Effect Of Free Surface Waves Research Articles

Turbulence structure under the free-surface wave around an advancing surface-piercing cylindrical body

Free-surface wave effects on the turbulent boundary layer and wake of a surface-piercing slender body are investigated by measuring the flow and free-surface elevation around the model in three different Froude number (Fr) conditions—0.126, 0.282, and 0.400. The flow is measured by a towed underwater stereoscopic particle image velocimetry system. At the lowest Fr of 0.126, there appears no significant wave pattern, but the flow retards under the free-surface. The free-surface hinders the flow underneath, as the solid surface does. The steady and smooth free-surface wave generated at the intermediate Fr of 0.282 affects the pressure gradient on the turbulent boundary layer. The momentum thickness increases due to the adverse pressure gradient from the wave trough to the wave crest. At the highest Fr of 0.400, a strong adverse pressure gradient results in flow separation and violent wave breaking. The wave-induced separation at the high Froude number condition stimulates momentum transfer and turbulence dissipation, resulting in isotropic turbulence.

Numerical Modeling of Free-Surface Wave Effects on Flexural Vibration of Floating Structures

To investigate flexural vibration of structures in a fluid, a numerical algorithm was developed to relate the added mass and damping effects of the fluid to each mode of vibration. These are separate from the traditional added mass associated with rigid body motion, such as the translational motion along Cartesian axes. In this formulation, small-amplitude free surface waves were accounted for by using a nonsingular implementation of the free-surface Green’s function for a potential flow solver based on Boundary Element Method. The formulation was applied to the forced vibration of structures, namely, a hemispherical shell and a reinforced half cylinder with typical dimensions of ships and offshore structures, to obtain their dynamic response at various excitation frequencies. It is observed that resonance frequency of the structure, in contact with water, decreases due to the added mass effect. The influence of the free-surface wave on the fluid loading was investigated for large structures. It is simpler to relate the fluid added mass to mode shapes rather than distribution of fluid load over wetted surface of the structure in engineering simulations. Moreover, it is found that the vibration energy radiated away by the fluid surface wave has little influence on the vibration response of the shell structures.

Instability of Viscoelastic Plates with Longitudinally Variable Speed and Immersed in Ideal Liquid

The instability of longitudinally variable speed viscoelastic plates in contact with ideal liquid is studied for the first time. The effect of free surface waves is taken into account in the present study. The viscoelasticity is considered by using the Kelvin–Voigt viscoelastic constitutive relations. The classical theory of thin plate is utilized to derive the governing equation of variable speed plates. The fluid is assumed to be incompressible, inviscid and irrotational. Additionally, the velocity potential and Bernoulli’s equation are utilized to describe the fluid pressure acting on the vibrating plates. The fluid effect on the vibrational plates is described as the added mass of the plates which can be formulated by the kinematic boundary conditions at the structure–fluid interfaces. Parametric instability is analyzed by directly applying the method of multiple scales to the governing partial-differential equations and boundary conditions. The unstable boundaries are derived from the solvability conditions and the Routh–Hurwitz criterion for principal parametric, sum-type and difference-type combination resonances. Based on the numerical simulation, the effects of some key parameters on the unstable boundaries are illustrated in the excitation frequency and excitation amplitude plane in detail.

Hydroelastic dynamic analysis of axially moving plates in continuous hot-dip galvanizing process

For the purpose of understanding the vibrational characteristics of moving plates in continuous hot-dip galvanizing process, the linear and nonlinear free vibrations of an axially moving rectangular plate coupled with dense fluid having a free surface are investigated. The fluid is assumed to be incompressible, inviscid and irrotational in this study. Effect of free surface waves of the fluid is taken into account in the analysis. The classical thin plate theory is adopted to formulate the equation of motion of the vibrating plate. The velocity potential and Bernoulli׳s equation are used to describe the fluid pressure acting on the moving plate. The effect of fluid on the vibrations of the plate may be equivalent to added mass of the plate. The system is solved by applying directly the method of multiple scales to the governing partial-differential equations and boundary conditions. Results show the immersion depth, moving speed, fluid-plate density ratio, stiffness ratio and aspect ratio all have significant effects on the natural frequencies of the immersed moving plate. The nonlinear frequencies of the plate-fluid system are influenced by initial amplitude, moving speed and nonlinear coefficient. It is also shown that the 1:1 and 1:3 internal resonances of the immersed moving plate can occur at certain speeds. Owing to the internal resonance, amplitude ratio of the two internal resonance modes shows multi-value characteristics. With the increase of nonlinear coefficient, the internal resonance phenomenon becomes more and more intense.

