Experimental study of interactions between multi-directional focused wave and vertical circular cylinder,part II: Wave force

We describe experiments with multi-directional focused waves interacted with a vertical circular cylinder in a 3D wave basin. The focus of this study is on the run-up of multi-directional focused waves, wave forces, and wave pressures on the cylinder. Part I, the study on wave run-up, has already been presented by Li et al. (2012). In this paper, the analysis of the wave force on the vertical cylinder is presented.In this experiment, a cylinder with 0.25 m in diameter was adopted and different wave parameters, such as focused wave amplitude, peak frequency, frequency bandwidth and directional spreading index, are considered. The model scale k_pa (k_p is the wave number corresponding to peak frequency, a is the radium of the cylinder) varies from 0.32 to 0.65. The maximum forces of multi-directional focused wave on cylinder were measured and investigated. The results showed that the wave parameters have a significant influence on the wave force, and that the spatial profile of the surface of multi-directional focused wave can also affect its force on the cylinder, which is different from two-dimensional wave. In addition, the ‘secondary loading cycle’ phenomenon was also observed and discussed. In our experiments, the ‘secondary loading cycles’ occur when kA > 0.36 for all cases. While in some referred small scale experiments, the secondary load cycles are observed even for kA = 0.2, when the waves are longer enough. To larger model scale, the pronounced secondary load cycle occurs with larger wave steepness waves.     

Focused wave evolution using linear and second order wavemaker theory

In this paper, the evolution of focused waves using different paddle displacements (piston type) under laboratory conditions is presented. It is well known that in intermediate water depths, linear paddle displacements will generate spurious, free, sub and super harmonics. Thus, a second order correction to suppress these spurious free sub and super harmonics was used to generate the focused waves. The focused waves were generated in the laboratory using a linear superimposition principle, in which the wave paddle displacement is derived based on the sum of a number of sinusoidal components at discrete frequencies, whose phases are accordingly set to focus at a particular location. For this method of generation, the second order wave maker theory proposed by Schäffer [24] can be easily adopted and was used in the present study. Two different centre frequencies (f_c = 0.68 Hz and 1.08 Hz) with three different bandwidth ratios (Δf/f_c = 0.5, 0.75 and 1.0) were tested in a constant water depth, to consider both narrow and broadband spectra. These test cases correspond to wave focusing packets propagating in intermediate and deep water regions. Further, for each wave packet, two different amplitudes were considered in order to analyze non-breaking and breaking cases. Thus, by systematically generating the wave packets using the linear and second order paddle displacements, the analysis was carried out for the spectral and temporal evolution of selected long waves. The temporal evolution of the selected harmonics was analyzed using the Inverse Fast Fourier Transform (IFFT), to show the propagation of the spurious, free, long waves. Further, the variations in energy for the lower, higher and primary frequency ranges are reported for different wave paddle displacements. The analysis revealed that for the broadband spectrum the differences are more pronounced when using linear paddle displacements. We have also noticed a shift in focusing/breaking location and time (i.e. premature) due to the increase in crest height using linear displacements. The experiment data used in this paper has been provided as a supplementary, which can be used to validate the numerical models.     

Prediction of geometrical properties of perfect breaking waves on composite breakwaters

Breaking wave loads on coastal structures depend primarily on the type of wave breaking at the instant of impact. When a wave breaks on a vertical wall with an almost vertical front face called the “perfect breaking”, the greatest impact forces are produced. The correct prediction of impact forces from perfect breaking of waves on seawalls and breakwaters is closely dependent on the accurate determination of their configurations at breaking. The present study is concerned with the determination of the geometrical properties of perfect breaking waves on composite-type breakwaters by employing artificial neural networks. Using a set of laboratory data, the breaker crest height, h_b, breaker height, H_b, and water depth in front of the wall, d_w, from perfect breaking of waves on composite breakwaters are predicted using the artificial neural network technique and the results are compared with those obtained from linear and multi-linear regression models. The comparisons of the predicted results from the present models with measured data show that the h_b, H_b and d_w values, which represent the geometry of waves breaking directly on composite breakwaters, can be predicted more accurately by artificial neural networks compared to linear and multi-linear regressions.     

