Bottom-mounted Circular Cylinder Research Articles

An Analytical Solution for the Interaction of Waves with Arrays of Circular Cylinders

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.

Numerical Simulation of Inline Forces on a Bottom-Mounted Circular Cylinder Under the Action of a Specific Freak Wave

Freak waves pose a great threat to the tension-leg platforms (TLPs) and monopile foundations of offshore wind turbines (OWTs), which necessitates comprehensive investigations on the characteristics of freak waves and the wave actions on those offshore renewable energy structures with circular cylinder. The recorded freak wave series “New Year Wave” (NYW) was numerically simulated using the Computational Fluid Dynamics methods. The compensation measure was adopted to effectively improve simulation accuracy. Under the action of the NYW, the inline forces and secondary load cycle (SLC) on a vertical-mounted cylinder, as the classic form for the TLPs and foundation of OWTs, were fully addressed. The simulation results were compared with the empirical formulations and experimental data to reveal the differences and the possible causes. The development of SLC was found to be closely related to the downstream vortex and return flow, which induces the reduction of the wall pressure and thus the inline force. The maximum inline forces vary with the cylinder position relative to the wave peak, and the simulation results reveal that the linear inline forces calculated by Morison formulation may be less than 65% of the total wave forces.

The secondary load cycle of a bottom-mounted circular cylinder at different Keulgan-Carpenter numbers and Froude numbers

The influences of the Keulgan-Carpenter number (KC) and Froude number (Fr) on the Secondary Load Cycle (SLC) on a bottom-mounted circular cylinder are comprehensively investigated using computational fluid dynamics simulations by OpenFOAM. The volume of fluid method is used to simulate the wave surface, and the shear stress transportation k-ω turbulence model is adopted for turbulence modelling. The sensitivity of the inline force to grid refinement is checked, and the numerical simulation accuracy is validated by the experimental results. The development of SLC is found to be closely related to the downstream vortex shedding, which induces the reduction of the wall pressure and thus the inline force. The studied ranges of KC and Fr are (11.01, 20.26) and (0.167, 0.534), respectively. It reveals that KC and Fr significantly affect the magnitude of SLC and the 3rd and 4th-order harmonic components dominate the SLC.

Diffraction of hydroelastic waves by multiple vertical circular cylinders

The diffraction problem of hydroelastic waves beneath an ice sheet by multiple bottom-mounted circular cylinders is considered. The elastic thin-plate theory is adopted to model the ice sheet, while the linearized velocity potential theory adopted for the fluid flow. The velocity potential corresponding to each cylinder is expanded into a series of eigenfunctions, and the total potential is expressed as a summation of these expansions over the entire NC number of cylinders. For each cylinder, the Green’s second identity is used outside its domain to obtain a set of linear equations. For each different cylinder, the domain used is different. NC cylinders give NC sets of coupled linear equations. Investigations are made for different arrangements of cylinders, piercing through ice sheets. Results for the wave forces on the cylinders with clamped and free conditions of the ice edge are obtained. Physical phenomena corresponding to cylinders arranged in square, in an array, in a double-array and in a staggered double array are discussed.

Forcing of a bottom-mounted circular cylinder by steep regular water waves at finite depth

AbstractForcing by steep regular water waves on a vertical circular cylinder at finite depth was investigated numerically by solving the two-phase incompressible Navier–Stokes equations. Consistently with potential flow theory, boundary layer effects were neglected at the sea bed and at the cylinder surface, but the strong nonlinear motion of the free surface was included. The numerical model was verified and validated by grid convergence and by comparison to relevant experimental measurements. First-order convergence towards an analytical solution was demonstrated and an excellent agreement with the experimental data was found. Time-domain computations of the normalized inline force history on the cylinder were analysed as a function of dimensionless wave height, water depth and wavelength. Here the dependence on depth was weak, while an increase in wavelength or wave height both lead to the formation of secondary load cycles. Special attention was paid to this secondary load cycle and the flow features that cause it. By visual observation and a simplified analytical model it was shown that the secondary load cycle was caused by the strong nonlinear motion of the free surface which drives a return flow at the back of the cylinder following the passage of the wave crest. The numerical computations were further analysed in the frequency domain. For a representative example, the secondary load cycle was found to be associated with frequencies above the fifth- and sixth-harmonic force component. For the third-harmonic force, a good agreement with the perturbation theories of Faltinsen, Newman & Vinje (J. Fluid Mech., vol. 289, 1995, pp. 179–198) and Malenica & Molin (J. Fluid Mech., vol. 302, 1995, pp. 203–229) was found. It was shown that the third-harmonic forces were estimated well by a Morison force formulation in deep water but start to deviate at decreasing depth.

