Numerical simulation of wave interaction with vertical circular cylinders of different submergences using immersed boundary method

Dynamic effects on a large vertical circular cylinder due to waves were measured experimentally in a wave tank in the form of pressures and forces. Because of the large diameter of the cylinder compared to the wave lengths, appreciable diffraction of the waves from the surface of the cylinder was expected. Expressions of pressures, forces and moments were developed following the classical analysis of Havelock on a fixed obstacle in waves which includes the wave diffraction effects. The viscous effects were neglected in the analysis on the assumption that for a cylinder of this size the inertia force is most predominant. Linear wave theory was applied in the study of the interaction of waves with a bottom-mounted object. The prediction results from this theory compared favorably with the experimentally measured data. INTRODUCTION In recent years, offshore oil fields are being discovered in remote locations far away from the coastal region. In areas where the field is distant from the shore, offshore storage is economically viable. This is evident by the installation of large offshore oil storage tanks in many parts of the world. Chicago Bridge and Iron successfully completed its third 500,000 barrel undersea storage tank for Dubai Petroleum Company off the coast of Dubai. In relatively shallow water (50 to 180 ft.) and sheltered environments (waves up to 40 ft. high), a cylindrical storage tank can be used with additional economic benefits. The tank is held in place by the weight of the tank and the product. This downward load prevents sliding and overturning of the structure from the wave action. In the past, vertical cylindrical piles have been used as members of marine structures more frequently than any other basic geometry. For example, an extensive application of this member is found in the construction of drilling platforms. A vertical cylinder whose diameter is small compared to the wave length (d/L < 0.1) may be treated by Morison's approach (8). The equation for the horizontal wave force proposed by Morison, et al (8) is divided into two parts inertial and drag linearly added together. The inertial part is based on the concept of added mass and is associated with a virtual mass coefficient. The drag term is nonlinear being proportional to the square of the water particle velocity. It is considered due to the viscous effect and has a drag coefficient as a multiplier. When the scattering of waves from the surface of the cylinder becomes significant, Morison's equation fails to predict the forces accurately due to the unknown properties of the virtual mass and drag coefficients. Havelock (4) studied the diffraction effects of two-dimensional waves in the presence of a fixed obstacle.

Wave force analysis of the two vertical cylinders by boundary element method

The diffraction of waves by three bottom fixed vertical circular cylinders is investigated by using the boundary element method. This method has been successfully applied to the isolated vertical circular cylinder and now is used to study the interaction between waves and multiple vertical cylinders. In this paper, a numerical analysis by the boundary element method is developed by the linear potential theory. The numerical analysis by the boundary element method is based on Green's second theorem and introduced to an integral equation for the fluid velocity potential around the vertical circular cylinders. To verify this method, the results obtained in present study are compared with the results computed by the multiple scattering method. The results of the comparisons show strong agreement. Also in this paper, several numerical examples are given to illustrate the effects of various parameters on the wave exciting force such are the separation distance, the wave number and the incident wave angle. This numerical computation method might be used broadly for the design of various offshore structures to be constructed in the future.

A complete semi-analytical solution is given for second-order diffraction of monochromatic waves by a truncated vertical circular cylinder in water of uniform finite depth. The methodology presented in detail elsewhere (Eatock Taylor & Huang 1996) is adopted to find a particular solution which exactly satisfies the governing equation, the inhomogeneous free-surface condition and the seabed condition. In order to satisfy the boundary condition on the cylinder bottom, the fluid domain around the cylinder is divided into two regions. First- and second-order velocity potentials are described separately in the two regions and matched on the interface by the pressure and normal-velocity continuity conditions. Based on the formulation, the second-order wave field in the vicinity of the cylinder and the corresponding wave forces and overturning moments on the cylinder are studied in detail. Numerical results for the double frequency forces obtained by using the present semi-analytical approach are compared with those computed with a higher-order boundary element method (Eatock Taylor & Chau 1992). As well as the exact solution, an approximate solution is also given for the second-order potential and the corresponding forces. Numerical results show that the approximate solution possesses excellent accuracy for the total second-order heave force over a wide range of conditions. When kb > 1.2 (where k, b are the incident wavenumber and the draught of the cylinder respectively), the accuracy for total second-order surge force and pitch moment is also satisfactory. These results could lead to the development of very efficient solutions and corresponding algorithms for the analysis of second-order wave diffraction by more complicated structures such as tension leg platforms. Numerical results based on the present solution show that in many cases, both the first- and the second-order-free surface elevation in the vicinity of a truncated cylinder is very close to that of a bottom-seated cylinder. For waves with larger amplitudes, the maximum free-surface elevation around a vertical cylinder predicted with the second-order theory can significantly exceed that given by linear theory. There is also a considerable difference in the location of the maximum elevation predicted by the linear and nonlinear theories.