Sediment transport and beach morphodynamics induced by free long waves,bound long waves and wave groups

New laboratory data are presented on the influence of free long waves, bound long waves and wave groups on sediment transport in the surf and swash zones. As a result of the very significant difficulties in isolating and identifying the morphodynamic influences of long waves and wave groups in field conditions, a laboratory study was designed specifically to enable measurements of sediment transport that resolve these influences. The evolution of model sand beaches, each with the same initial plane slope, was measured for a range of wave conditions, firstly using monochromatic short waves. Subsequently, the monochromatic conditions were perturbed with free long waves and then substituted with bichromatic wave groups with the same mean energy flux. The beach profile changes and net cross-shore transport rates were extracted and compared for the different wave conditions, with and without long waves and wave groups. The experiments include a range of wave conditions, e.g. high-energy, moderate-energy, low-energy waves, which induce both spilling and plunging breakers and different turbulent intensities, and the beaches evolve to form classical accretive, erosive, and intermediate beach states. The data clearly demonstrate that free long waves influence surf zone morphodynamics and promote increased onshore sediment transport during accretive conditions and decreased offshore transport under erosive conditions. In contrast, wave groups, which can generate both forced and free long waves, generally reduce onshore transport during accretive conditions and increase offshore transport under erosive conditions. The influence of the free long waves and wave groups is consistent with the concept of the relative fall velocity, H/w_sT, as a dominant parameter controlling net beach erosion or accretion. Free long waves tend to reduce H/w_sT, promoting accretion, while wave groups tend to increase the effective H/w_sT, promoting erosion.     

多向不规则波浪的确定性模拟

波浪波动时间过程及波列的模拟,对于开展实际波浪对于工程建筑物的作用具有重要的意义。本文采用线性叠加的单叠加模型,建立了多向不规则波浪的确定性模拟方法。基于理论模拟的规则波、单向不规则波和多向不规则波,验证了波浪确定性模拟方法的有效性。定性地对比分析了模拟波列和已知波列的一致性;定量地研究了模拟波浪在空间范围rr/L_s的误差分布情况(rr表示指定位置与给定位置的空间距离,L_s为有效波长)。并且建议,采用本文方法进行波浪确定性模拟时,最佳的浪高仪间距应小于0.12L_s。     

Response based design metocean conditions for a permanently moored FPSO during tropical cyclones: Estimation of greenwater risk

Response based analysis (RBA) is used to establish the design metocean conditions (DMCs) of a generic weather-vaning FPSO off the North West Shelf (NWS) of Australia for determining greenwater severity. A vessel heading prediction tool, an essential component of the RBA analysis for weather-vaning vessels, is developed and evaluated by comparing with full-scale measurements from an operating FPSO. Locations at the bow, amidships and the stern of the vessel are found to be susceptible to greenwater risks and the vessel is often exposed to oblique waves during tropical cyclones. Long-term extrapolation is performed to estimate 1 in. N-year return relative wave-vessel motions represented by both the most probable maximum relative wave-vessel motion within a storm r_mp, and the maximum individual relative wave-vessel motion r_Max. It is observed that r_Max ˜ (1.1–1.2) r_mp. The use of r_Max allows for the variability of the short term maxima per storm and also the fact that the peak in response might not come in the most severe sea-state. Given the focus on greenwater rather than wave severity, the slightly larger value of r_Max at a given return period is used for assessment of greenwater risk. The sea-states that lead to r_Max at a 1 in 100 year level are identified and subsequently used for characterising the wave groups causing maximum relative wave-vessel motion at various locations around the vessel. For a given location, the shapes of the wave time histories which give rise to extreme relative wave-vessel motions in a set of design metocean conditions are similar, indicating that a ‘design wave’, derived within the framework of linear wave theory, may be a useful approach to tackle highly nonlinear and complex greenwater overtopping problems.     