Vibrations of Axially Moving Vertical Rectangular Plates in Contact with Fluid

The vibration characteristics of an axially moving vertical plate immersed in fluid and subjected to a pretension are investigated, with a special consideration to natural frequencies, complex mode functions and critical speeds of the system. The classical thin plate theory is adopted for the formulation of the governing equation of motion of the vibrating plates. The effects of free surface waves, compressibility and viscidity of the fluid are neglected in the analysis. The velocity potential and Bernoulli’s equation are used to describe the fluid pressure acting on the moving plate. The effect of fluid on the vibrations of the plate may be regarded as equivalent to an added mass on the plate. The formulation of added mass is obtained from kinematic boundary conditions of the plate–fluid interfaces. The effects of some system parameters such as the moving speed, stiffness ratios, location and aspect ratios of the plate and the fluid-plate density ratios on the above-mentioned vibration characteristics of the plate–fluid system are investigated in detail. Various different boundary conditions are considered in the study.

Study on the dynamic behavior of axially moving rectangular plates partially submersed in fluid

The natural frequencies, complex modes and critical speeds of an axially moving rectangular plate, which is partially immersed in a fluid and subjected to a pretension, are investigated. The effects of free surface waves, compressibility and viscidity of the fluid are neglected in the analysis. The subsection functions are used to describe the discontinuous characteristics of the system due to partial immersion. The classical thin plate theory is adopted to formulate the equations of motion of a vibrating plate. The velocity potential and Bernoulli's equation are used to describe the fluid pressure acting on the moving plate. The effect of fluid on the vibrations of the plate may be equivalent to the added mass on the plate. The effects of distance ratio, moving speed, immersed-depth ratio, boundary conditions, stiffness ratio and aspect ratio of the plate as well as the fluid-plate density ratios on the free vibrations of the moving plate-fluid system are investigated.

Natural vibration analysis of rectangular bottom plate structures in contact with fluid

A simple and effective procedure for the natural vibration analysis of rectangular bottom plate structures in contact with fluid is presented. Structural part of the coupled hydroelastic problem covers thin and thick rectangular plates and stiffened panels with different framing types. The eigenvalue problem is formulated using Lagrange׳s equation of motion and taking into account potential and kinetic energies of a plate structure and fluid kinetic energy, respectively. Natural frequencies and modes are obtained applying the assumed mode method using the characteristic polynomials of a Timoshenko beam. Potential flow theory assumptions are adopted for the fluid and the effect of free surface waves is ignored. From the boundary conditions for the fluid and structure the fluid velocity potential is derived and it is utilized for the calculation of added mass using the assumed modes. The developed theoretical model is verified with several numerical examples dealing with the natural vibration of bare plates and stiffened panels in contact with different fluid domains. A comparison of the results with those obtained by a general purpose FEA software showed very good agreement, especially for the lowest natural frequencies that are actually most relevant for the structural design from the viewpoint of vibration.

Coupled vibrations of a partially fluid-filled cylindrical container with an internal body including the effect of free surface waves

Internal bodies (baffles) are used as damping devices to suppress the fluid sloshing motion in fluid–structure interaction systems. An analytical method is developed in the present article to investigate the effects of a rigid internal body on bulging and sloshing frequencies and modes of a cylindrical container partially filled with a fluid. The internal body is a thin-walled and open-ended cylindrical shell that is coaxially and partially submerged inside the container. The interaction between the fluid and the structure is taken into account to calculate the sloshing and bulging frequencies and modes of the coupled system using the Rayleigh quotient, Ritz expansion and Galerkin method. It is shown that the present formulation is an appropriate and new approach to tackle the problem with good accuracy. The effects of fluid level, number of nodal diameters, internal body radius and submergence ratio on the dynamic characteristics of the coupled system are also investigated.