Water wave diffraction by a bottom-mounted circular cylinder clamped to an elastic plate floating on a two-layer fluid

The wave diffraction by a bottom-mounted circular cylinder, which is clamped to the center of a floating circular thin elastic plate, in the two-layer fluid of finite depth is investigated for the time-harmonic incident waves of the surface and interfacial wave modes. Each fluid layer is inviscid, incompressible and of constant density. The flexural–gravity waves are composed of the propagating, decaying propagating and evanescent wave modes. Within the framework of the linear potential flow theory, a closed system of simultaneous linear equations is derived to solve the undetermined expansion coefficients with the methods of the angular eigenfunction expansion and the inner product. Explicit numerical computations are employed to test the convergence of the two series for the angular expansions and the evanescent wave modes. The horizontal forces and the moments exerted on the circular cylinder due to different wave modes are discussed in the case of the incident waves of either the surface or interfacial wave mode. It is obtained that the evanescent wave modes are appreciable parts for a high frequency.

Simulation of second-order wave interaction with fixed and floating structures in time domain

The application of a time-domain second-order method in the numerical simulation of the nonlinear wave interaction with surface piercing fixed and floating circular cylinders is described. In this approach, Taylor series expansions are applied to the boundary conditions on the instantaneous free water surface and body surface, and Stokes perturbation procedure is then used to establish corresponding boundary value problems at the first and second order of wave steepness with respect to a time-independent fluid domain. A boundary element method based on a B-spline function expansion is adopted to calculate the wave field at each time step, and the time stepping scheme is implemented to predict the boundary conditions at the next time step. The combined diffraction–radiation problem is solved when the wave interaction with a floating body is considered, unlike treating them separately in the conventional frequency domain method. Additionally, a mathematical transformation is derived to remove the second-order spatial derivative appearing in the body boundary condition that may lead to the potential loss of accuracy. As an illustration, numerical results of the wave diffraction around a bottom-mounted circular cylinder, wave radiation by a circular cylinder undergoing specified motions and wave interaction with a freely and moored floating circular cylinder are presented. Comparisons of the wave forces on the fixed and floating structures with the second-order frequency domain and fully nonlinear solutions indicate that the present numerical method is accurate, efficient and stable.

Hydroelastic wave diffraction by a vertical cylinder

A linear three-dimensional problem of hydroelastic wave diffraction by a bottom-mounted circular cylinder is analysed. The fluid is of finite depth and is covered by an ice sheet, which is clamped to the cylinder surface. The ice stretches from the cylinder to infinity in all lateral directions. The hydroelastic behaviour of the ice sheet is described by linear elastic plate theory, and the fluid flow by a potential flow model. The two-dimensional incident wave is regular and has small amplitude. An analytical solution of the coupled problem of hydroelasticity is found by using a Weber transform. We determine the ice deflection and the vertical and horizontal forces acting on the cylinder and analyse the strain in the ice sheet caused by the incident wave.

