Wave forces acting on a submerged horizontal circular cylinder in oblique waves and on a vertical cylinder in deep waves

This paper presents a method of estimating wave forces acting on a submerged horizontal circular cylinder fixed in oblique waves.The experiments show that drag and inertia coefficients in beam sea are available for calculating the wave forces in oblique waves.Wave forces exerted on a vertical circular cylinder in deep waves are also investigated.The experimental results show that wave forces acting on the vertical cylinder coincide approximately with hydrodynamic forces acting on a submerged circular cylinder in an oscillating fluid.     

Interaction of linear waves with an array of infinitely long horizontal circular cylinders in a two-layer fluid of infinite depth

The linear water wave scattering and radiation by an array of infinitely long horizontal circular cylinders in a two-layer fluid of infinite depth is investigated by use of the multipole expansion method. The diffracted and radiated potentials are expressed as a linear combination of infinite multipoles placed at the centre of each cylinder with unknown coefficients to be determined by the cylinder boundary conditions. Analytical expressions for wave forces, hydrodynamic coefficients, reflection and transmission coefficients and energies are derived. Comparisons are made between the present analytical results and those obtained by the boundary element method, and some examples are presented to illustrate the hydrodynamic behavior of multiple horizontal circular cylinders in a two-layer fluid. It is found that for two submerged circular cylinders the influence of the fluid density ratio on internal-mode wave forces is more appreciable than surface-mode wave forces, and the periodic oscillations of hydrodynamic results occur with the increase of the distance between two cylinders; for four submerged circular cylinders the influence of adding two cylinders on the wave forces of the former cylinders is small in low and high wave frequencies, but the influence is appreciable in intermediate wave frequencies.     

Wave forces acting on a vertical circular cylinder with a constant forward velocity

Wave forces acting on submerged circular cylinders moving forward with a constant velocity in regular waves are investigated experimentally. Hydrodynamic forces acting on the cylinder forced to surge in a steady are also measured and hydrodynamic coefficients were obtained. Wave force coefficients obtained from wave force measurements are compared with the hydrodynamic coefficients from surging tests, and the similarity and difference between them are discussed. Experiments show that these coefficients are quite different from those of the cylinder without a forward velocity.     

On diffraction and radiation problem for two cylinders in water of finite depth

The hydrodynamic problem arising form the interaction of linear water waves with a wave energy device consisting of two coaxial vertical cylinders of different radii is investigated. One cylinder is riding in waves, while another is submerged in fluid. By use of the method of separation of variables and the method of matched eigenfunction expansion, analytical expressions for the potentials are obtained. Using the expressions for the potentials, analytical expressions for the hydrodynamic coefficients and exciting forces/moments on the device are obtained. Numerical results of the hydrodynamic coefficients and exciting forces/moments are presented for some ratios of the radius of the submerged cylinder to that of the riding one. It is found that the radius of the submerged cylinder has a significant influence on the hydrodynamic coefficients and exciting forces/moments for relatively bigger radius of the submerged cylinder at low frequencies.     

Computation of the inertia-dominated forces on horizontal circular cylinders placed in oblique waves and near to a plane bed

Wave-force coefficients of horizontal circular cylinders inclined with respect to the incoming waves, are studied numerically under conditions when the effects of flow separation are insignificant. The mathematical model is set in terms of a boundary-value problem for the velocity potential of the wave, which is formulated under the assumption of the linear diffraction theory, and solved numerically by the boundary element method. The numerical calculations are performed in the vertical plane, assuming uniform water depths in the direction along the axis of the cylinder. A first-order correction to the pressures is introduced to take account of the asymmetry of the velocity field around the cylinder when it is close to the plane bed. The correction procedure is found to be highly effective in computing the transverse forces for small gap ratios. The numerical results show that irrespective of the values of the gap ratio, the in-line forces are always sensitive to the wave directionality. The transverse forces, however, show sensitivity only for the smaller gap ratios. It is also shown that by accounting for the wave directionality effects in the wave kinematics only, the forces could be estimated to a certain extent by using the hydrodynamic force coefficients of inertia and lift corresponding to the normal waves.     

