An analytical approach for the calculation of random wave forces on submerged tunnels

This paper deals with the random forces produced by high ocean waves on submerged horizontal circular cylinders. Arena [Arena F, Interaction between long-crested random waves and a submerged horizontal cylinder. Phys Fluids 2006;18(7):1–9 (paper 076602)] obtained the analytical solution of the random wave field for two dimensional waves by extending the classical Ogilvie solution [Ogilvie TF, First- and second-order forces on a cylinder submerged under a free surface. J Fluid Mech 1963;16:451–472; Arena F, Note on a paper by Ogilvie: The interaction between waves and a submerged horizontal cylinder. J Fluid Mech 1999;394:355–356] to the case of random waves. In this paper, the wave force acting on the cylinder is investigated and the Froude Krylov force [Sarpkaya T, Isaacson M, Mechanics of wave forces on offshore structures, Van Nostrand Reinhold Co.; 1981], on the ideal water cylinder, is calculated from the random incident wave field. Both forces represent a Gaussian random process of time. The diffraction coefficient of the wave force is obtained as quotient between the standard deviations of the force on the solid cylinder and of the Froude Krylov force. It is found that the diffraction coefficient of the horizontal force C_do is equal to the C_dv of the vertical force. Finally, it is shown that, since a very large wave force occurs on the cylinder, it may be calculated, in time domain, starting from the Froude Krylov force. It is then shown that this result is due to the fact that the frequency spectrum of the force acting on the cylinder is nearly identical to that of the Froude–Krylov force.     

Triangular Configuration Tension Leg Platform behaviour under random sea wave loads

This study investigates the dynamic response of a Triangular Configuration Tension Leg Platform (TLP) under random sea wave loads. The random wave has been generated synthetically using the Monte-Carlo simulation with the Peirson–Moskowitz (P–M) spectrum. Diffraction effects and second-order wave forces have not been considered. The evaluation of hydrodynamic forces is carried out using the modified Morison equation with water particle kinematics evaluated using Airy's linear wave theory. Wave forces are taken to be acting in the surge degree-of-freedom. The effect of coupling of various structural degrees-of-freedom (surge, sway, heave, roll, pitch and yaw) on the dynamic response of the TLP under random wave loads is studied. Parametric studies for random waves with different H_s and T_z under the presence of current have also been carried out. For the orientation of the TLP, surge, heave and pitch degrees-of-freedom responses are influenced significantly. The surge power spectral density function (PSDF) indicates that the mean square response is affected by the amplification at the natural frequency of the surge degree-of-freedom and also at the peak frequency of the wave loading. The PSDF of the heave response shows higher peak values near the surge frequency and near the peak frequency of the wave loading. Surge response, therefore, influences heave response to the maximum. Variable submergence seems to be a major source of nonlinearity and significantly enhances the responses in surge, heave and pitch degrees-of-freedom. In the presence of current, the response behaviour of the TLP is altered significantly introducing a non-zero mean response in all degrees-of-freedom.     

Regular wave pressures and forces on submerged pipelines near a sloping boundary

The hydrodynamic pressures induced by regular waves around the circumference of a pipeline normal to the wave direction and near a rigid bed of slope 1:10 have been investigated in a wave flume. The pressures were integrated to obtain the force time history, from which the peak horizontal and vertical forces are evaluated. The maximum and root mean square horizontal and transverse force coefficients are correlated with the Keulegan–Carpenter (KC) number. The effect of the distance between the sloping bed and the pipeline on the force coefficients is discussed. The force coefficients are found to decrease with an increase in KC number and with the decrease in the relative clearance of the pipeline from the boundary. In addition, the reflection characteristics of the sloping bed in the presence of the pipeline as a function of surf similarity parameter and their comparison with the results from existing literature are also reported. The details of the model setup, experimental procedure, results and discussion are presented in this paper.     

