Systems of Hydrodynamic Type that Approximate Two-Dimensional Ideal Fluid Equations

Statistical properties of different finite-dimensional approximations of two-dimensional ideal fluid equations are studied. A special class of approximations introduced by A.M. Obukhov (systems of hydrodynamic type) is considered. Vorticity distributions over area and quasi-equilibrium coherent structures are studied. These coherent structures are compared to structures occurring in a viscous fluid with random forcing.     

大深度分层流体中二维淹没浮体的波浪力分析

研究了大深度分层流体中二维任意形状淹没浮体的波浪力特性。首先基于一种合适的格林函数,采用边界积分方程法研究了流体中浮体对水波散射问题,然后通过单个淹没圆柱体的透射能和反射能与解析方法结果的比较,对所提出的方法进行了验证,最后分析了在不同的几何和物理条件下几种形状的浮体对波浪力的特有影响,得到了一些有意义的结果,这对分层海洋中淹没浮体的设计具有重要的参考价值。     

Formation of sand waves on unconsolidated sediments

Assuming that the long waves of sand found in sea beds are generated by stationary internal waves, a simplified analysis is made of their stability. The fluid model considered is two-dimensional and two-layered, with the lower layer stratified due to suspended sediments. Finally, the effect of flow over developed sand waves is studied under homogeneous conditions.     

On the properties of decaying two-dimensional turbulence: Characteristic features of the spectra and coherent structures

The main results of the theory of two-dimensional turbulence are presented. The conventional approach based on the Karman-Howarth equation is used to describe anomalous properties (sharply differing from those inherent in usual three-dimensional turbulence) of two-dimensional turbulent motion of an incompressible fluid, in particular, energy transfer across the spectrum from larger toward smaller wave numbers and the formation of coherent structures, whose origin is associated with the fact that the spectrum attains the form of a δ-function. A uniform method is proposed for obtaining self-similar spectra of two-dimensional turbulence in inertial ranges. The problem of turbulent diffusion of a passive tracer in the two-dimensional case is also considered. It turns out that the corresponding quantities, as well as those related to the dynamic characteristics of motion, can exhibit anomalous properties under certain conditions. The relationship between the results and experimental data is discussed. In particular, the experimentally observed inversion of the spectra of two-dimensional turbulence in the atmosphere is explained.     

Numerical modeling of dynamic responses and mooring forces of submerged floating breakwater

Using the volume of fluid (VOF) method, a numerical model is developed to estimate the nonlinear dynamics of a pontoon type moored submerged breakwater under wave action and the forces acting on the mooring lines, for both the vertical and inclined mooring alignments. The model is developed for a two-dimensional wave field in a vertical plane. The finite displacements of the breakwater such as sway, heave and roll in a very small time step are considered and the numerical grid cells intersected by the breakwater surfaces for changing its position due to wave action are treated using the concept of porous body model. Also, two-dimensional experimental studies are carried out to investigate the performance of the proposed model. The comparison of the computed and measured results reveals that the developed numerical model can reproduce well the dynamics of the floating body and the mooring line forces.     

Wave-induced surge motion of a tension leg structure

An analytical solution for the coupling problem of a two-dimensional tension leg structure interacting with a monochromatic linear wave train in an inviscid and incompressible fluid is presented. The tension legs are considered to be linearly elastic. The flow is further assumed to be irrotational and single-valued velocity potentials can then be defined.The boundary value problem is incorporated into a scattering and a radiation problem. The boundary value problems are then solved separately and combined to resolve all unknowns. The complete solutions of the velocity potentials are represented by the series of eigen-functions, and the surge motion of the structure is described in terms of the incident wave properties.The analytical solution is compared with a computer-coded numerical solution utilizing the boundary element method. The solutions agree very well, and both predict a resonant frequency for a specific structure which is different from the natural frequency of the structure due to the presence of the evanescent waves caused by the structure.     

High aspect ratio (L/D) riser VIV prediction using wake oscillator model

A two-dimensional (2-D) vortex-induced vibration (VIV) prediction model for high aspect ratio (L/D) riser subjected to uniform and sheared flow is studied in this paper. The nonlinear structure equations are considered. The near wake dynamics describing the fluctuating nature of vortex shedding is modeled using classical van der Pol equation. A new approach was applied to calibrate the empirical parameters in the wake oscillator model. Compared the predicted results with the experimental data and computational fluid dynamic (CFD) results. Good agreements are observed. It can be concluded that the present model can be used as simple computational tool in predicting some aspects of VIV of long flexible structures.     