This paper deals with the hydrodynamic forces exerted on a rigid object describing harmonic oscillations under or on a free surface as well as the forces resulting from the interaction of the object held fixed in a train of regular surface waves. The problem is formulated for a body of arbitrary shape in water of finite depth and the development of a numerical scheme for carrying out the calculations is described. An energy balance as well as Haskind's relations are used as a check on the accuracy of the numerical results. Numerical results are presented for a floating sphere, a vertical circular cylinder, and a practical semi-immersed caisson configuration. Nomenclature a = characteristic dimension of body Ct = wave force (or moment) coefficient, Eqs. (36) and (40) Cij - ith component force (or moment) coefficient associated with the ;th component motion d_ = depth of submergence dS - differential surface area / = source strength function G = Green's function g - acceleration of Gravity gi = (i = 1,2,...7) See Eq. (19) h_ - water depth H = wave height i,l,k = cartesian unit vectors L = wavelength Mtj — added mass coef., (added mass /pa3) or (added mom. of I/pa5) Ntj = damping coef., (linear damping/po-a3) or (angular damp

ABSTRACTA hybrid-grid 3D non-hydrostatic model is developed and applied to predict nonlinear regular waves interacting with vertical circular cylinders. The model solves unsteady momentum and mass conservation equations by means of an explicit projection method. The 3D grid system is built from a 2D horizontal grid by adding several horizontal layers. The Perot’s scheme is used to treat horizontal advection terms in momentum equations. The 2D horizontal domain is discretized using a hybrid-grid system comprising rectangular and triangular cells to improve the model boundary-flexibility. The numerical model is validated against the experimental data of Stokes wave and cnoidal wave interacting with a vertical circular cylinder, respectively. Good agreement between model results and the experimental data are achieved, demonstrating the capability of the model on resolving interactions between nonlinear regular waves and vertical circular cylinders.

Based on the methods of variable separation and matching eigenfunction expansions for velocity potential, analytical solutions are developed for a water wave radiation problem by a submerged vertical cylinder in finite water depth. They are validated by comparison with results from the higher order boundary element method and convergent examinations on the number of expansion models. Numerical analysis is carried out to investigate the influence submerged depth, cylinder length, and water depth on added mass and radiation damping. When the submerged depth is large, the added mass approaches a stable value, and radiation damping tends to zero. At high frequency range, the heave and pitch of the added mass of the submerged cylinder is about twice that of the floating cylinder. The influence of cylinder length for hydrodynamic coefficients is quite complex and shows the various properties at different frequency ranges. Added mass can be increased with the decrease of water depth, whereas the effect o...

Large-scale random-wave experiments (H a = 1.3 m, f p = 0.17 Hz, d = 5.0 m) on a vertical circular cylinder (D = 0.5 m), having a roughness of about 60 µm, were performed to study the wave loading in the free surface zone at high Reynolds numbers (up to 7·105). Some first results of the data analysis are presented, showing characteristic time histories of velocities, pressures, and in-line forces. Also the energy density spectra and exceedance curves for both the maximum and the minimum inline-force are compared at two levels, z = 0 m and z = −2.3 m..

Numerical simulation of three-dimensional breaking waves and its interaction with a vertical circular cylinder

When bottom mounted structures of large displacement are immersed in the sea, earthquake induced hydrodynamic loads may become important design factors. Specifically, as the earth oscillates, a bottom mounted structure is forced to describe time dependent motion in an otherwise still fluid. As a result, hydrodynamic loads in addition to the inertial loads of the structure itself are induced. In this paper, a theoretical approach to the calculation of these hydrodynamic loads is outlined and numerical results are presented for several submerged configurations. Practical geometries considered include a submerged oil storage tank configuration and a conical configuration as has been proposed for offshore drilling rig designs for deployment in the Arctic. Also, computations were carried out for a sphere and vertical circular cylinder and various comparisons with classical results are made. Numerical results for these submerged structures are presented in the form of a dimensionless hydrodynamic load parameter or added mass coefficient. Results corresponding to a number of different water depths are presented to show the rather sizable effect of the relative water depth on the hydrodynamic force. It is shown that for typical earthquake frequencies, the effect of the free water surface is to reduce the hydrodynamic loads in comparison to the corresponding infinite depth values. Experimental results obtained by vibration testing are presented for a submerged sphere and a vertical circular cylinder. These results show excellent agreement with the theoretical results. Introduction Principal attention in earthquake engineering has been given to the generation of tsunami waves, the shoreline run-up and damage caused by these waves, as well as the damage to dry land structures caused by strong ground motion. With the increased deployment of large submerged structures, an additional facet of this important problem has come to light. Namely, if a large bottom mounted structure submerged in the ocean is excited by oscillatory ground motion, hydrodynamic loads in addition to its own inertia forces come into play. As the structure is caused to move through the water, hydrodynamic forces which are dependent upon the size and shape of the structure as well as the water depth and frequency of oscillation arise. On account of the large density of water, these forces are often quite large and have considerable influence on the structural design. When an object is accelerated through a fluid there are, in general, two types of forces that are recognized, one being a drag component and a second, the inertial component. The nature of the flow produced by the time dependent motion of the immersed rigid object and the relative contribution of these two components of force is generally considered to be strongly dependent on the amplitude of the relative fluid motion in comparison to the characteristic lineal dimension of the object. For example, Keulegan and Carpenter [1] found that, for harmonic motion of the fluid past a fixed circular cylinder, major flow separation did not occur and the forces were well represented by potential flow values provided the amplitude of the motion was less than about a half diameter.