Tsunami simulation with Green–Naghdi theory

Green–Naghdi (GN) theory is a fully nonlinear wave theory which has been used with success to simulate nonlinear water waves. In previous applications of GN theory to water wave problems the ocean bottom was assumed to be time invariant. In this work no such restriction is made and GN theory is used to simulate tsunami caused by bottom fluctuation. As first test cases we simulate two-dimensional nonlinear surface waves generated by positive bottom movements. The results in the generation region for three different seabed movements compare well against earlier experimental data. The results in the downstream region for impulsive seabed movements show some discrepancies in wave phase and amplitude compared with earlier experimental values. It is suspected that the viscous effects may have played a role. The GN theory is then used to study three-dimensional near-field tsunami amplitudes caused by submarine landslides and slumps spreading in two orthogonal directions. The GN results agree with previous linear solution very well when the ratio of the velocities is v₁/v₂=1.0. But GN theory give more believable results for the case of v_T/v=0.1 and v₁/v₂=0.1.     

Shoreline response to a single shore-parallel submerged breakwater

Submerged breakwaters (SBWs) are becoming a popular option for coastal protection, mainly due to their low aesthetic impact on the natural environment. However, SBWs have rarely been employed for coastal protection in the past and therefore, their efficacy remains largely unknown. The main objective of the present study was to investigate the structural and environmental conditions that govern the mode of shoreline response (i.e shoreline erosion vs shoreline accretion) to SBWs. The relative importance of the key structural and environmental parameters governing the response mode to a single shore parallel SBW is investigated through a combination of theoretical analysis and numerical modelling. Using physical considerations, a theoretical response-function model is derived under several simplifying assumptions including parallel depth contours, linear wave theory, shore normal waves, and no wave–current interaction. Numerical modelling is undertaken with the Mike21 model suite to simulate the depth averaged velocity fields (without morphological updating) due to waves acting on a single shore-parallel SBW located on a schematised beach with parallel depth contours. In total 92 coupled wave–current simulations were undertaken. The results indicate that the mode of shoreline response to the SBW can be expressed in terms of the two non-dimensional parameters h_B/H₀ and (s_B/h_B)^3/2(L_B/h_B)²(A³/h_B)^1/2 (variables defined in the text).     

The effect of statistical wind corrections on global wave forecasts

The skill of modern wave models is such that the quality of their forecasts is, to a large degree, determined by errors in the forcing wind field. This work explores the extent to which large-scale systematic biases in modelled waves from a third generation wave model can be attributed to the forcing winds. Three different sets of winds with known global bias characteristics are used to force the WAVEWATCH III model. These winds are based on the Australian Bureau of Meteorology’s ACCESS model output, with different statistical corrections applied. Wave forecasts are verified using satellite altimeter data. It is found that a negative bias in modelled Significant Wave Height (H_s) has its origins primarily in the forcing, however, the reduction of systematic wind biases does not result in universal improvement in modelled H_s. A positive bias is present in the Southern Hemisphere due primarily to an overestimation of high H_s values in the Southern Ocean storm tracks. A positive bias is also present in the east Pacific and East Indian Ocean. This is due both to the over-prediction of waves in the Southern Ocean and lack of swell attenuation in the wave model source terms used. Smaller scale features are apparent, such as a positive bias off the Cape of Good Hope, and a negative bias off Cape Horn. In some situations, internal wave model error has been compensated for by error in the forcing winds.     