Water entry of a finite width wedge near a floating body

The problem of a two-dimensional finite-width wedge entering water near a freely floating body is considered through the velocity potential theory for the incompressible liquid with the fully nonlinear boundary conditions on the free surface. The problem is solved by using the boundary element method in the time domain. The numerical process is divided into two phases based on whether the interaction between the wedge and floating body is significant. In the first phase, when the single wedge enters water at initial stage, only a small part near its tip is in the fluid, the problem is studied in a stretched coordinate system and the presence of the floating body has no major effect. In the second phase, the disturbance by water entry of the wedge has reached the floating body, and both are considered together in the physical system. The auxiliary function method is adopted to decouple the nonlinear mutual dependence between the motions of the wedge and floating body, both in three degrees of freedom, and the fluid flow, as well as the interaction effects between them. Case studies are undertaken for a wedge entering water in forced or free fall motion, vertically or obliquely. Results are provided for the accelerations, velocities, pressure distribution and free surface deformation, and the interaction effects are discussed.     

Simulation of Gravity Currents Using VOF Model

By the Volume of Fluid (VOF) rnultiphase flow model two-dimensional gravity currents with three phases including air are numerically simulated in this article. The necessity of consideration of turbulence effect for high Reynolds numbers is demonstrated quantitatively by LES (the Large Eddy Simulation) turbulence model. The gravity currents are simulated for h ≠ H as well as h= H, where h is the depth of the gravity current before the release and H is the depth of the in-truded fluid. Uprising of swell occurs when a current flows horizontally into another lighter one for h ≠ H. The problems under what condition the uprising of swell occurs and how long it takes are considered in this article. All the simulated results are in reasonable agreement with the experimental results available.     

Numerical modelling of liquefaction in loose sand deposits subjected to ocean waves

The failure of marine structures is often attributed to liquefaction in loose sand deposits that are subjected to ocean waves. In this study, a two-dimensional integrated numerical model is developed to characterize the liquefaction behaviours of loosely deposited seabed foundations under various types of ocean waves. In the present model, Reynolds-Averaged Navier–Stokes (RANS) equations are used to simulate the surface wave motion, and Biot's consolidation equations are used to link the solid-pore fluid interactions in a porous medium. A poro-elasto-plastic solution is used to reproduce foundation behaviour under cyclic shearing. Unlike previous investigations, both oscillatory and residual soil responses were considered; they are coupled in an instantaneous approach. Verification of the model results to the previous centrifugal wave tests is carried out, obtaining fairly good agreement. Numerical examples show that foundation behaviour under various types of wave loading, particularly standing waves or a solitary wave, embodies a completely two-dimensional process in terms of residual pore pressure development. The parametric studies demonstrate that liquefaction caused by the build-up of pore pressures is more likely to occur in loosely deposited sand foundations with poor drainage and under large waves.     

Wave radiation and diffraction by a two-dimensional floating body with an opening near a side wall

The radiation and diffraction problem of a two-dimensional rectangular body with an opening floating on a semi-infinite fluid domain of finite water depth is analysed based on the linearized velocity potential theory through an analytical solution procedure. The expressions for potentials are obtained by the method of variation separation, in which the unknown coefficients are determined by the boundary condition and matching requirement on the interface. The effects of the position of the hole and the gap between the body and side wall on hydrodynamic characteristics are investigated. Some resonance is observed like piston motion in a moon pool and sloshing in a closed tank because of the existence of restricted fluid domains.     

Combined effects of current and waves on fluid force

Combined effects of current and waves on the force exerted on an element of a cylinder in a random gravity wave field in deep water are studied. Wave-current interactions are taken into account. Statistical quantities of the fluid force such as force spectrum and root mean square value of the force are obtained numerically and presented in graphical forms. Comparisons are made of the cases in which wave-current interactions are considered and ignored. It is shown that wave-current interactions contribute to changes in fluid force to an appreciable extent and therefore should be considered in the evaluation of fluid forces on objects.     

Floating pontoon breakwaters

The hydrodynamic properties of a pair of long floating pontoon breakwaters of rectangular section are investigated theoretically. The structures are partially restrained by linear symmetric moorings fore and aft. The fluid motion is idealized as linearized, two-dimensional potential flow. The breakwater motions are assumed to be two-dimensional, in surge, heave and pitch. The solution for the fluid motion is obtained by the boundary integral equation method using an appropriate Green's function. Numerical results are presented that illustrate the effects of the various wave and structural parameters on the efficiency of the breakwaters as barriers to wave action. It is found that the wave reflection properties of the structures depend strongly on their width, draft and spacing and the mooring line stiffnesses, while their excess buoyancy is of lesser importance.     

On two-dimensional moonpool resonance for twin bodies in a two-layer fluid

This paper studies the moonpool resonance of two heaving rectangular bodies in a two-layer fluid. A mathematical model is proposed based on an eigenfunction matching approach. The motion of the two-dimensional bodies is assumed to be vertical and harmonic. Heave added mass and damping coefficients are computed to examine the hydrodynamic behavior of the twin bodies. The free surface and internal wave elevations are obtained near the resonant frequencies. The presented results and analyses reveal that there exist both Helmholtz and higher-order resonances in the two-layer fluid system, which is similar to the single-layer fluid case. It is also found that the resonances are closely associated with the free surface elevation inside moonpool gap, not the wave elevation at the interfacial surface. In addition, parametric studies have been performed to identify the dependencies of hydrodynamic behavior on geometry and density stratification.     

