Surface-piercing Circular Cylinder Research Articles

On the scattering of focused wave by a finite surface-piercing circular cylinder: A numerical investigation

For nonlinear wave–structure interactions, the high-frequency scattered waves can be identified within the drag-inertia regime, especially in steep incident waves where viscous effects are not negligible. According to previous studies, this unexpected phenomenon is highly associated with the local flow field, posing challenges to the existing harmonic-based diffraction solutions (mostly up to second-order). To overcome these shortcomings in potential flows, we establish a high-fidelity numerical wave tank to solve this two-phase free surface flow in the open source computational fluid dynamics framework OpenFOAM. We implement the ghost fluid method to eliminate the spurious velocities, mostly reported in two-phase volume of fluid solvers, in the vicinity of the free surface and preserve a sharp air–water interface. A modified generating–absorbing boundary condition is employed to achieve high computational efficiency without passive relaxation zones. Good agreement with experimental data demonstrates the reliability and accuracy of the present numerical wave tank in extreme wave conditions. On this basis, this paper numerically investigates the wave scattering of the focused wave by a finite surface-piercing circular cylinder, with emphasis on the flow mechanism. Three types of high-frequency scattered waves are identified in the near field, namely, Type-1, Type-2, and Type-1* waves. The typical mechanisms of each type are analyzed in depth with detailed flow field data, which confirms and complements the observations from previous experiments. More importantly, the primary vortical structures involved in scattering are extracted by the Liutex vortex identification method. The behaviors of these vortical structures could characterize the evolution of the high-frequency scattered waves and provide new insights into this strongly nonlinear phenomenon. An overall schematic of the wave scattering evolution in this complex condition is summarized for a straightforward understanding.

자유 수면이 벽 근처 원형 실린더 주위 유동에 미치는 영향

The flow past a surface-piercing circular cylinder near a moving wall at Re=100 and Fr=0.6 has been investigated using direct numerical simulation. The gap distance between the cylinder and the wall is G/D=∞, 1.5, and 0.6, where G/D=∞represents the isolated cylinder without the wall. The periodic shedding of organized vortices from the cylinder is observed in the deep flow which is far from the free surface. However, the vortex shedding is attenuated beneath the free surface, showing the flow past the cylinder is almost steady. Suppression of vortical development causes the decreases in the temporal fluctuations of the drag and lift coefficients beneath the free surface. The decreases in the coefficient fluctuations for G/D=0.6 are more dominant than those for G/D=1.5. The transition of hydrodynamic forces beneath free surface depends on the gap distance between the cylinder and the moving wall.

Hydrodynamic scaling and wave force estimation of offshore structures

Offshore structures are generally classified as small-scale structures or large-scale structures. Their wave forces are then estimated by Morison equation and diffraction/radiation theories, respectively. However, the classification criterion is not well quantified. In the present paper, a numerical wave flume is established to simulate the wave forces acting on a fixed and vertical surface-piercing circular cylinder under linear waves. By solving Navier–Stokes equation and Euler equation with free surface involved, respectively, the viscous force and inertia force are separated accurately. The variation of viscous force and inertia force with the cylinder diameter to wave length ratio is discussed in detail. The scale intervals for significant viscous and diffraction effects are given. The error caused by neglecting viscous and diffraction forces is quantitatively analyzed. Based on these analysis, the concept of medium-scale structure is proposed and the classification criteria for small-, medium- and large-scale structures are given. In the meantime, the estimation methods of wave forces for different scales of structures are suggested. A numerical wave flume is established to simulate the wave forces on a fixed and vertical surface-piercing circular cylinder under linear waves. By solving Navier–Stokes and Euler equations, respectively, the viscous and inertia forces are separated accurately. The dependence of these two force components on the cylinder diameter to wave length ratio is quantified. A new concept of medium-scale structure is proposed and the novel classification criteria for small-, medium- and large-scale structures are given. Correspondingly, the estimation methods of wave forces for different scales of structures are suggested

Numerical Study on Wave Run-up of a Circular Cylinder with Various Diffraction Parameters and Body Drafts

Wave run-up is an important phenomenon that should be considered in ocean structure design. In this study, the wave run-up of a surface-piercing circular cylinder was calculated in the time domain using the three-dimensional linear and fully nonlinear numerical wave tank (NWT) techniques. The NWT was based on the boundary element method and the mixed Eulerian and Lagrangian method. Stokes second-order waves were applied to evaluate the effect of the nonlinear waves on wave run-up, and an artificial damping zone was adopted to reduce the amount of reflected and re-reflected waves from the sidewall of the NWT. Parametric studies were conducted to determine the effect of wavelength, wave steepness, and the draft of the cylinder on the wave run-up of the cylinder. The maximum wave run-up value occurred at 0°, which was in front of the cylinder, and the minimum value occurred near the circumferential angle of 135°. As the diffraction parameter increased, the wave run-up increased up to 1.7 times the wave height. Furthermore, the wave run-up was 4% higher than the linear wave when the wave steepness was 1/35. In particular, the crest height of the wave run-up increased by 8%.