Depth inversion in shallow water based on nonlinear properties of shoaling periodic waves

Two depth inversion algorithms (DIA) applicable to coastal waters are developed, calibrated, and validated based on results of computations of periodic waves shoaling over mild slopes, in a two-dimensional numerical wave tank based on fully nonlinear potential flow (FNPF) theory. In actual field situations, these algorithms would be used to predict the cross-shore depth variation h based on sets of values of wave celerity c and length L, and either wave height H or left–right asymmetry s₂/s₁, simultaneously measured at a number of locations in the direction of wave propagation, e.g., using video or radar remote sensing techniques. In these DIAs, an empirical relationship, calibrated for a series of computations in the numerical wave tank, is used to express c as a function of relative depth k_oh and deep water steepness k_oH_o. To carry out depth inversion, wave period is first predicted as the mean of observed L/c values, and H_o is then predicted, either based on observed H or s₂/s₁ values. The celerity relationship is finally inverted to predict depth h. The algorithms are validated by applying them to results of computations for cases with more complex bottom topography and different incident waves than in the original calibration computations. In all cases, root-mean-square (rms)-errors for the depth predictions are found to be less than a few percent, whereas depth predictions based on the linear dispersion relationship—which is still the basis for many state-of-the-art DIAs—have rms-errors 5 to 10 times larger.     

Wave evolution over submerged sills: tests of a high-order Boussinesq model

A Boussinesq model accurate to O(μ)⁴, μ=k₀h₀ in dispersion and retaining all nonlinear effects is derived for the case of variable water depth. A numerical implementation of the model in one horizontal direction is described. An algorithm for wave generation using a grid-interior source function is derived. The model is tested in its complete form, in a weakly nonlinear form corresponding to the approximation δ=O(μ²), δ=a/h₀, and in a fully nonlinear form accurate to O(μ²) in dispersion [Wei, G., Kirby, J.T., Grilli, S.T., Subramanya R. (1995). A fully nonlinear Boussinesq model for surface waves: Part 1. Highly nonlinear unsteady waves. J. Fluid Mech., 294, 71–92]. Test cases are taken from the experiments described by Dingemans [Dingemans, M.W. (1994). Comparison of computations with Boussinesq-like models and laboratory measurements. Report H-1684.12, Delft Hydraulics, 32 pp.] and Ohyama et al. [Ohyama, T., Kiota, W., Tada, A. (1994). Applicability of numerical models to nonlinear dispersive waves. Coastal Engineering, 24, 297–313.] and consider the shoaling and disintegration of monochromatic wave trains propagating over an elevated bar feature in an otherwise constant depth tank. Results clearly demonstrate the importance of the retention of fully-nonlinear effects in correct prediction of the evolved wave fields.     

Experimental Study on Crescent Waves Diffracted by A Circular Cylinder

Crescent waves often observed on the sea surface are unusual wave pattern induced by the instability of Stokes wave. The paper presents the experimental results of the wave field around a circular cylinder generated by the diffraction of crescent wave in order to examine the difference of diffracted crescent waves from the commonly-used diffracted Stokes waves. The results show that with the existence of the cylinder, the crescent wave pattern can still get fully developed, and with the presence of this type of wave pattern, the symmetry breaking of the wave amplitude distribution occurs and there are extra wave components at the frequencies of 0.5ω₀, 1.5ω₀ and 2.5ω₀ (ω₀ is the frequency of Stokes waves) appearing in the wave amplitude spectrum.     