Total horizontal and vertical forces of irregular waves on partially perforated caisson breakwaters

In this study, the total horizontal and vertical forces as well as the phase differences of irregular waves on a partially perforated caisson breakwater are investigated. The partially perforated caisson is located on a rubble fill foundation and filled with rock. Based on linear potential theory, a simple semi-analytical solution of the present problem is developed by means of the matched eigenfunction method and the finite element method. Experimental tests are also conducted to validate the theoretical model and examine the wave forces acting on the perforated caisson. The effects of some of the main factors on the total wave forces and the phase difference are examined. Theoretical and experimental studies show that when the total horizontal force reaches its peak, the simultaneous total vertical (upward) force is rather small or even becomes downwards. This is due to the existence of an obvious phase difference between the time histories of the total horizontal and vertical forces, which is an important advantage of perforated caissons over traditional structures.     

Measurements of higher harmonic wave forces on a vertical truncated circular cylinder

Wave forces on a vertical truncated circular cylinder in Stokes waves with the wave slopes ranging from 0.06 to 0.24, are measured in a wave tank. The higher harmonic wave forces are compared with the available values from theories of the FNV (Faltisen–Newman–Vinje) model and Varyani solution. The first harmonic horizontal forces measured are much larger than the theoretical values from the FNV model, while the first harmonic vertical forces are well predicted by the Varyani theory. It was also found that the FNV model significantly overpredicts the second harmonic horizontal forces in high frequency waves, but under predicts the third harmonic forces. The differences between the actual measurement and the theory, in the second and third harmonic horizontal forces, become smaller at low wave frequencies as the wave slope increases. In addition, the transverse instabilities in the incoming waves with high wave slope were observed, which is due to the nonlinear modulation. Measurements were, thus, carried out before the instability occurred.     

Calculating the Wave Force on Partially Immersed Large-Scale Horizontal Cylinders

Large-scale interceptors constitute the main structure of offshore self-driven floating marine litter collection devices,and the structural stability of such interceptors under the action of waves directly influences the overall safety of the device.When the ratio of the diameter of a horizontal cylinder in such interceptors to the incident wavelength is larger than 0.25,the wave force can be calculated by using the diffraction theory,by considering the problem as that of the interaction between the waves and a partially immersed large-scale horizontal cylinder.In this study,an analytical approach to calculate the wave force on a partially immersed large-scale horizontal cylinder was formulated by using the stepwise approximation method.Physical model tests were conducted to investigate the effects of different factors(wave height,period,and immersion depth)on the wave force on a large-scale horizontal cylinder under conditions involving short-period waves.The results show that both horizontal and vertical wave forces on the cylinder increase as the wave height(immersion depth)increases in most cases.The vertical wave force decreases with the decrease of the period.For the horizontal wave force,it increases with the decrease of the period when the wavelength is larger than the diameter of the cylinder and decreases with the decrease of the period when the wavelength is smaller than the diameter of the cylinder.     

Wave forces on vertical cylinders upon shoals

An experimental study has been carried out on the forces from plunging breaking regular and irregular waves on a vertical cylinder on a shoal. Total as well as local wave forces have been measured. Engineering formulae for the calculation of the horizontal forces and overturning moments have been derived. The duration of the impact forces have been measured and compares fairly well with theoretical values.     

Breaking wave forces and velocity fields

A new technique for measuring velocities under breaking wave crests using laser-doppler anemometry has been developed. Results may be obtained at positions up to 4 millimeters from the crest.A wave field is presented for a vertically fronted wave and comparison is made with an appropriate numerical study. The velocities obtained are far in excess of the predictions of linear theory.Measurements have also been made of the forces produced by this wave on a horizontal cylinder. Comparisons with large regular waves indicate that forces due to breaking waves can be up to five times greater for similar wave heights.     

