Forced Heave Motion Research Articles

Effects of nonlinearity and viscous damping on the resonant responses in two-dimensional moonpools with a recess

In this study, the nonlinear and viscous damping effects on the resonant responses inside a two-dimensional moonpool with a recess are investigated. Based on a fully nonlinear potential flow (FNPF) model, the nonlinear moonpool responses excited by the forced heave motion of a structure are simulated in the time domain. To consider the vortex-shedding damping effects in the moonpool, induced by both piston-type motion (the first harmonic component) and sloshing-type motions (the higher harmonic components), a pressure drop model based on Faltinsen and Timokha (2015) is developed and applied to the FNPF model (FNPF-V for short). The response amplitude operators (RAOs) of the higher harmonics, by which the nonlinear effects are evaluated, are computed by the FNPF-V model. It is found that the higher harmonics are noticeable for the excitation frequencies ω n / ( n + 1 ) ( n is a positive integer, ω n denotes the natural frequency of the n th sloshing mode), where secondary resonances can be triggered. In addition, it is found that the strong nonlinear interactions between the first and higher harmonics can lead to a significant amplification of the moonpool responses. The nonlinear behaviour (the secondary resonance, as well as the sum- and difference-frequency effects) is substantially enhanced as the excitation amplitude increases. Moreover, it is found that the length of the recess influences the nonlinear response of the moonpool in two respects with opposite effects. On the one hand, for the same first harmonic response, moonpools with a longer recess are more apt to exhibit nonlinearity. On the other hand, for the same external excitation, as the length of the recess increases, the first harmonic responses at the excitation frequencies ω n / ( n + 1 ) decrease, suppressing the nonlinear behaviour. • This paper studies the secondary resonances triggered by the piston-mode response. • A pressure drop model is introduced to consider the vortex-shedding effects. • The nonlinear interactions between different harmonic components are investigated. • The effects of recess length on the nonlinear response are analysed and interpreted.

A potential flow method combining immersed boundaries and overlapping grids: Formulation, validation and verification

A fully nonlinear two-dimensional numerical method based on potential-flow theory for water waves and their interaction with surface-piercing rigid bodies is presented. The harmonic polynomial cell (HPC) method is used to solve the Laplace equation for the velocity potential and its time derivative. The HPC method, which is a high-order method using analytical expressions (harmonic polynomials) to represent the solution inside overlapping cells, has previously been shown accurate and efficient. Supplementary research has shown that, in order to maximally benefit from the method’s accuracy, it is a requirement that square or close-to-square cells are used. Here, we use an immersed boundary method to model non-stationary boundaries such as the free surface or the surface of a rigid body, and overlapping, body-fixed grids that are locally Cartesian to refine the solution near moving bodies. Combining these two modelling concepts with the HPC method represents the main novel contribution in the present work. With this combined method, denoted as an immersed-boundary overlapping grid method (IBOGM), the challenge of generating boundary-fitted grids for complex boundaries is avoided. Moreover, square cells can be used throughout the domain and the solution can be refined locally without increasing the number of unknowns unnecessarily. The method is systematically validated and verified against analytical, experimental and numerical reference results. The cases studied include propagation of steep waves, forced heave motions of a semi-submerged circular cylinder and a fixed and freely floating ship section in beam-sea waves. For the freely floating ship section, the present method is compared in detail with results from a dedicated study performed with a fully nonlinear boundary element method for cases with roll motions up to 30°. The present results are generally in good agreement with reference results, even for the most challenging wave–body interaction cases considered. Based on this, we later intend to use the method to examine in depth the importance of nonlinear effects in the interaction between waves and rigid bodies.

Comparative analysis on various components of heave damping for sandglass-type floating body

For the new sandglass-type floating body with special shape, this paper mainly analyzes the characteristic of the heave damping and its corresponding components by computational fluid dynamics (CFD) method. Firstly, the convergence and accuracy of the CFD method to simulate the heave motion of sandglass-type floating body were validated along with the experimental data. Secondly, the numerical model of the floating body with forced heave motion was created, and then hydrodynamic forces versus different motion amplitudes were calculated and compared to discuss its nonlinear characteristics. Afterwards, based on time-history curves of the hydrodynamic force, heaving added mass and damping of the floating body versus various frequencies were achieved and found to be different from the potential results. In order to analyze the reasons that cause difference, numerical simulations using inviscid and k-ε turbulent models were performed respectively, and the results indicate that the effect of fluid viscosity on the hydrodynamic coefficients is small and not decisive. Then equivalent radiation damping based on CFD wave elevation was deduced to explore the relation between the inviscid CFD and potential models. Finally, local wave field and vorticity field near the body were mainly discussed, and quantitative analysis based on mechanical energy was performed to verify that the vortex due to the body shape is the essential cause of the heave damping for the sandglass-type floating body.