Experimental validation of a CFD model using a narrow wave flume

Induced swell is characterized in an experimental wave flume that is used to validate the corresponding computational model. The experiments and the numerical simulations are performed in water at several depths (h [m] of 0.2, 0.1, and 0.07), using a piston-type wave maker at set amplitudes (0.015 < A_p [m] < 0.15), accelerations-decelerations (0.3 < a_p < 1.0) and average velocities (0.03 < U [m/s] < 0.3) that control the propagation velocity, the period and the wavelength of the waves. The physical effects are modelled with a 2D computational model (STAR-CCM + v11.02) with a mesh of around 630,000 cells of different adaptive sizes, depending on the region under consideration. The physical model is based on a two-phase Eulerian “Volume of Fluid” unsteady model, accounting for gravity and surface tension, that characterizes turbulence with a k-ε model. A user-defined function, based on the period and the amplitude of the vertical paddle in the wave maker, describes the cyclic motion of the linear induction motor. Both the experimental and the computational results are analyzed taking the validity limits of various wave theories as a reference (Le Méhauté). As a result, the experiments are classified within the intermediate water depth regime that corresponds to the second-order Stokes’ wave theory. In addition, both the wave propagation velocity and the period are represented as a function of the wavelength and compared with the analytical solutions from the wave theories. The experimental and the computational test campaign yielded results that confirmed the validity of the computational model and that defined the most appropriate conditions for a high-quality CFD simulation.     

Wave height parameter for damage description of rubble-mound breakwaters

In this paper it will be shown that the wave height parameter H₅₀, defined as the average wave height of the 50 highest waves reaching a rubble-mound breakwater in its useful life, can describe the effect of the wave height on the history of the armor damage caused by the wave climate during the structure's usable life.Using Thompson and Shuttler (Thompson, D.M., Shuttler, R.M., 1975. Riprap design for wind wave attack: A laboratory study on random waves. HRS Wallingford, Report 61, UK) data it will be shown that H₅₀ is the wave parameter that best represents the damage evolution with the number of waves in a sea state. Using this H₅₀ parameter, formulae as van der Meer (van der Meer, J.W., 1988. Rock slopes and gravel beaches under wave attack. PhD Thesis. Technical University of Delft) and Losada and Giménez-Curto (Losada, M.A., Gimenez–Curto, L.A., 1979. The joint effect of the wave height and period on the stability of rubble mound breakwaters using Iribarren's number. Coastal Engineering, 3, 77–96) are transformed into sea-state damage evolution formulae. Using these H₅₀-transformed formulae for regular and irregular sea states it will be shown how damage predictions are independent of the sea state wave height distribution.To check the capability of these H₅₀-formulae to predict damage evolution of succession of sea states with different wave height distributions, some stability tests with regular and irregular waves have been carried out. After analysing the experimental results, it will be shown how H₅₀-formulae can predict the observed damage independently of the sea state wave height distribution or the succession of sea states.     

Evaluation of two WAM white capping parameterizations using parallel unstructured SWAN with application to the Northern Gulf of Mexico, USA

The performance of two well accepted formulations for white capping and wind input of third generation wave models, viz., WAM-3 and WAM-4, were investigated using parallel unstructured SWAN (PunSWAN). Several alternative formulations were also considered to evaluate the effects of higher order steepness and wave number terms in white capping formulations. Distinct model configurations were calibrated and validated against available in situ measurements from the Gulf of Mexico. The results showed that some of the in situ calibrated models outperform the saturation level calibrated models in reproducing the idealized wave growth curves. The simulation results also revealed that increasing the power of the steepness term can enhance the accuracy of significant wave height (H_s), at the expense of a higher bias for large waves. It also has negative effects on mean wave period (T_a) and peak wave period (T_p). It is also demonstrated that the use of the quadratic wave number term in the WAM-3 formulation, instead of the existing linear term, ameliorates the T_a underestimation; however, it results in the model being unable to reach any saturation level. In addition, unlike H_s and T_p, it has been shown that T_a is sensitive to the use of the higher order WAM-4 formulation, and the bias is decreased over a wide range of wave periods. However, it also increases the scatter index (SI) of simulated T_a. It is concluded that the use of the WAM-4 wind input formulation in conjunction with the WAM-3 dissipation form, is the most successful case in reproducing idealized wave growth curves while avoiding T_a underestimation of WAM-3 and a potential spurious bimodal spectrum of WAM-4; consequently, this designates another perspective to improve the overall performance of third generation wave models.     