Mooring forces in horizontal interlaced moored floating pipe breakwater with three layers

The paper presents the results from model scale experiments on the study of forces in the moorings of horizontally interlaced, multi-layered, moored floating pipe breakwaters. The studies are conducted with breakwater models having three layers subjected to waves of steepness H_i/L (H_i is the incident wave height and L the wavelength) varying from 0.0066 to 0.0464, relative width W/L (W is the width of breakwater) varying from 0.4 to 2.65, and relative spacing S/D (S is the spacing of pipes and D the diameter of pipe) of 2 and 4. The variation of measured normalized mooring forces on the seaward side and leeward side are analyzed by plotting non-dimensional graphs depicting f/γW² (f is the force in the mooring per unit length of the breakwater, γ the weight density of sea water) as a function W/L for various values of H_i/d (d is the depth of water). It is found that the force in the seaward side mooring increases with an increase in H_i/L for d/W values ranging between 0.081 and 0.276. The experimental results also reveal that the forces in the seaward side mooring decrease as W/L increases, up to a value of W/L=1.3, and then increases with an increase in W/L. It is also observed that the wave attenuation characteristics of breakwater model with relative spacing of 4 is better than that of the model with relative spacing of 2. The maximum force in the seaward side mooring for model with S/D=4 is lower compared to that for the breakwater model with S/D=2. A multivariate non-linear regression analysis has been carried out for the data on mooring forces for the seaside and leeside.     

Wave climate off northern Norway

The wave climate off northern Norway is considered and the investigation is based on wave measurements made at Tromsøflaket by means of a waverider buoy during the years 1977–1981. Data quality of waverider measurements is briefly commented upon; however, more emphasis is given to an evaluation of the long-term representativity of the actual measuring period and to a procedure accounting approximately for a lack of representativity. The wave climate is presented in terms of a smoothed joint probability density function of the significant wave height, H_s, and the spectral peak period, T_p. Based on this distribution a consistent design curve in the H_s, T_p space is established.     

Numerical modeling of wave runup and forces on an idealized beachfront house

In this paper, a numerical wave model based on the incompressible Reynolds-averaged Navier–Stokes (RANS) and k–ε equations is used to estimate the impact of a solitary wave on an idealized beachfront house located at different elevations on a plane beach. The locations of the free surface are reconstructed by volume of fluid (VOF) method. The model is satisfactorily tested against the experimental data of wave runup, and the analytical solution of wave forces on vertical walls. The time histories of wave profiles, forces, and overturning moments on the idealized house are demonstrated and analyzed. The variations of wave forces and overturning moments with the elevation of the idealized beachfront house are also investigated.     

波群作用于垂直桩柱的峰力统计性质

研究具有窄谱和Ｗｅｉｂｕｌｌ波高分布的波群对非线性桩柱系统作用力的统计性质。求得了桩桩的波浪峰力的各种特征值及其比值。指出这些数值不仅是阻力一惯性力参数ｂＨ的函数，也随着波群因子而变化。本文模式更具广泛性。文中给出了一系列计算图表，可从理论计算波群作用于桩柱的波浪峰力。     

Ice forces acting on inclined wedges and cones

The increased off shore activity in arctic waters has occasioned investigations with the purpose of reducing the ice forces acting on structures. Korzhavin (Novosibirsk, U.S.S.R.) has shown that the ice forces will be appreciably reduced if wedges or cones are introduced instead of vertical faces. A theory is presented in the following, by which it is possible to calculate the forces based on the physical and geometrical parameters of the system. The theory has been verified by model tests with both artificial and natural ice.For an inclined wedge or cone the force will F = C_FF_max where C_F is a reduction factor and F_max = r_ced is the maximum force acting on a vertical faced structure, r_c being the compression strength, e the thickness of the ice and d the width of the structure. The value of C_F will normally vary from 0·1 to 0·4.     