Hydro- and aerodynamic analysis for the design of a sailing yacht

The results of the design analysis for a sailing yacht’s hull and sails are reported. The results were used to confirm the design of a 30 ft long sloop, which was planned, designed, and built in Korea for the first time in history. Flows around a sailing yacht above and under the free surface were analyzed separately using both computational and experimental methods. For the underwater flow analysis, turbulent flow simulations with and without free surface wave effects were carried out for the canoe hull with keel/rudder. The computed drag and side forces on the hull model were compared with the measurement data obtained from the towing tank experiments. In order to assess the sail performance, another set of computations was carried out for the flow around a sail system composed of main and jib sails with a mast. The present study demonstrates that, for the design analysis of a sailing yacht, computational fluid dynamics techniques can be utilized with a reasonable level of confidence.

Rans Predictions of Free Surface Effects on Axisymmetric Underwater Body

The aim of the current investigation is an assessment of the influence of free surface effects on hydrodynamic coefficients of an axisymmetric underwater body using Reynolds averaged Navier-Stokes (RANS) solver. This paper mainly focuses on the capabilities of turbulence models (two high-Re models, k-ε Re-Normalised Group (RNG) and k-ε Realizable and one low-Re model, k-ε Abe-Kondoh-Nagano (AKN) together with volume of fluid (VOF) method for hydrodynamic investigation of an axisymmetric underwater body (Afterbody1) near the free surface. The current numerical investigations were carried out to simulate measured hydrodynamic coefficients for several experimental parameters, including velocity variations from 0.4 m/s (Reynolds number Rev = 2.12x 105) to m/s (Reynolds number Rev = 7.42 x 105), depth of submergence from 0.75D to 4.0D, and angle of attack 0° to 15° on a vertical plane for a model scale of 1:2. Numerical simulations were conducted at a grid density of 0.46 million cells based on grid independence study using a commercial flow solver, Fluent. Free surface effects on hydrodynamic coefficients (drag, lift and pitching moment) are studied from the calculated hydrodynamic forces and moments. Validation of numerical results is done by comparing with measured values. The present investigation found that the k-ε Realizable model together with VOF method consistently provided superior performance in predicting the free surface wave effects on hydrodynamic coefficients for Afterbody1 model. This study is expected to aid the selection of apposite RANS model for simulation of hydrodynamic forces over underwater bodies near the free surface.

Experimental studies on the slowly varying drift motion of a berthed container ship model

The effects of free-surface waves on the floating structures are of great importance in the offshore industry. Among the six degrees of motions of a surface ship the absence of restoring forces in surge, sway and yaw led to critical situations for moored ships in the recent times. The order of forces in horizontal plane and their exciting frequencies are matters of interest. The resonance with the presence of moored chains led to many accidents in the recent past. The lines in dry conditions may not give good damping and in wet condition they may trigger the system to chaotic motions and jumps. Two different loading conditions of a container ship model are tested with waves in laboratory conditions in two different drafts. The mooring lines are chosen as per scale law and the energy under the response spectrum is determined from the plots. The results give new insights into the movement of a berthed ships subjected to waves. Response of the moored ship to different loading conditions in different water depths are discussed in this paper. The paper gives the order of energy due to first-order and slowly varying movement of a berthed container model in a towing tank.

배플을 갖는 원통형 유체저장 탱크의 연성진동해석

The coupled vibration characteristics for the fluid-structure interaction systems are investigated through the finite element method. The present paper is focused on vibration characteristics of the cylindrical fluid-storage tank with a baffle. The tank is partially filled with an inviscid and irrotational fluid having a free surface. A baffle is assumed here to have the shape of a thin annular plate and a conical shell, attached to the cylindrical tank and positioned below the fluid surface. The liquid domain is limited by a rigid flat bottom. As the effect of free surface waves is taken into account in the analysis, the bulging and sloshing modes are studied. To demonstrate the validity of present results, they are compared with the published ones. The effect of positions and inner-to-outer radius ratio of annular baffle and setting angles of conical baffle on coupled vibration characteristics is investigated.