Experimental studies of a damaged ship section in beam sea waves

This work presents a series of experiments performed with a prismatic hull form in a small wave flume. The model is a midship section with rectangular damage opening on the side. It is slightly smaller than the flume breadth to achieve predominantly two-dimensional behavior in the experiments. Freely-floating tests in regular beam-sea waves have been carried out on the model section in intact and damaged conditions. Free-roll decay tests were performed for intact and damaged sections to understand the effect of floodwater on roll natural period and roll damping of the model. Video recordings and measurements of wave elevation inside the damaged compartment were performed in all experiments. Linear response amplitude operators (RAOs) for water elevation inside the model and model motions are presented and discussed. Effect of wave steepness, wave period, initial loading condition, damage-compartment division about centerplane (symmetric/asymmetric flooding), damage-opening size and air compressibility in the damaged compartment are examined. The presented results demonstrate occurrence of sloshing and piston mode resonances and their influence on damaged ship motions in waves is highlighted. A linearized strip theory method based on viscous flow is implemented to cross-check and complement the experimental results. The latter method can estimate the damaged ship motions with reasonable accuracy. The initial loading condition determines the equilibrium flooding state and, therefore, the sloshing and piston mode resonance frequencies of the flooded water. The damage-opening size mainly affects the roll damping behavior of the section. Air compressibility in the airtight compartment acts as a coupled spring system with the floodwater and restricts the free-surface motion in the damaged compartment. Cases of transient flooding for a damaged section are also presented, where the freely floating model moves in beam-sea waves and subsequent flooding takes place. Effect of wave steepness, incident wave period and initial intact stability of the model is examined for this scenario. The present work is a continuation of a series of experiments conducted on the same damaged section for forced heave motions documented in [1]. Due to the complex and nonlinear behavior of floodwater inside a damaged ship section, as the previous analysis, the current work is aimed to contribute in the physical understanding of ship behavior in damaged conditions, as well as to serve as a database for the research community to be used for numerical validation.

A Two-Dimensional Experimental and Numerical Study of Moonpools With Recess

Abstract Moonpool resonance is investigated in a two-dimensional setting in terms of regular, forced heave motions of a model with moonpool with different rectangular-shaped recess configurations. A recess is a reduced draft zone in the moonpool. Dedicated experiments were carried out. The model consisted of two boxes of 40 cm width each, with a distance of 20 cm between them. Recess configurations varying between 5 cm and 10 cm in length and 5 cm in height were tested. Different drafts were also tested. A large number of forcing periods and five forcing amplitudes were tested. A time-domain boundary element method (BEM) code based on the linear potential flow theory was implemented to investigate the resonance periods, mode shapes, as well as the moonpool response as predicted by the (linear) potential flow theory. Dominant physical effects were discussed, in particular damping due to flow separation from the sharp corners of the moonpool inlet and recess. The effect of the recess on the piston-mode behavior is discussed. The nondimensional moonpool response suggests strong viscous damping at the piston-mode resonance. The viscous BEM (VBEM) simulations demonstrate improvement over inviscid BEM, although further improvement of the method is needed. The VBEM simulations are, in general, in good agreement with the experiments. For the largest recess case, some discrepancies are observed in the amplitude-dependent response amplitude operators (RAOs). The piston-mode shapes are clearly different from the near flat free-surface elevation for a moonpool without recess, consistent with the recently published theory.

Experimental studies of a damaged ship section in forced heave motion

This work documents a detailed series of experiments performed in a wave flume on a thin walled prismatic hull form. The model consists of a rectangular opening located on the side. The length of the model is slightly smaller than the flume breadth to achieve two-dimensional (2D) behavior in the experiments. Forced oscillatory heave tests in calm water have been carried out by varying the model-motion parameters and examining both intact and damaged conditions. Video recordings, measurements of the wave elevation inside the damaged compartment and of the force on the model were performed in all the experiments. The effect of damage opening in the model on hydrodynamic loads is examined by comparing with an intact section. A theoretical analysis is used to explain the behavior of added mass and damping coefficients in heave for a 2D damaged section. The presented results demonstrate occurrence of sloshing and piston mode resonances in the tests and their influence on the hydrodynamics loads of a damaged ship. Detailed physical investigations are presented at these resonance frequencies for the damaged section. Effect of filling level in the damage compartment, damage-opening length and air compressibility in the airtight compartment is examined. Nonlinear effects are documented and appear dominant, especially, for lowest filling level where we have shallow-water depth conditions in the damaged compartment. Resonance phenomena that can lead to significant local loads are identified for the shallow water condition. Air compressibility in the airtight compartment and floodwater act as a coupled system and influence inflow/outflow of floodwater in the compartment. It has a significant effect on local floodwater behavior in the damaged compartment.