Wave interaction with a wave absorbing double curtain-wall breakwater

This study examines the hydrodynamic performance of a wave absorbing double curtain-wall breakwater. The breakwater consists of a seaward perforated wall and a shoreward impermeable wall. Both walls extend from above the seawater to some distance above the seabed. Then the below gap allows the seawater exchange, the sediment transport and the fish passage. By means of the eigenfunction expansion method and a least square approach, a linear analytical solution is developed for the interaction of water waves with the breakwater. Then the reflection coefficient, the transmission coefficient and the wave forces acting on the walls are calculated. The numerical results obtained for limiting cases agree very well with previous predictions for a single partially immersed impermeable wall, the double partially immersed impermeable walls and the bottom-standing Jarlan-type breakwater. The predicted reflection coefficients for the present breakwater also agree reasonable with previous experimental results. Numerical results show that with appropriate structure parameters, the reflection and transmission coefficients of the breakwater may be both below 0.5 at a wide range of the relative water depth. At the same time, the magnitude of wave force acting on each wall is small. This is significant for practical engineering.     

Validation of a time-domain strip method to calculate the motions and loads on a fast monohull

The paper presents a comparison between experimental data and numerical results of the hydrodynamic coefficients and also of the wave induced motions and loads on a fast monohull model. The model with 4.52 m length was constructed in Fibre Reinforced Plastic (FRP), and made up of 4 segments connected by a backbone in order to measure sectional loads. The objective of the investigation was to assess the capability of a nonlinear time domain strip method to represent the nonlinear and also the forward speed effects on a displacement high speed vessel advancing in large amplitude waves. With this objective in mind the experimental program included forced oscillation tests in heaving and pitching, for a range of periods, three different amplitudes and several speeds of advance. In head regular waves comprehensive ranges of wave periods, wave steepness and speeds, were tested in order to measure heave, pitch and loads in three cross sections.

The numerical method assumes that the radiation and diffraction hydrodynamic forces are linear and the nonlinear contributions arise from the hydrostatics and Froude–Krilov forces and the effects of green water on deck. The assumption of linearity of the radiation forces is validated by comparing calculated hydrodynamic coefficients with experimental data for three different amplitudes of the forced oscillations. Both global coefficients and sectional coefficients are compared. The motions and loads in waves are compared in terms of first and higher harmonic amplitudes and also in terms of sagging and hogging peaks.     

Hydrodynamic forces on a partially submerged cylinder at high Reynolds number in a steady flow

The hydrodynamic forces on the stationary partially submerged cylinder are investigated through towing test with Reynolds number ranging from 5 × 10⁴ to 9 × 10⁵. Three test groups of partially submerged cylinders with submerged depths of 0.25 D, 0.50 D, and 0.75 D and one validation group of fully submerged cylinders are conducted. During the experiments, the hydrodynamic forces on the cylinders are measured using force sensors. The test results show a considerable difference in the hydrodynamic coefficients for the partially submerged cylinders versus the fully submerged cylinders. A significant mean downward lift force is first observed for the partially submerged cylinders in a steady flow. The maximum of the mean lift coefficients can reach 1.5. Two distinct features are observed due to the effects of overtopping: random distributions in the mean drag coefficients and a clear quadratic relationship between the mean lift coefficients and the Froude number appear in the non-overtopping region. However, the novel phenomenon of a good linear relationship with the Froude number for the mean hydrodynamic coefficients is clearly shown in the overtopping region. In addition, fluctuating hydrodynamic coefficients are further proposed and investigated. These results are helpful to have a better understanding of the problem and to improve related structural designs.     

Diffraction of oblique waves by thick rectangular barriers

A complete analytical solution is presented for the linear diffraction of oblique waves by horizontal rectangular cylinders either fixed at the free surface or mounted on the sea bed in a finite-depth of water. Helmholtz equation is employed as the governing differential equation obtained by reducing the 3-D oblique wave scattering problem to a 2-D case. According to the method proposed, the fluid region is divided into three sub-regions in which the governing differential equation is solved by the separation of variables. The solutions for each region are then matched on the common boundaries of sub-regions to determine the unknowns of the eigen series expansions and Fourier series. Thus transmitted and reflected waves are obtained in the far-field, and forces and moments acting on the rectangular cylinder fixed at the free surface are also given. Comparisons are made in order to check the accuracy of the method.     