Coupled vibration analysis of liquid-filled rigid cylindrical storage tank with an annular plate cover

An analytical approach is developed to investigate the vibration characteristics of the sloshing and bulging modes for a liquid-filled rigid circular cylindrical storage tank with an elastic annular plate in contact with sloshing surface of liquid. The cylindrical tank is filled with a non-viscous and incompressible liquid having a free surface. The free surface of liquid is partially covered by an annular plate with outer clamped edge and inner free edge. The liquid domain is limited by a rigid cylindrical wall and a rigid flat bottom. As the effect of free surface waves is taken into account in the analysis, the bulging and sloshing modes are studied. The solution for the velocity potential of liquid movement is assumed as a suitable harmonic function that satisfies the Laplace equation and the relevant boundary conditions. The Rayleigh–Ritz method is used to derive the frequency equation of the liquid-filled rigid cylindrical tank with an annular plate on the sloshing surface. The effects of inner-to-outer radius ratio, thickness of annular plate and liquid volume on vibration characteristics of the storage tank are studied. To demonstrate the validity of the present analytical approach, the published results are compared for the rigid cylindrical tank without a plate and the finite element analysis is performed for the tank with a plate.

Coupled vibration of partially fluid-filled cylindrical shells with ring stiffeners

A theoretical method is developed to investigate the coupled vibration characteristics of the ring-stiffened cylindrical shells partially filled with an inviscid, incompressible and irrotational fluid having a free surface. As the effect of free surface waves is taken into account in the analysis, the bulging and sloshing modes are studied. The Rayleigh–Ritz method is used to derive the frequency equation of the ring-stiffened and partially fluid-filled shells based on Love's thin shell theory. The solution for the velocity potential of fluid movement is assumed as a sum of two sets of linear combinations of suitable harmonic functions that satisfy Laplace equation and the relevant boundary conditions. The effect of fluid level, stiffener's number and position on the coupled vibration characteristics is investigated. To demonstrate the validity of present theoretical method, the published results are compared for simply supported shell and the finite element analysis is performed for unstiffened/stiffened, partially fluid-filled shells with clamped–free boundary condition.

VIBRATIONS OF CIRCULAR PLATES RESTING ON A SLOSHING LIQUID: SOLUTION OF THE FULLY COUPLED PROBLEM

Vibrations of circular plates resting on a sloshing liquid free surface are studied. The fully coupled problem between sloshing modes of the free surface and bulging modes of the plate is solved by using the Rayleigh–Ritz method. The sloshing boundary condition is directly inserted into the eigenvalue problem. The liquid domain is limited by a rigid cylindrical surface and a rigid flat bottom. The fluid is considered inviscid and incompressible; it is described by the velocity potential expanded in a series. The present model has as limit cases: (1) circular plates resting on half-infinite liquid domain and (2) circular plates completely covering the liquid in a circular cylindrical tank. The theory is suitable for all axisymmetric plate boundary conditions. The effect of free surface waves on the plate natural frequencies is significant when the fundamental bulging mode of the plate has its natural frequency close to those of the first sloshing modes of the free surface. The present original solution allows the study of plates having a very strong coupling between sloshing and bulging modes to be studied to a high level of accuracy. The convergence of the method is shown. The natural frequencies and mode shapes for different system parameters are given.