Backscattering reduction for resonating obstacle in water-wave channel

We consider the propagation of water waves in a waveguide with a surface-piercing circular cylinder. A plane wave interacting with the cylinder leads to a Fano resonance resulting in strong scattering with a large reflection coefficient. Using a smoothly varying bathymetry whose shape is optimized, we show both numerically and experimentally that broadband and robust backscattering reduction can be obtained below the first cutoff frequency.

Benchmarking of a Computational Fluid Dynamics-Based Numerical Wave Tank for Studying Wave Load Effects on Fixed and Floating Offshore Structures

A computational fluid dynamics (CFD) based numerical wave tank (NWT) is developed and verified to study wave load effects on fixed and free floating offshore structures. The model is based on solving Navier–Stokes equations on a structured grid, level set method for tracking the free surface, and an immersed boundary method for studying wave–structure interaction. This paper deals with establishing and verifying a CFD-based NWT. Various concerns that arise during this establishment are discussed, namely effects of wave reflection which might affect the structure response, damping of waves in downstream, and three-dimensional (3D) effects of the waves. A method is described and verified to predict the time when incoming waves from wave generator are affected by reflecting waves from the structure which can help in better designing the dimensions of NWT. The model is then used to study sway, heave, and roll responses of a floating barge which is nonuniform in density and limited in sway direction by a spring and damper. Also, it is used to study wave loads on a fixed, large diameter, surface piercing circular cylinder. The numerical results are compared with the experimental and other numerical results, and in general very good agreement is observed in all range of studied wave frequencies. It is shown that for the studied fixed cylinder, the Morison equation leads to promising results for wavelength to diameter ratio larger than 2π (kD < 1), while for shorter wavelengths results in considerable over prediction of wave loads, due to simplification of wave diffraction effects.

Oscillations of elastically mounted cylinders in regular waves

Under the assumption of potential flow and linear wave theory, a semi-analytic method based on eigenfunciton expansion is proposed to predict the hydrodynamic forces on an array of three bottom-mounted, surface-piercing circular cylinders. The responses of the cylinders induced by wave excitation are determined by the equations of motion coupled with the solutions of the wave radiation and diffraction problems. Experiments for three-cylinder cases are then designed and performed in a wave flume to determine the accuracy of this method for regular waves.

An efficient domain decomposition strategy for wave loads on surface piercing circular cylinders

A fully nonlinear domain decomposed solver is proposed for efficient computations of wave loads on surface piercing structures in the time domain. A fully nonlinear potential flow solver was combined with a fully nonlinear Navier–Stokes/VOF solver via generalized coupling zones of arbitrary shape. Sensitivity tests of the extent of the inner Navier–Stokes/VOF domain were carried out. Numerical computations of wave loads on surface piercing circular cylinders at intermediate water depths are presented. Four different test cases of increasing complexity were considered; 1) weakly nonlinear regular waves on a sloping bed, 2) phase-focused irregular waves on a flat bed, 3) irregular waves on a sloping bed and 4) multidirectional irregular waves on a sloping bed. For all cases, the free surface elevation and the inline force were successfully compared against experimental measurements.

Experimental study on water-wave trapped modes

We present an experimental study on the trapped modes occurring around a vertical surface-piercing circular cylinder of radius a placed symmetrically between the parallel walls of a long but finite water waveguide of width 2d. A wavemaker placed near the entrance of the waveguide is used to force an asymmetric perturbation into the guide, and the free-surface deformation field is measured using a global single-shot optical profilometric technique. In this configuration, several values of the aspect ratio a/d were explored for a range of driving frequencies below the waveguide's cutoff. Decomposition of the obtained fields in harmonics of the driving frequency allowed for the isolation of the linear contribution, which was subsequently separated according to the symmetries of the problem. For each of the aspect ratios considered, the spatial structure of the trapped mode was obtained and compared to the theoretical predictions given by a multipole expansion method. The waveguide–obstacle system was further characterized in terms of reflection and transmission coefficients, which led to the construction of resonance curves showing the presence of one or two trapped modes (depending on the value of a/d), a result that is consistent with the theoretical predictions available in the literature. The frequency dependency of the trapped modes with the geometrical parameter a/d was determined from these curves and successfully compared to the theoretical predictions available within the frame of linear wave theory.