Finite water depth effect on wave-body problems solved by Rankine source method

Finite water depth effect for wave-body problems are studied by continuous Rankine source method and non- desingularized technique. Free surface and seabed surface profiles are represented by continuous panels rather than a discretization by isolated points. These panels are positioned exactly on the fluid boundary surfaces and therefore no desingularization technique is required. Space increment method is applied for both free surface source and seabed source arrangements to reduce computational cost and improve numerical efficiency. Fourth order Runge-Kutta iteration scheme is adopted on the free surface updating at every time step. The finite water depth effect is studied quantitatively for a series of cylinders with different B/T ratios. The accuracy and efficiency of the proposed model are validated by comparison with published numerical results and experimental data. Numerical results show that hydrodynamic coefficients vary for cylinder bodies with different ratios of B/T. For certain set of B/T ratios the effect of finite water depth increases quickly with the increase of motion frequency and becomes stable when frequency is relatively large. It also shows that water depths have larger hydrodynamic effects on cylinder with larger breadth to draft ratios. Both the heave added mass and damping coefficients increase across the frequency range with the water depths decrease for forced heave motion. The water depths have smaller effects on sway motion response than on heave motion response.

On damping of two-dimensional piston-mode sloshing in a rectangular moonpool under forced heave motions

Based on a similarity between piston-mode sloshing in a two-dimensional rectangular moonpool and a small-amplitude oscillatory flow through a slot in a slatted screen, a pressure discharge in the moonpool opening is theoretically quantified. This makes it possible, by changing the dynamic free-surface condition inside the moonpool, to account for the vortex-induced damping and thereby to modify the inviscid potential flow statement. The present paper is an important addition to Faltinsenet al.(J. Fluid Mech., vol. 575, 2007, pp. 359–397), where a discrepancy between the potential flow theory and model tests was reported. New calculations are supported by the earlier model tests of the authors as well as numerical data obtained by means of viscous solvers.

Influences of Vorticity to Vertical Motion of Two-Dimensional Moonpool under Forced Heave Motion

Potential (inviscid-irrotational) and Navier-Stokes equation-based (viscous) flows were numerically simulated around a 2D floating rectangular body with a moonpool; particular emphasis was placed on the piston mode through the use of finite volume method (FVM) and boundary element method (BEM) solvers. The resultant wave height and phase shift inside and outside the moonpool were compared with experimental results by Faltinsen et al. (2007) for various heaving frequencies. Hydrodynamic coefficients were compared for the viscous and potential solvers and sway and heave forces were discussed. The effects of the viscosity and vortex shedding were investigated by changing the gap size, corner shape, and viscosity. The viscous flow fields were thoroughly discussed to better understand the relevant physics and shed light on the detail flow structure at resonant frequency. Vortex shedding was found to account for the most of the damping. The viscous flow simulations agreed well with the experimental results, showing the actual role and contribution of viscosity compared to potential flow simulations.

문풀을 가지는 2차원 부유체의 강제 상하동요에 대한 CFD 해석

A two-dimensional floating body with a moon pool under forced heave motion, including a piston mode, is numerically simulated. A dynamic CFD simulation is carried out to thoroughly investigate the flow field around a two-dimensional moon pool over various heaving frequencies. The numerical results are compared with experimental results and a linear potential program by Faltinsen et al. (2007). The effects of vortex shedding and viscosity are investigated by changing the corner shapes of the floating body and solving the Euler equation, respectively. The flow fields, including the velocity, vorticity, and pressure fields, are discussed to understand and determine the mechanisms of wave elevation, damping, and sway force.

Discussion on Venkat & Spaulding: “Numerical Simulation of Nonlinear Free-Surface Flows Generated by a Heaving Body of Arbitrary Cross Section”

A method has been presented recently by Venkat and Spaulding to solve the nonlinear boundary-value problem of oscillating two-dimensional cylinders of arbitrary cross section on the free surface of a fluid. The method relies on a second-order finite-difference technique with a modified Euler method for the time domain and a successive over-relaxation procedure for the spatial domain. The authors compare their numerical results with those of other authors (theoretical and experimental), as they have published data for specialized forms like a wedge, circular cylinders, and ship-like sections in forced heave motion (references [4] to [7] and [22], [23] of the paper).

The dynamics of wave induced ship motions in shallow water

The analytical method developed by Svendsen (1968) for a forced heave motion is extended to the general problem of wave induced heave, roll and sway motions of a long ship at a depth of water which is only slightly larger than the draught of the ship. This corresponds, for example, to the situation of a fully loaded ship in a harbour area. After linearization of the problem, the water motion is considered for each of the three individual motions and for the wave reflection-transmission problem for a fixed ship. The ensuing results for the forces on the ship are then synthesized to form the equations of motion, which are presented with all coefficients given, including mooring forces. Analytical and numerical results are given for the three components of motion, for the associated resonance frequencies, and for the hydrodynamic masses and moments of inertia. Finally, the assumptions used are analyzed and evaluated by comparison with measurements and with other results for a special case.

On The Forces Induced On A Rectangular Cylinder By A Forced Heave Motion With Draught-Depth Ratio Close To Unity

On The Forces Induced On A Rectangular Cylinder By A Forced Heave Motion With Draught-Depth Ratio Close To Unity