Vibration of vertical rectangular plate in contact with water on one side

In this paper, the vibratory characteristics of a rectangular plate in contact with water on one side are studied. The elastic plate is considered to be a part of a vertical rectangular rigid wall in contact with water, the edges of which are elastically restrained and parallel to those of the rigid wall. The location and size of the plate on the rigid wall may vary arbitrarily. The water with a free surface is in a rectangular domain infinite in the length direction. The effects of free surface waves, compressibility of the water and the hydrostatic water pressure are neglected in the analysis. An analytical-Ritz method is developed to analyse the interaction of the plate–water system. First of all, by using the method of separation of variables and the method of Fourier series expansion, the exact expression of the motion of water is derived in the form of integral equations including the dynamic deformation of the plate. Then the Rayleigh–Ritz approach is used to derive the eigenfrequency equation of the system via the variational principle of energy. By selecting beam vibrating functions as the admissible functions of the plate, the added virtual mass incremental (AVMI) matrices for plate vibration are obtained. The convergency studies are carried out. The effects of some parameters such as the depth and width of water, the support stiffnesses, location and aspect ratio of the plate and the plate–water size and density ratios on the eigenfrequencies of the plate–water system are investigated. Several numerical examples are given. The validity of AVMI factor approach is also confirmed by comparing the AVMI factor solutions with the analytical-Ritz solutions. The results show that the approach presented here can also be used as excellent approximate solutions for rectangular plates in contact with water of infinite width and/or infinite depth. Copyright © 2000 John Wiley & Sons, Ltd.

VIBRATION OF CIRCULAR PLATES ON A FREE FLUID SURFACE: EFFECT OF SURFACE WAVES

The effect of free-surface waves on free vibrations of circular plates resting on a free fluid surface is studied. The solution is achieved by using a perturbation technique and the Hankel transformation method which give a couple of dual integral equations of Titchmarsh type. The fluid is considered inviscid and incompressible and the velocity potential describes its motion. The Kirchhoff theory of plates is used to model the elastic thin plate. The theory is suitable for all axisymmetric plate boundary conditions. Numerical results are given in non-dimensional form for modes up to five nodal circles and diameters for clamped, simply supported and free-edge plates, to be ready-to-use in applications. The effect of free-surface waves on the plate's natural frequencies is significant only when bluging and sloshing modes of the system have close natural frequencies.

VIBRATIONS OF CIRCULAR TUBES AND SHELLS FILLED AND PARTIALLY IMMERSED IN DENSE FLUIDS

Free vibrations of circular cylindrical shells and tubes completely filled with a dense fluid and partially immersed in a different fluid (liquid) having a free surface are studied. Elastic shell constraints, varying from simply supported to clamped ends are assumed. Fluids are assumed to be stationary, inviscid and incompressible. The fluid outside the shell is assumed to be unlimited in the radial direction and limited in the vertical direction by a rigid bottom and a free surface. The effect of free surface waves is considered, so that both sloshing and bulging modes of the system are investigated. A velocity potential is used to describe the fluid oscillations, and the Rayleigh–Ritz method has been extended to the case of fluid–structure interaction to obtain the solution of the coupled system.

Low-frequency second-order wave-drift forces and damping

The effects of free-surface waves on floating structures is of great importance in the offshore industry. Besides the first-order responses to the waves, second-order effects play a role, in particular, in the situation where an anchored object may be excited in its frequency of resonance. This paper concerns the study of this phenomenon. Two different approaches are developed, each with its own advantage. Special attention is paid to excitation forces generating the motions; also a theory to determine the inviscid damping that limits the extreme excursions is treated. Numerical results are presented for a sphere, while numerical results for two classes of tankers, namely for a VLCC and a LNG-carrier, and a semi-submersible are compared with experimental data obtained at the Maritime Research Institute in the Netherlands (MARIN).

VIBRATIONS OF PARTIALLY FILLED CYLINDRICAL TANKS WITH RING-STIFFENERS AND FLEXIBLE BOTTOM

The dynamics of a tank partially filled with a liquid having a free surface is investigated. In the analysis, the effect of free surface waves is taken into account, so that both bulging and sloshing modes are studied. The structure is completely flexible, and is composed of a vertically standing circular cylindrical shell, with ring-stiffeners, and a flexible bottom that consists of a circular plate generally resting on an elastic Winkler foundation. The plate edge is joined to the shell by coupling rotational springs distributed around the edge. The interaction between the plate and the shell via the fluid is included. Application to the particular cases of (i) a flexible shell with a rigid bottom and (ii) a flexible bottom plate with a rigid cylinder are also presented, as well as a comparison with approximate theories. Ring-stiffeners on the shell are also considered in order to investigate their influence on the modal behaviour of the tank. Solution of this problem is achieved via the method of artificial springs within the framework of the Rayleigh–Ritz theory, while considering strong fluid–structure interaction.