Additional flux arising from unresolved scales in eddying ocean models

An analysis is presented of snapshot data (eastward and northward velocity components: u and v; tracer such as potential temperature: τ) from an eddy-resolving (Rgrid: 1/12°) ocean model experiment, in order to explore a method for improving eddy-permitting model performance. Horizontal 3 × 3 R-grid averages give the eddy-permitting grid (P-grid: 1/4°) variables: 〈u〉, 〈v〉, and 〈τ〉, where 〈〉 denotes the spatial P-grid scale average. The difference between the horizontal tracer flux across the boundary face of a P-grid and that across the corresponding faces of R-grids is estimated as F_2E. It is found that the correlations among the gradients of u, v, and τ give a good approximation F_2C to the estimated flux F_2E. The approximated flux is a function of these gradients and the grid size. A method is presented for implementing the F_2C for density to an eddying ocean model as an additional advection. Practical experiments were conducted with a realistic configuration. It is shown that the zonal mean isotherms in the Kuroshio extension region are more flattened in the run using the proposed method than in another run using the conventional horizontal biharmonic operator, suggesting that the additional flux correction leads to an enhancement of sub-basin scale mixing.     

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.     

The inception of sheet flow in oscillatory flow

A simple model is developed to study the inception of sheet flow in oscillatory flow based on the available experimental data. The inception of sheet flow in oscillatory flow is well defined by the simple model: A/d=KA²ω/ν+B, where A is the semi-excursion of wave orbital motion near the bed, d is the grain size, ω is the angular frequency, ν is the kinematic viscosity of water, and K and B are the coefficients and dependent on sediment properties only. The inception velocity of sheet flow derived from the model is shown to be the function of grain size d, oscillatory period T and specific sediment density s. For a given sediment, the inception velocity is found to increase sharply initially with T and then approach a constant at T>6.0 s. The present model is quite simple and gives good agreement with the available experimental data.     

Inertial wave loads on horizontal cylinders: A field experiment

A long submerged horizontal circular cylinder of .90 m diameter was assembled off the beach at Reggio Calabria where the wind waves typically have significant height ranging within 0.20 and 0.40 m and dominant period within 1.8 and 2.6 s. Three ultrasonic probes recorded the waves, and two sets of pressure transducers, the first one at the cylinder and the second one in the undisturbed wave field, enabled to compare the force amplitude on the cylinder to the force amplitude on an equivalent mass of water in the undisturbed wave field (Froude-Krylov F-K force). After ten days of measurements, the experiment was repeated with a cylinder of .45 m diameter. The Keulegan-Carpenter number was within 2.5, and the wave forces proved to be inertial. The following general features emerged: (i) the force spectrum is usually very narrow even if the wave spectrum is broad; (ii) the vertical diffraction coefficient is somewhat smaller than the horizontal diffraction coefficient; (iii) the positive extremes of F_z (vertical force referred to the buoyancy force) markedly exceed the negative extremes; (iv) the pressure fluctuations induced by the highest waves at the cylinder are very similar to the measured pressure-surface displacement covariances. In each of the 580 records obtained in the course of the experiment it was found that the propagation speed reduces to about a half at the cylinder, and the amplitude of the pressure fluctuations increases of 10–15% at the upper half of the cylinder and decreases of about the same percentage at the lower half. These phenomena fully explain why the force amplitude on the cylinder is larger than the F-K force amplitude.     

Finite element analysis of two-dimensional nonlinear sloshing problems in random excitations

A two-dimensional nonlinear random sloshing problem is analyzed by the fully nonlinear wave velocity potential theory based on the finite element method. A rectangular container filled with liquid subjected to specified horizontal random oscillations is studied. Both wave elevation and hydrodynamic force are obtained. The spectra of random waves and forces have also been investigated, and the effects of the peak frequencies and spectral width of the specified spectrum used for the generation of the random oscillations are discussed. It is found that the energy mainly concentrates at the natural frequencies of the container and is dominant at the ith order natural frequency when the peak frequency is close to the ith order natural frequency. Some results are compared between the fully nonlinear solutions, the linear solutions and the linear plus second-order solutions.     