Simulation of freefall water entry of a finite wedge with flow detachment

A two-dimensional finite wedge entering water obliquely in freefall with three degrees of freedom is considered through the velocity potential theory for the incompressible liquid. The problem is solved by using the boundary element method in the time domain. The scheme of the stretched coordinate system is adopted at the initial stages when only a small part of the wedge near its tip has entered water. The auxiliary function method is adopted to decouple the nonlinear mutual dependence between the body motions in three degrees of freedom and the fluid flow. When the liquid has detached from the knuckle of the wedge, the free jet is treated through the momentum equation. The developed method is verified through existing results for one degree of freedom in vertical motion. Various case studies are undertaken for a wedge entering water vertically, obliquely and with rotational angles. Results are provided the accelerations, velocities, pressure distribution and free surface deformation, and the physical implications are discussed.     

Study of non-linear wave motions and wave forces on ship sections against vertical quay in a harbor

The resonance phenomenon of fluid motions in the gap between ship section, seabed and vertical quay wall is studied numerically and experimentally. The natural frequency of the fluid motions in the gap is derived. A two-dimensional time-domain coupled numerical model is developed to calculate the non-linear wave forces acting on a ship section against vertical quay in a harbor. The fluid domain is divided into an inner domain and an outer domain. The outer domain is the area between the left side of ship section and the incident boundary, where flow is expressed by Boussinesq equations. The rest area is the inner domain, which is the domain beneath the ship section plus the domain between the right side of ship section and vertical quay wall. The flow in the inner domain is expressed by Newton's Second Law. Matching conditions on the interface between the inner domain and the outer domain are the continuation of volume flux and the equality of pressures. The numerical results are validated by experimental data.     

Simplified CFD modeling for bilge keel force and hull pressure distribution on a rotating cylinder

A horizontal, circular cylinder fitted with one bilge keel is forced to rotate harmonically around its axis. The bilge keel load and hull pressure distribution are investigated. A fully submerged condition (infinite fluid), and three partly-submerged conditions are considered. A two-dimensional numerical study is performed, and the results are validated against recently published experimental data by van’t Veer et al. [30]. In addition, comparisons for mass and drag coefficients are also made with experimental data for plate in infinite fluid (Keulegan and Carpenter [8]), and wall-mounted plate (Sarpkaya and O’Keefe [9]) in oscillatory flow.A Navier–Stokes solver based on the Finite Volume Method is adopted for solving laminar flow of incompressible water. The free-surface condition is linearized by neglecting the nonlinear free-surface terms and the influence of viscous stresses in the free surface zone, while the body-boundary condition is exact. This simplified modeling of the problem required the mesh to be fine only around the bilge keels, leading to a total number of cells around N ≃ 1 ×10⁴, which reduced computational cost significantly.The influence of draft and amplitude of oscillations on the bilge keel force and hull pressure distribution are considered. The bilge keel force is presented in terms of non-dimensional drag and mass coefficients including higher harmonic components. The numerical results are also compared with the industry standard empirical method for calculation of roll damping proposed by Ikeda et al. [4]. In general, a good agreement between the results of the present numerical method and the experimental data is obtained and the differences with those predicted by the empirical method are addressed.     

Effect of ice cover on oscillations of fluid in a closed basin

Within the framework of the linear theory of long waves, the problem of the effect of ice cover on seiche oscillations of fluid in a two-dimensional constant-depth basin is solved. The eigenfrequencies and eigenfunctions of seiche oscillations are obtained for different boundary conditions at ice edges: rigid coupling and free edges. The forced oscillations of fluid and ice under the action of a moving disturbance of atmospheric pressure are investigated. The change in the stress of ice bending is considered and it is shown that the coast ice can be broken.     

Influence of wave-current interactions on fluid force

Influence of wave-current interactions on fluid force on a single cylinder and on two cylinders in tandem is examined. Quantities obtained are spectra of fluid force on elements, total fluid force and its statical moment. Comparisons are made of the cases in which wave-current interactions are considered and ignored. It is shown that interactions affect spectra of element force and statical moment more than do total force spectrum. For two cylinders in tandem, for cylinder spacings, current conditions and frequency range of practical interest, the effect of interactions is slightly reduced.     

A note on boundary layers and wakes in rotating fluids

The flow induced by the two-dimensional line vortex moving in a rotating fluid is discussed. The governing vorticity equation is linearized adopting the Oseen approximation.First, the problem is considered on a constantf-plane. The solution shows that the Stewartson E^1/4 layer is transformed into the Oseen wake as the role of the advection becomes important.Second, the problem is considered on a-plane. When the line vortex moves westward, the solution shows a pattern of Rossby lee waves decaying downstream of the vortex and alternating flows far upstream. When the line vortex moves eastward, the inviscid solution shows definite alternating jets downstream. In a viscous case, however, the jets become less definite and identical with the above mentioned alternating flows in the far field. Far upstream, there are no disturbances because of the special propagation characteristics of Rossby waves.