堤的反射特性进行研究。基于势流理论,建立规则波和不规则波对局部开孔沉箱防波堤作用的三维解析解,采用二次压力损失边界条件考虑沉箱开孔墙对波浪运动的影响,利用周期性边界条件考虑防波堤结构沿长度方向的周期性变化。开展相应规则波和不规则波物理模型试验,验证理论模型的合理性。通过算例分析,研究不同波浪要素和结构参数对防波堤反射特性的影响。研究表明:双消浪室局部开孔沉箱相对消浪室宽度取值为0.08~0.20,沉箱前墙开孔率大于后墙开孔率时,防波堤在较大波浪频率范围内消波效果显著;当前后墙的开孔率相等时,防波堤反射系数的最小值随着开孔率增大而减小。