Investigating the impact of surface wave breaking on modeling the trajectories of drifters in the northern Adriatic Sea during a wind-storm event

An accurate numerical prediction of the oceanic upper layer velocity is a demanding requirement for many applications at sea and is a function of several near-surface processes that need to be incorporated in a numerical model. Among them, we assess the effects of vertical resolution, different vertical mixing parameterization (the so-called Generic Length Scale –GLS– set of k–ε, k–ω, gen, and the Mellor–Yamada), and surface roughness values on turbulent kinetic energy (k) injection from breaking waves.First, we modified the GLS turbulence closure formulation in the Regional Ocean Modeling System (ROMS) to incorporate the surface flux of turbulent kinetic energy due to wave breaking. Then, we applied the model to idealized test cases, exploring the sensitivity to the above mentioned factors. Last, the model was applied to a realistic situation in the Adriatic Sea driven by numerical meteorological forcings and river discharges. In this case, numerical drifters were released during an intense episode of Bora winds that occurred in mid-February 2003, and their trajectories compared to the displacement of satellite-tracked drifters deployed during the ADRIA02-03 sea-truth campaign.Results indicted that the inclusion of the wave breaking process helps improve the accuracy of the numerical simulations, subject to an increase in the typical value of the surface roughness z₀. Specifically, the best performance was obtained using α_CH = 56,000 in the Charnok formula, the wave breaking parameterization activated, k–ε as the turbulence closure model. With these options, the relative error with respect to the average distance of the drifter was about 25% (5.5 km/day). The most sensitive factors in the model were found to be the value of α_CH enhanced with respect to a standard value, followed by the adoption of wave breaking parameterization and the particular turbulence closure model selected.     

Wave forces on, and water-surface fluctuations around a vertical cylinder encircled by a perforated square caisson

The wave forces and moments on and the water surface fluctuations around a vertical circular cylinder encircled by a perforated square caisson were experimentally investigated. The porosity of the outer square caisson was varied from 4.24 to 14.58%. The in-line wave forces on the inner vertical cylinder are influenced by changing the porosity of the outer caisson, whereas the variations in the water surface fluctuations are less influenced in this porosity range. The in-line moment on the vertical cylinder is relatively less sensitive when the porosity is increased from 4.24 to 8.75%, but varies substantially when it is increased from 8.75 to 14.58%. The force and moment ratio (i.e. the ratio of the force or moment on the vertical cylinder, when it is encircled by the perforated caisson to the force or moment on the cylinder without any protection around it) reduces with increased wave height, H, and wave length, L, whereas the wave height ratio (ratio of the wave height at a point in the vicinity of the structure to the incident wave height) is less sensitive for the varying H and L. A new non-dimensional parameter, p^1.5 (D/L)/(H/d), is introduced to predict the in-line force and moment on the inner vertical cylinder, where d is local water depth, D is the diameter of the inner cylinder and p is the porosity of the outer caisson in percentage. Simple predictive equations for forces, moments and water surface fluctuations are provided.     

