Energy dissipation in sloshing waves in a rolling rectangular tank—I. Mathematical theory

The hydrodynamic aspects of the motion of a viscous fluid having a free surface in a rolling tank have been investigated. In a sequence of three papers, an analytical technique together with a numerical solution method will be presented to describe the sloshing phenomenon accurately and efficiently. This first paper introduces a linear theory of viscous liquid sloshing and formulates a boundary value problem subject to appropriate conditions. Viscosity is included in the problem formulation and its effects are properly accounted for. The second paper will describe a solution of the problem in function space by the truncation of infinite series. Boundary conditions are satisfied through the use of Fourier series expansions. However, the no-slip condition at the side walls can also be treated in a least-squares sense.Among the results that will be reported in the third paper are the effects of viscosity on liquid sloshing phenomenon and the dependence of viscous dissipation on the Reynolds and Froude numbers. Furthermore, the influence of the tank aspect ratio on viscous dissipation has been explored. These results demonstrate some unknown features of the functional relationships that exist between the dissipated energy and the Reynolds and Froude numbers. Similarly, the dependence of the dissipated energy on the aspect ratio has been analytically studied. The results obtained agree with the physical laws for the range of parameters investigated.     

Energy dissipation in sloshing waves in a rolling rectangular tank—II. Solution method and analysis of numerical technique

In Part I of these series of papers, the complete problem formulation in a linearized form was presented. In order to provide the engineer with an in-depth knowledge about the exact solution of the problem, it is natural and essential to start with a linear solution. This will be the objective of Part II, together with an exposition to the analysis of numerical technique utilized.A truncated infinite Fourier series-type solution is adopted for the linearized boundary value problem. It is shown that such a solution is mathematically consistent and represents the phenomenon properly by satisfying all of the field equations and the imposed boundary conditions. The dependence of the Fourier coefficients on the truncation limit has been investigated. The best lower and upper “cutoff limits” for the truncation of an infinite series are determined. An error analysis of the solution technique is performed.     

Nonlinear modeling of liquid sloshing in a moving rectangular tank

A nonlinear liquid sloshing inside a partially filled rectangular tank has been investigated. The fluid is assumed to be homogeneous, isotropic, viscous, Newtonian and exhibit only limited compressibility. The tank is forced to move harmonically along a vertical curve with rolling motion to simulate the actual tank excitation. The volume of fluid technique is used to track the free surface. The model solves the complete Navier–Stokes equations in primitive variables by use of the finite difference approximations. At each time step, a donor–acceptor method is used to transport the volume of fluid function and hence the locations of the free surface. In order to assess the accuracy of the method used, computations are verified through convergence tests and compared with the theoretical solutions and experimental results.     

双浮板液舱晃荡特性的数值研究

基于黏性流理论,采用动网格技术和6自由度模型,以及动量源方法,建立了双浮板液舱晃荡的数值模型。分别采用3种不同空间步长的网格离散计算区域,进行了网格收敛性验证。通过光滑液舱晃荡的模型试验和解析得到的爬高最大值,验证了数值模型的精确性。在载液率为50%,激励幅值为2 mm条件下,对双浮板液舱晃荡进行了数值计算,与光滑液舱相比,双浮板液舱晃荡的最大爬高明显减小。通过一个激励周期内双浮板液舱晃荡的波面显示发现,液舱晃荡模式由光滑液舱的驻波模式变为U管模式,晃荡模式的改变起到了明显地抑制液舱晃荡的效果。     

Numerical simulation of sloshing waves in a 3D tank based on a finite element method

The sloshing waves in a three dimensional (3D) tank are analysed using a finite element method based on the fully non-linear wave potential theory. When the tank is undergoing two dimensional (2D) motion, the calculated results are found to be in very good agreement with other published data. Extensive calculation has been made for the tank in 3D motion. As in 2D motion, in addition to normal standing waves, travelling waves and bores are also observed. It is found that high pressures occur in various circumstances, which could have important implications for many engineering designs.     