Effect of bed roughness on turbulent boundary layer and net sediment transport under asymmetric waves

This paper presents an investigation of the roughness effects in the turbulent boundary layer for asymmetric waves by using the baseline (BSL) k–ω model. This model is validated by a set of the experimental data with different wave non-linearity index, N_i (namely, N_i = 0.67, N_i = 0.60 and N_i = 0.58). It is further used to simulate asymmetric wave velocity flows with several values of the roughness parameter (a_m/k_s) which increase gradually, namely from a_m/k_s = 35 to a_m/k_s = 963. The effect of the roughness tends to increase the turbulent kinetic energy and to decrease the mean velocity distribution in the inner boundary layer, whereas in the outer boundary layer, the roughness alters the turbulent kinetic energy and the mean velocity distribution is relatively unaffected. A new simple calculation method of bottom shear stress based on incorporating velocity and acceleration terms is proposed and applied into the calculation of the rate of bed-load transport induced by asymmetric waves. And further, the effect of bed roughness on the bottom shear stress and bed-load sediment transport under asymmetric waves is examined with the turbulent model, the newly proposed method, and the existing calculation method. It is found that the higher roughness elements increase the magnitude of bottom shear stress along a wave cycle and consequently, the potential net sediment transport rate. Moreover, the wave non-linearity also shows a big impact on the bottom shear stress and the net sediment transport.     

Experimental study of unidirectional irregular wave slamming on the three-dimensional structure in the splash zone

The experimental investigation of unidirectional random wave slamming on the three-dimensional structure in the splash zone is presented. The experiment is conducted in the marine environment channel in the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The test wave is unidirectional irregular wave. The experiments are carried out with perpendicular random waves (β=0°) and oblique random waves (β=15°, 30°, 45°), the significant wave heights H_1/3 ranging from 7.5 to 20 cm with 2.5 cm increment, the peak wave periods T_p ranging from 0.75 to 2.0 s with 0.25 s increment, and the clearance of the model with respect to the significant wave height s/H_1/3 ranging from 0.0 to 0.5 with 0.1 increment. The statistical analysis results of different test cases are presented. The statistical distribution characteristics of the perpendicular irregular wave impact pressures are compared with that of the oblique irregular wave on the underside of the structure. The effect of the wave direction β on the wave impact forces on the underside of the structure is determined. The relation between the impact forces and the parameters such as the significant wave height, the relative structure width and the relative clearance of the structure is also discussed.     

A higher-order σ-coordinate non-hydrostatic model for nonlinear surface waves

A higher-order non-hydrostatic model in a σ-coordinate system is developed. The model uses an implicit finite difference scheme on a staggered grid to simultaneously solve the unsteady Navier-Stokes equations (NSE) with the free-surface boundary conditions. An integral method is applied to resolve the top-layer non-hydrostatic pressure, allowing for accurately resolving free-surface wave propagation. In contrast to the previous work, a higher-order spatial discretization is utilized to approximate the large horizontal pressure gradient due to steep surface waves or rapidly varying topographies. An efficient direct solver is developed to solve the resulting block hepta-diagonal matrix system. Accuracy of the new model is validated by linear and nonlinear standing waves and progressive waves. The model is then used to examine freak (extreme) waves. Features of downshifting focusing location and wave asymmetry characteristics are predicted on the temporal and spatial domains of a freak wave.     

Effects of high-order nonlinear interactions on unidirectional wave trains

Numerical simulations of gravity waves with high-order nonlinearities in two-dimensional domain are performed by using the pseudo spectral method. High-order nonlinearities more than third order excite apparently chaotic evolutions of the Fourier energy in deep water random waves. The high-order nonlinearities increase kurtosis, wave height distribution and H_max/H_1/3 in deep water and decrease these wave statistics in shallow water. Moreover, they can generate a single extreme high wave with an outstanding crest height in deep water. High-order nonlinearities (more than third order) can be regarded as one cause of freak waves in deep water.