A pressure impulse model is presented for wave impact on vertical circular cylinders. Pressure impulse is the time integral of the pressure during an impact of short time scale. The model is derived for a simplistic geometry and has relative impact height, crest length and cylinder radius as effective variables. The last parameter, the maximum angle of impact, is free and can be calibrated to yield the right force impulse. A progression of simpler pressure impulse models are derived in terms of a three-dimensional box generalization of the two-dimensional wall model and an axisymmetric model for vertical cylinders. The dependence on the model parameters is investigated in the simpler models and linked to the behaviour of the three-dimensional cylinder model. The model is next validated against numerical results for a wave impact for a phase- and direction-focused wave group. The maximum impact angle is determined by calibration against the force impulse. A good match of the pressure impulse fields is found. Further comparison to the force impulse of two common models in marine engineering reveals improved consistency for the present model. The model is found to provide a promising representation of the pressure impulse field, based on a limited number of input parameters. Its further validation and potential as a robust tool in force and response prediction is discussed.

Surface gravity wave interaction with a submerged horizontal flexible porous plate

This paper presents an analytical method to investigate the multiple scattering problem within arrays of vertical bottom-mounted circular cylinders subjected to linear incident waves. Based on the Laplace equation and boundary conditions on the seabed and surface, a formulation of a two-dimensional multiple scattering problem is first obtained by using the variable separation method. Furthermore, the analytical solution of the wave forces on multiple circular cylinders is derived, which consists of the incident wave force due to the linear incident wave and the scattered wave forces considering multiple scattering waves. The presented analytical solution is validated by comparing its results with a numerical method, and the result shows that the analytical solution is in good agreement with the numerical one. Finally, the multiple scattering analysis is conducted on arrays of cylinders with different incident wave numbers, distances between cylinders, and quantities.

On the distribution of second order pressure on a vertical circular cylinder

Purpose. Reference data in technical literature about such characteristics as stiffness and elasticity of rail threads at torsion are rather contradictory. This is due to the absence of correct analytical solution of this task and because of complexity of experimental studies of this problem under combined action of vertical and horizontal lateral forces. The article presents the new method of solving the task of calculation the real values of stiffness and elasticity modules of rail thread when torsion under combined action of vertical and horizontal lateral forces. Methodology. The complex method of solving the problem was used in the paper. It includes the theoretical part and the results of the experiment. Using the experiment the characteristics of horizontal lateral stiffness of the rail threads of the head and bottom at different constructions of the modern rail fastenings was measured. The second task of determination of the real values of elasticity and stiffness modulus of the rail thread when torsion was solved by the theoretical methods. Findings. New values of stiffness characteristics and elasticity module of rail thread when torsion for the modern rail constrictions Р65, UIC60, Р50 and the rail fastenings of the types КПП, КБ, ДО, under combined action of vertical and horizontal forces and torsion were obtained. Originality. For the first time in domestic research it was carried out the experimental and theoretical solution of the task for determination of the real values of the stiffness and elasticity modulus of rail thread when torsion under combined action of vertical and horizontal forces of the wheel loadings. The solution is represented in the functional dependence on the correlation of vertical and horizontal forces ; acting together. The solution is carried out for the modern designs of the rails of domestic standard P65, P50, P43 and international standard UIC60. Practical value. The obtained results make it possible to carry out calculations of the real acting strains and deformations in the rails of different types under the joint action of vertical and horizontal forces for railway track on concrete and wooden sleepers with different designs of rigid and elastic rail fastenings КБ, КПП, Д0 with higher accuracy as compared to the current calculation methodology.