Effects of perforated baffle on reducing sloshing in rectangular tank: Experimental and numerical study

A liquid sloshing experimental rig driven by a wave-maker is designed and built to study liquid sloshing problems in a rectangular liquid tank with perforated baffle. A series of experiments are conducted in this experimental rig to estimate the free surface fluctuation and pressure distribution by changing external excitation frequency of the shaking table. An in-house CFD code is also used in this study to simulate the liquid sloshing in three-dimensional （3D） rectangular tank with perforated baffle. Good agreements of free surface elevation and pressure between the numerical results and the experimental data are obtained and presented. Spectral analysis of the time history of free surface elevation is conducted by using the fast Fourier transformation.     

液舱三维晃荡运动二阶共振理论解及特性分析

从无旋运动的理论出发,并利用微扰法,推导了液舱三维晃荡运动二阶共振问题的理论解。考虑纵荡和横荡运动情况,对液舱三维晃荡二阶共振问题进行了分析。当两个晃荡方向的和频(即其外部激发频率的和)或差频(即其外部激发频率的差值)等于液舱固有频率时,二阶共振发生;当某一晃荡方向(横荡或纵荡)外部激发频率与另一晃荡方向(纵荡或横荡方向)液舱某一固有频率的和或差值等于液舱另一固有频率时,二阶共振也会发生。进一步研究了各个二阶共振激发频率下水深变化对晃荡振幅的影响。结果表明,对于两个晃荡方向外部激发频率的和频和单一晃荡方向(纵荡或横荡)某一个激发频率与另一晃荡方向(横荡或纵荡)某一个属于奇模的固有频率的和频所引发的共振情况,水深变化对共振振幅大小的影响比较大;而对于相应差频所引发的共振情况,水深变化对共振振幅大小的影响比较小。     

液舱晃荡载荷计算与强度校核系统研究

随着薄膜型LNG运输船的需求量不断增加,晃荡载荷已成为船舶安全性研究的重要内容之一。本文结合中国船级社规范所推荐的公式,对薄膜型LNG船晃荡水平的载荷进行研究,提出载荷计算方法和流程,在MSC.patran的基础上结合二次开发语言PCL,设计了一套晃荡载荷计算与校核系统。该系统对有限元模型进行前后处理,设计了舱室识别算法来搜索晃荡载荷的作用域,实现薄膜型LNG船晃荡载荷的自动计算与施加,完成屈服强度评估。通过算例测试证明本系统自动计算结果的有效性和准确性,可以为工程设计人员大大节约工作量,大幅度提高工作效率。     

Sloshing in a three-dimensional rectangular tank: Numerical simulation and experimental validation

Pressure variations and three-dimensional effects on liquid sloshing loads in a moving partially filled rectangular tank have been carried out numerically and experimentally. A numerical algorithm based on the volume of fluid (VOF) technique is used to study the non-linear behavior and damping characteristics of liquid sloshing. A moving coordinate system is used to include the non-linearity and avoid the complex boundary conditions of moving walls. The numerical model solves the complete Navier–Stokes equations in primitive variables by using of the finite difference approximations. In order to mitigate a series of discrete impacts, the signal computed is averaged over several time steps. In order to assess the accuracy of the method used, computations are compared with the experimental results. Several configurations of both baffled and unbaffled tanks are studied. Comparisons show good agreement for both impact and non- impact type slosh loads in the cases investigated.     

Vortex generation and evolution in water waves propagating over a submerged rectangular obstacle: Part I. Solitary waves

Interactions between a solitary wave and a submerged rectangular obstacle are investigated both experimentally and numerically. The Particle Image Velocimetry (PIV) technique is used to measure the velocity field in the vicinity of the obstacle. The generation and evolution of vortices due to flow separation at the corners of the obstacle are recorded and analyzed. It is found that although the size of the vortex at the weatherside of the obstacle is smaller than that at the leeside, the turbulence intensity is, however, stronger. A numerical model, based on the Reynolds Averaged Navier–Stokes (RANS) equations with a k–ϵ turbulence model, is first verified with the measurements. Overall, the agreement between the numerical results and laboratory velocity measurements is good. Using the RANS model, a series of additional numerical experiments with different wave heights and different heights of the rectangular obstacle are then performed to test the importance of the energy dissipation due to the generation of vortices. The corresponding wave transmission coefficient, the wave reflection coefficient and the energy dissipation coefficient are calculated and compared with solutions based on the potential flow theory. As the height of the obstacle increases to D/h=0.7, the energy dissipation inside the vortices can reach nearly 15% of the incoming wave energy.     