Nonlinear wave interaction with a vertical circular cylinder: wave forces

Second-order wave forces on a large diameter vertical circular cylinder, computed according to a semi-analytic nonlinear diffraction theory, are compared to results of 22 laboratory experiments with regular waves. In general, predicted forces agree quite well with measured forces. In most tests, both measured and predicted maximum forces exceeded linear theory by 5 to 15%. In a few cases, however, the measured forces were less than those predicted by linear theory, in contrast to the second-order predictions. It is shown that these results are related to the phasing of various linear and nonlinear wave force components, and are consistent with those obtained by other investigators.     

Experimental study of reflection coefficient and wave forces acting on perforated caisson

The reflection coefficient and the total horizontal forces of regular waves acting on theperforated caisson are experimentally investigated. The empirical relationship between reflection coefficient and the ratio of the total horizontal forces acting on the perforated caisson to those on solid vertical walls with the relative chamber width, relative water depth and porosity of perforated wall, etc. are given. Moreover, the results of the ratio of the total horizontal forces are also compared with formulas given by Chinese Harbour Design Criteria and Takahashi, which may be useful for the practical engineering application.     

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.     

Wave forces on a large, horizontal submerged cylinder

A numerical boundary integral equation method combined with a non-linear time stepping procedure is used for the calculation of wave forces on a large, submerged, horizontal circular cylinder. As the method is based on potential theory, all computations are performed in the inertia dominated domain, that is, for small Keulegan-Carpenter numbers. Computations are carried out for the Eulerian mean current under wave trough level equal to zero. When the cylinder is moved towards the sea bed the computations show that the inertia coefficients increase significantly, which is associated with a blockage effect. Furthermore, the effect of the wave steepness is reduced when the submergence of the cylinder is increased. In the vicinity of the free water surface the vertical inertia coefficient is highly dependent upon the wave steepness, which tends to reduce it, whereas the horizontal inertia coefficient is only slightly dependent on the wave steepness. Computations are also carried out for cylinder diameters comparable with the wave length. Finally, inertia coefficients computed by the present method are compared with some analytical results by Ogilvie [(1963), First and second order forces on a cylinder submerged under a free surface. J. Fluid Mech. 16, 451–472]. As long as the assumptions leading to Ogilvie's theory are fulfilled (cylinder radius small compared to the wave length), the results are quite similar.     

Wave motion over a breakwater system of a horizontal plate and a vertical porous wall

A two-dimensional analytical solution is presented to study the reflection and transmission of linear water waves propagating past a submerged horizontal plate and through a vertical porous wall. The velocity potential in each fluid domain is formulated using three sets of orthogonal eigenfunctions and the unknown coefficients are determined from the matching conditions. Wave elevations and hydrodynamic forces acting on the porous wall are computed. Reflection and transmission coefficients are presented to examine the performance of the breakwater system. The present analytical solutions are found in fairly good agreement with the available laboratory data. The results indicate that the plate length, the porous-effect, the gap between plate and porous wall, and the submerged depth of the plate all show a significant influence on the reflected and transmitted wave fields. It is also interesting to note that the submerged plate plays an important role in reducing the transmitted wave height, especially for long incident waves.     

Wave reflection from partially perforated-wall caisson breakwater

In 1995, Suh and Park developed a numerical model that computes the reflection of regular waves from a fully perforated-wall caisson breakwater. This paper describes how to apply this model to a partially perforated-wall caisson and irregular waves. To examine the performance of the model, existing experimental data are used for regular waves, while a laboratory experiment is conducted in this study for irregular waves. The numerical model based on a linear wave theory tends to over-predict the reflection coefficient of regular waves as the wave nonlinearity increases, but such an over-prediction is not observed in the case of irregular waves. For both regular and irregular waves, the numerical model slightly over- and under-predicts the reflection coefficients at larger and smaller values, respectively, because the model neglects the evanescent waves near the breakwater.