The mechanism of the development of wind-wave spectra

The mechanism of the development of wind-waves will be proposed on the basis of the observed wave spectra in the wind tunnels and at Lake Biwa (Imasato, 1976). It consists of two aspects: One is that the air flow over the wind-waves transfers momentum concentratively to the steepest component waves and the other is that the upper limit of the growth of a wave spectral density is given by the ultimate value in the slope spectral density. The first aspect means that the wave field has the momentum transfer filter on receiving the momentum from the air flow. Wind-waves in the stage of sea-waves receive the necessary amount of momentum by the form drag,e.g. according to the Miles' (1960) inviscid mechanism, through a very narrow frequency region around a dominant spectral peak. On the other hand, wind-waves in the stage of initial-wavelets receive it according to the Miles' (1962a) viscous model through a fairly broad frequency region around the peak. The upper limit ofS _max developing according to viscous mechanism is given byS _max =6.40×10^–4 k _max ^–2cm²s andS _max =2.03C(f _max )^–2cm²s(S _max is the power density of the wave spectral peak with the frequencyf _max ,k _max is the wave number corresponding to the frequencyf _max andC is the phase velocity).From the second aspect, the upper limit of the growth of wave spectral density is given by 33.3f ^–4cm²s in the frequency region of late stage of sea-waves. Therefore, the spectral peak, which has the largest value in the slope spectral density in the component waves of the wave spectrum, rises high over the line 4.15f ^–5cm²s. The energy is transported from the spectral peak to the high frequency part and to the forward face of a wave spectrum by nonlinear wave-wave interaction. This nonlinearity is confirmed by the bispectra calculated from the observed wind-wave data. In the stage of sea-waves, nonlinear rearrangement of the wave energy comes from a narrow momentum transfer filter, and, in the stage of initial-wavelets, it comes mainly from small corrugations and small steepness of the wave field.     

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.     

An experimental investigation of hydrodynamic coefficients for a vertical truncated rectangular cylinder due to regular and random waves

The research into hydrodynamic loading on ocean structures has concentrated mostly on circular cross-section members and relatively limited work has been carried out on wave loading on other cross-sections such as rectangular sections. These find applications in many offshore structures as columns and pontoons in semi-submersibles and tension-leg platforms. The present investigation demonstrates the behaviour of rectangular cylinders subject to wave loading and also supplies the hydrodynamic coefficients for the design of these sections.This paper presents the results of wave forces acting on a surface piercing truncated rectangular cylinder set vertically in a towing tank. The experiments are carried out in a water depth of 2.2 m with regular and random waves for low Keulegan–Carpenter number up to 6. The rectangular cylinder is of 2 m length, 0.2 m breadth and 0.4 m width with a submergence depth of 1.45 m from still water level. Based on Morison equation, the relationship between inertia and drag coefficients are evaluated and are presented as a function of KC number for various values of frequency parameter β, for two aspect ratios of cylinders, equals to 1/2 and 2/1. Drag and inertia coefficients obtained through regular wave tests are used for the random wave analysis to compute the in-line force spectrum.The results of the experiments show the drag and inertia coefficients are strongly affected by the variation in the aspect ratios of the cylinder. The drag coefficients decreases and inertia coefficients increases with increase in Keulegan–Carpenter number up to the range of KC number tested. The random wave results show a good correlation between measured and computed force spectrums. The transverse forces in both regular and random waves are found to be small compared to in-line forces.     

A note on the joint distribution of wave height and period during the growth phase of a storm

In this note the effect of changes in sea-state, as measured by the significant wave heigh H_s, on the joint distribution of individual wave height and period are considered. Wave data, obtained from a Waverider buoy during the growth phase of a storm, are used in the analysis. It is found that, by correctly scaling the individual heights and periods, the form of the joint distribution does not depend on H_s, but is dependent on the bandwidth of the spectrum. The results obtained also give some indication of the period of individual, high zero-upcrossing waves.     

Hydrodynamic behavior of a straight floating pipe under wave conditions

This paper examines the hydrodynamic behavior of a floating straight pipe under wave conditions. The main problem in calculating the forces acting on a small-sized floating structure is obtaining the correct force coefficients C_n and C_t, which differ from a submerged structure. For a floating straight pipe of small size, we simplify it into a 2D problem, where the pipe is set symmetrically under wave conditions. The force equations were deduced under wave conditions and a specific method proposed to resolve the wave forces acting on a straight floating pipe. Results of the numerical method were compared to those from model tests and the effects of C_n and C_t on numerical results studied. Suggestions for the selection of correct C_n and C_t values in calculating wave forces on a straight floating pipe are given. The results are valuable for research into the hydrodynamic behavior of the gravity cage system.