The radiation and diffraction of water waves by a bottom-mounted circular cylinder in a two-layer fluid

An analytical method was proposed to analyze the radiation and diffraction of water waves by a bottom-mounted circular cylinder in a two-layer fluid. Analytical expressions for added mass and damping coefficients, as well as the wave excitation forces of the circular cylinder were obtained by an eigenfunction expansion method. The hydrodynamic forces on the bottom-mounted circular cylinder in a two-layer fluid include not only the added mass and damping coefficients, but also the wave forces of the surface and internal-wave modes. This is different from the case of a homogenous fluid. Some examples were given, showing that density stratification can have a relative large effect on these hydrodynamic forces over a wide range of frequencies.

Wave propagation through arrays of unevenly spaced vertical piles

The interaction of regular waves with arrays of bottom-mounted circular cylinders is considered. This subject has been thoroughly investigated in the recent past, but most of the time under the assumption of regular and spatially periodic arrangements. Unlike these authors, we consider here arrays of unevenly spaced cylinders, displaced randomly from a regular array according to a disorder parameter. Focus is put on two effects of this spacing irregularity: reduction of peak forces associated to trapped mode phenomena, and regularization of the transmission coefficient for waves propagating through the arrays.

Water wave diffraction by a cylinder array. Part 2. Irregular waves

Diffraction of irregular waves, focused wave groups, and random seas by an array of vertical bottom-mounted circular cylinders is investigated using theoretical, computational and experimental methods. This is an extension of our study of such an array in regular waves, reported in Part 1. Linear focused wave group theory is reviewed as a method for predicting the probable shape of extreme events from random wave spectra. Measurements are presented of the free surface elevation distribution in the vicinity of a multi-column structure in an offshore basin when subjected to irregular waves having peak frequencies and significant wave heights in the range 0.449 < kpa < 0.555 and 0.114 < Hs < 0.124 respectively, where a is the cylinder radius. Analytical linear diffraction theory is extended for application to focused wave groups and random seas. Experimental irregular wave data are analysed for comparison with this theory. Linear diffraction theory for random seas is shown to give an excellent prediction of incident wave spectral diffraction, while linear diffraction theory for focused wave groups works well for linearized extreme events. Due to the phase shifting of incident wave spectral components, diffraction is shown to generate focused wave groups in the vicinity of the cylinder array.

Water wave diffraction by a cylinder array. Part 1. Regular waves

Diffraction of regular waves by arrays of vertical bottom-mounted circular cylinders is investigated using theoretical, computational, and experimental methods. Experiments in an offshore wave basin are designed to measure free surface elevation η at multiple locations in the vicinity of a multi-column structure subjected to regular waves of frequency 0.449 < ka < 0.524 and steepness 0.122 < kA < 0.261, where k is the wavenumber, a the cylinder radius and A the wave amplitude. Results from regular wave data analysis for first-order amplitudes are compared with those from analytical linear diffraction theory, which is shown to be accurate for predicting incident waves of low steepness. Second- and third-order terms are also estimated from the measured time series, and the effects near a second-order near-trapping frequency are compared to semi-analytical second-order diffraction theory. Linear diffraction theory is shown to be very accurate at predicting the global surface elevation features, even for waves of high steepness. However, violent events and significant nonlinear interactions, including breaking induced by wave scattering, have been observed. Furthermore, second-order near-trapping was observed to affect the magnitude of local free surface oscillations as well as scattered far-field radiation.

Evaluation of non-linear wave forces on a fixed body by the higher-order boundary element method

A numerical method is developed to evaluate non-linear waves diffracted by a fixed body and the resulting wave force acting on it. The entire boundary and all the variables are discretized by means of isoparametric biquadratic elements. To resolve the difficulty of describing the flow in the intersection region with regular elements, discontinuous elements are introduced. The generalized minimal residual (GMRES) algorithm is implemented to solve the reduced linear algebraic equations. Non-linearity of the free surface is integrated accurately in time by the fourth-order Runge-Kutta method with minimum truncation error. A new numerical radiation condition for non-linear diffraction is suggested, which resolves accurate non-linear diffraction around the body. Non-linear diffraction problem of a bottom-mounted circular cylinder is exemplified. The non-linear wave used by Rienecker and Fenton is chosen as the incident wave field. Based on the numerical results, it is believed that the present method is quite promising.