Chromium(III) — chromium(VI) interconversions in seawater

The stable form of dissolved chromium in oxygenated seawater is Cr(VI). But Cr(III)-species are also present at an analytically significant level. It is shown that Cr(III) is oxidized only slowly by dissolved oxygen, and that manganese oxide is a strong catalyst for such oxidation. However, the low oceanic concentration of suspended MnO₂ impedes the conclusion that the former process is quantitatively less important than the latter one. The distribution of dissolved chromium among Cr(VI)- and Cr(III)-species is probably kinetically controlled.     

Linear and nonlinear irregular waves and forces in a numerical wave tank

A time-domain higher-order boundary element scheme was utilized to simulate the linear and nonlinear irregular waves and diffractions due to a structure. Upon the second-order irregular waves with four Airy wave components being fed through the inflow boundary, the fully nonlinear boundary problem was solved in a time-marching scheme. The open boundary was modeled by combining an absorbing beach and the stretching technique. The proposed numerical scheme was verified by simulating the linear regular and irregular waves. The scheme was further applied to compute the linear and nonlinear irregular wave diffraction forces acting on a vertical truncated circular cylinder. The nonlinear results were also verified by checking the accuracy of the nonlinear simulation.     

Interactions between fully nonlinear water waves and cylinder arrays in a wave tank

Fully nonlinear interactions between water waves and vertical cylinder arrays in a numerical tank are studied based on a finite element method (FEM). The three-dimensional (3D) mesh is constructed through an extension of a 2D Delaunay surface grid along the vertical line. The velocity potential is obtained by solving a linear matrix system of FEM, and a difference scheme is then used to calculate the velocity on the free surface to track its movement. Waves and hydrodynamic forces are obtained for both bottom mounted and truncated cylinders. The simulations have provided many results to show the nature of mutual interference between cylinders in arrays and its effects on waves and forces at the nearly trapped mode frequency. The effect of the tank wall on waves and forces has been investigated, and the nonlinear features of waves and forces have also been discussed.     

Vortex generation and evolution in water waves propagating over a submerged rectangular obstacle: Part II: Cnoidal waves

Vortex generation and evolution due to flow separation around a submerged rectangular obstacle under incoming cnoidal waves is investigated both experimentally and numerically. The Particle Image Velocimetry (PIV) technique is used in the measurement. Based on the PIV data, a characteristic velocity, phrased in terms of incoming wave height, phase speed, dimension of the obstacle, and a local Reynolds number are proposed to describe the intensity of vortex. The numerical model, which solves the two dimensional Reynolds Averaged Navier Stokes (RANS) equations, is used to further study the effects of wave period on the vortex intensity. Measurements for the mean and turbulent velocity fields further indicate that the time history of the intensity of fluid turbulence is closely related to that of the vortex intensity.     

Sloshing waves and resonance modes of fluid in a 3D tank by a time-independent finite difference method

A 3D time-independent finite difference method is developed to solve for wave sloshing in a three-dimensional tank excited by coupled surge and sway motions. The 3D equations of fluid motion are derived in a moving coordinate system. The three-dimensional tank, with an arbitrary depth and a square base, is subjected to a range of excitation frequencies with motions that exhibit multiple degrees of freedom. For demonstration purposes the numerical scheme is validated by a benchmark study. Five types of sloshing waves were observed when the tank is excited by various excitation frequencies. A spectral analysis identified the resonant frequencies of each type of wave and the results show a strong correlation between resonant modes and the occurrence of the sloshing wave types. The method can be used to simulate fluid sloshing in a 3D tank with six-degrees of freedom.     

Oil and fishing — conflict in the North Sea

A programme of research is underway at Aberdeen University on the conflicts arising in the North Sea between the fishing and offshore oil and gas industries. This research note summarizes the work that has been undertaken to date and the main aspects on which future work will concentrate.