Strongly nonlinear internal soliton load on a small vertical circular cylinder in two-layer fluids

The solutions of MCC theory are used to investigate larger-amplitude strongly nonlinear internal soliton load on a small surface-piercing circular cylinder in two-layer fluids. By comparing the wave profiles and instantaneous horizontal velocities calculated by MCC theory with those of KdV theory and experimental data, we verify the validity of MCC theory for larger-amplitude strongly nonlinear internal soliton. The accelerations are computed, and then force and torque on a small cylinder are estimated based on Morison’s formula for both MCC and KdV theories. Computed results show that the internal soliton force and torque become more and more large and wide with the increase of amplitude for MCC theory. The location of torque crest calculated by MCC theory departs from origin (moving to the right) as the amplitude grows and whenever the inertial term is included or not, the wave forces computed based on the two theories both have small discrepancies for the same amplitude, but when the inertial term is included, the torque obtained by MCC theory will be much larger and the torque obtained by KdV still have a small discrepancy. The reasons are presented in detail. The internal wave force will be underestimated if the traditional KdV theory is used. Therefore, ocean engineers should consider the large-amplitude strongly nonlinear internal soliton load on marine construct carefully.

Second-order sum-frequency wave diffraction by a truncated surface-piercing cylinder in bichromatic waves

This work considers the second-order sum-frequency diffraction problem for a stationary truncated surface-piercing circular cylinder in bichromatic waves. The solution method was based on a semianalytical formulation of the second-order sum-frequency diffraction potential. The boundary conditions were properly satisfied by introducing the “locked” and the “free” wave components of the nonlinear velocity potential. The method was validated by comparing the calculated results with numerical data previously reported by other authors. Particular attention was paid to the second-order sum-frequency heave forces and the change in the wave run-up configuration due to the existence of the lower fluid domain underneath the truncated cylinder.

Pseudospectral element model for free surface viscous flows

PurposeThe research is directed at development of an efficient and accurate technique for modelling incompressible free surface flows in which viscous effects may not be neglected. The paper describes the methodology, and gives illustrative results for simple geometries.Design/methodology/approachThe pseudospectral matrix element method of discretisation is selected as the basis for the CFD technique adopted, because of its high spectral accuracy. It is implemented as a means of solving the Navier‐Stokes equations coupled with the modified compressibility method.FindingsThe viscous solver has been validated for the benchmark cases of uniform flow past a cylinder at a Reynolds number of 40, and 2D cavity flows. Results for sloshing of a viscous fluid in a tank have been successfully compared with those from a linearised analytical solution. Application of the method is illustrated by the results for the interaction of an impulsive wave with a surface piercing circular cylinder in a cylindrical tank.Research limitations/implicationsThe paper demonstrates the viability of the approach adopted. The limitation of small amplitude waves should be tackled in future work.Practical implicationsThe results will have particular significance in the context of validating computations from more complex schemes applicable to arbitrary geometries.Originality/valueThe new methodology and results are of interest to the community of those developing numerical models of flow past marine structures.

Numerical implementation of the harmonic modified mild-slope equation

A numerical solver is presented of the modified time-independent mild-slope equation, which incorporates energy dissipation. Using a second-order parabolic approximation, the following external boundary conditions are modelled: open and fully transmitting to both incoming and outgoing waves; partially reflecting, and; fully absorbing. Discretisation of the governing equation and boundary conditions is by means of a second-order accurate central difference scheme. The resulting sparse-banded matrix is solved using an inexpensive banded solver with Gaussian elimination. The numerical predictions are in excellent agreement with the analytical solution for the interaction of non-breaking waves with an array of vertical surface-piercing circular cylinders on a horizontal bed. Results are compared with those for the same array on various seabed topographies. The model is robust and can be used for wave propagation in complex geometries. It has fewer restrictions associated with wave obliqueness at boundaries than traditional models based on the mild-slope equation.

Steady flow past a vertical surface-piercing circular cylinder

This paper describes experiments in which a vertical surface-piercing circular cylinder with a large draught was towed at steady speeds through water initially at rest. The cylinder diameter d was 210 mm, and measurements were made of pressures around its circumference at elevations between 2.4 d below still water level to 0.7 d above, at Froude numbers (based on d) up to 1.67. The tests were carried out at a constant ratio of Reynolds number to Froude number of 2.79×10 5. The total resistance coefficient reached a maximum at a Froude number of about 1, when that part of the loading that can be attributed to the presence of the free surface was equivalent to the submerged form drag on a length of cylinder of about 0.9 d. Measurements are also presented of the run-up on the front of the cylinder and of the depth of the depression at the back. Previous measurements by Hay (Flow about Semi-submerged Cylinders of Finite Length. Princeton University Report, Princeton, NJ, 1947) for the case of a cylinder with a submerged free end, and by Hsieh (Proc. Am. Soc. Civil Eng. 90 (1964) 161) of forces on cylinders standing on the floor of an open channel, are reanalysed. In most respects these results are found to be compatible with the present data for a cylinder of large draught.