Second-order water wave diffraction by an array of vertical cylinders

The problem of second-order water wave diffraction of an incident monochromatic wave field by an array of bottom-mounted circular cylinders is solved by a semi-analytical approach. The solution for the second-order potential is obtained by combining eigenfunction expansions with an integral representation. Unlike the indirect approach for second-order forces (Lighthill 1979; Molin 1979), this approach gives complete information about local flow characteristics (pressure, velocities, wave elevation, etc.) thus providing a basis for solving the third-order problem. The results obtained are compared with other published data, and new detailed results, useful for benchmarking purposes, are given. Finally the influences of wave incidence, cylinder radius and cylinder configuration are considered. This leads to the suggestion that there exists a near-trapping phenomenon for the second-order wave in an array of cylinders, at half the wave frequency at which the corresponding linear near-trapped mode occurs.

Trapped modes about multiple cylinders in a channel

The trapped modes which can occur in a long narrow wave channel containing any number of different-sized bottom-mounted circular cylinders arbitrarily spaced along the centreline of the channel are considered. The modes, all of which are antisymmetric with respect to the centreplane of the channel are of two types: Neumann modes, in which the fluid has normal velocity zero on the channel walls corresponding to a localized sloshing near the cylinders, and Dirichlet modes, in which the dynamic pressure vanishes on the channel walls. These latter modes have no physical meaning in the water-wave context but have been observed in a related acoustic context where the same governing equations and boundary conditions apply.It is shown that in general there are [les ]N trapped modes for any configuration of N cylinders, the precise number depending critically on the geometry of the configuration. Both types are of importance in predicting the exciting forces on individual cylinders within a large but finite periodic arrangement of cylinders.

Wave diffraction by a long array of cylinders

Water wave diffraction by an array of bottom-mounted circular cylinders is analysed under the assumptions of linear theory. The cylinders are identical, and equally spaced along the array. When the number of cylinders is large, but finite, near-resonant modes occur between adjacent cylinders at critical wavenumbers, and cause unusually large loads on each element of the array. These modes are associated with the existence of homogeneous solutions for the diffraction by an array which extends to infinity in both directions. This phenomenon is related to the existence of trapped waves in a channel. A second trapped wave is established, corresponding to Dirichlet boundary conditions on the channel walls, as well as a sequence of higher wavenumbers where ‘nearly trapped’ modes exist.

Near-trapping of waves by circular arrays of vertical cylinders

The effect of incident waves on arrays of identical bottom-mounted circular cylinders arranged in a circle are considered. The present paper is motivated by the recent work of Maniar and Newman [Maniar, H.D. and Newman, J.N., Journal of Fluid Mechanics, 1997, 339, 309–330], who show how large forces can be generated on long linear arrays at certain frequencies corresponding to trapped modes present in an infinite periodic linear array. Their analysis for the linear array motivates the idea of ‘near trapping’ in circular arrays in which it is assumed that adjacent cylinders only differ by a change in phase characterised by an integer, p. By using the interaction theory of Linton and Evans [Linton, C.M. and Evans, D.V., Journal of Fluid Mechanics, 1990, 215, 549–569] it is shown how large peaks in the forces on circular arrays of four, five and six cylinders develop as the gap between the cylinders is reduced, and the peaks are shown to correspond to particular values of p by examining the complex roots of the near-trapping system. The largest forces appear to arise when the near-trapped mode corresponds to a standing wave motion, in agreement with the largest forces in linear arrays.

Wave Forces on Bottom-Mounted Large-Diameter Cylinder

A systematic set of experimental results for the wave force acting on a large horizontal bottom-mounted circular cylinder is presented and compared with a simple theory based on an unseparated flow model. The horizontal force is represented as an inertial component while the uplift force is represented by a lift force arising from the flow velocity plus a Froude-Krilov component. The test results show agreement with the theory.