Water wave interaction with an array of bottom-mounted surface-piercing porous cylinders

The interaction of water waves with arrays of bottom-mounted, surface-piercing circular cylinders is investigated theoretically. The sidewall of each cylinder is porous and thin. Under the assumptions of potential flow and linear wave theory, a semi-analytical solution is obtained by an eigenfunction expansion approach first proposed for impermeable cylinders by Spring and Monkmeyer (1974), and later simplified by Linton and Evans (1990). Analytical expressions are developed for the wave motion in the exterior and all interior fluid regions. Numerical results are presented which illustrate the effects of various wave and structural parameters on the hydrodynamic loads and the diffracted wave field. It is found that the porosity of the structures may result in a significant reduction in both the hydrodynamic loads experienced by the cylinders and the associated wave runup.

Parallel finite element computation of free-surface flows

In this paper we present parallel 2D and 3D finite element computation of unsteady, incompressible free-surface flows. The computations are based on the Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) finite element formulation, which takes automatically into account the motion of the free surface. The free-surface height is governed by a kinematic free-surface condition, which is also solved with a stabilized formulation. The meshes consist of triangles in 2D and triangular-based prism elements in 3D. The mesh update is achieved with general or special-purpose mesh moving schemes. As examples, 2D flow past spillway of a dam and 3D flow past a surface-piercing circular cylinder are presented.

A numerical procedure for time domain nonlinear surface wave calculations

A numerical procedure based upon Boundary Integral Method for time domain wave-making problems is developed. The free-surface boundary condition is expressed in a Lagrangian notation to follow the free-surface's motion in time. The corresponding material derivative term is approximated by a finite difference expression, and the velocity terms are extrapolated in time for the completion of the formulations. Solutions with a linear free-surface condition and a non-linear free-surface condition are obtained at small time increment values. Modelling of surface wave problems are studied in two dimensions as well as in three dimensions, and comparisons are made to results based on linear theory as well as to published experimental results. Good agreements between the numerical solutions and measured values are found. Applications are on the simulations of surface waves in a towing tank, ship's bow wave calculation, and wave diffraction problem for a vertical, surface piercing circular cylinder. For the modelling of the three-dimensional wave diffraction problem, results at high wave amplitude are restricted because of the use of quadrilateral elements. The near cylinder region of the free-surface is not considered to be well represented because of the coarse element size. Wave forces calculated on the vertical cylinder are found to be affected by the modeled tank length. When the simulated wave length is comparable to the wave tank's length, numerical results are found to be less than the experimental measurements. However, when the wave length is much shorter than the wave tank's length, solutions are obtained with very good precision.

Drift forces on a pair of vertical cylinders

An approximate method is presented here for estimating the hydrodynamic second-order drift forces on a pair of bottom-mounted, surface-piercing circular cylinders in water of arbitrary uniform depth. The theoretical results are based upon the large-spacing approximation. Results are presented to illustrate the influence of the various wave and structural parameters on the hydrodynamic forces on each of the cylinders.

Time‐Domain Second‐Order Wave Diffraction in Three Dimensions

A time-domain second-order method is developed to study nonlinear wave diffraction around a surface-piercing body of arbitrary shape in three dimensions. The free-surface boundary conditions and the radiation condition are satisfied to second order by a numerical integration in time, and the field solution at each time step is obtained by an integral equation method based on Green’s theorem. The solution is separated into a known incident potential and a scattered potential to be determined. The radiation condition applied to the scattered potential is modified to incorporate a time-dependent celerity to account for the transient and second-order effects. With initial conditions corresponding to a Stokes second-order wave field in the domain, the scattered potential is allowed to develop im time and space through the imposition of the body-surface boundary condition. The computed wave force and free-surface elevation components at second order are compared with previous theoretical results for the case of a surface-piercing circular cylinder and a favorable agreement is indicated.

Second order wave diffraction forces and runup by finite—infinite element method

This paper presents a finite-infinite element method for solving the second order wave diffraction problem. The formulation of the problem is based on the inhomogeneous far field condition and its corresponding higher order asymptotic solutions for the second order diffracted potential suggested by Li 9 , and follows the finite-infinite element method as used by Lau and Ji 18 . The interpolation in the infinite element along the radial direction has a far field decay rate compatible diffraction by a vertical surface-piercing circular cylinder. Numerical solutions of the second order diffracted potential, satisfying all the governing and boundary conditions as well as the inhomogeneous far field condition, are obtainable with the method. Numerical tests of the method give very good results of wave forces and runup profiles when compared with those published by other researchers.