MPS方法数值模拟液舱晃荡问题

基于无网格粒子法MPS方法(moving particle semi-implicit method)研究了液舱晃荡问题。针对二维矩形液舱晃荡问题进行了数值验证,结果表明MPS方法能够很好地计算晃荡产生的拍击压力。同时将MPS方法应用到带隔板的液舱晃荡问题计算中,分析了二维和三维带隔板液舱晃荡问题。计算结果表明:隔板的存在很大程度地限制了流体的水平运动,隔板附近出现了自由面的翻卷、破碎和融合现象,MPS方法能够很好地模拟这些流动现象。计算得到的波高与实验测得的波高吻合较好,表明MPS方法模拟带隔板的晃荡问题具有一定的可靠性。     

Three-dimensional liquid sloshing in a tank with baffles

A numerical model has been developed to study three-dimensional (3D) liquid sloshing in a tank with baffles. The numerical model solves the spatially averaged Navier-Stokes equations, which are constructed on a non-inertial reference frame having six degree-of-freedom (DOF) of motions. The large-eddy-simulation (LES) approach is employed to model turbulence by using the Smagorinsky sub-grid scale (SGS) closure model. The two-step projection method is employed in the numerical solutions, aided by the Bi-CGSTAB technique to solve the pressure Poisson equation for the filtered pressure field. The second-order accurate volume-of-fluid (VOF) method is used to track the distorted and broken free surface. The baffles in the tank are modeled by the concept of virtual boundary force (VBF) method. The numerical model is first validated against the available analytical solution and experimental data for two-dimensional (2D) liquid sloshing in a tank without baffles. The 2D liquid sloshing in tanks with baffles is then investigated. The numerical results are compared with other results from available literatures. Good agreement is obtained. Finally, the model is used to study 3D liquid sloshing in a tank with vertical baffles. The effect of the baffle is investigated and discussed.     

Effect of anti-slosh baffles on free liquid oscillations in partially filled horizontal circular tanks

An exact two dimensional hydrodynamic analysis based on the linear potential theory is introduced to study the free liquid sloshing characteristics of transverse oscillation modes in a non-deformable horizontal circular cylindrical baffled container which is filled to an arbitrary depth with an inviscid incompressible liquid. Three common baffle configurations are considered: a pair of internal rigid horizontal side baffles of arbitrary extension installed at the free liquid surface, and a surface-piercing or a bottom-mounted vertical rigid baffle of arbitrary extension positioned along the tank vertical axis of symmetry. The problem solution is obtained by the method of successive conformal coordinate transformations, leading to standard truncated matrix eigenvalue problems on simple (rectangular) regions which are then solved numerically for the resonance eigen-frequencies. The effects of liquid fill level, baffle arrangement and length upon the three lowest antisymmetric and symmetric sloshing frequencies and the associated hydrodynamic pressure mode shapes are examined. Also, convergence of the adopted approach with respect to the fill condition, and baffle type/extension is discussed. Limiting cases are considered and the validity of results is established in comparison with the data in the existing literature.     

On the analysis of sloshing of water in rectangular containers: Numerical study and experimental validation

In this work the analysis of sloshing of water in rectangular open tanks has been extensively carried out. Two mathematical models are employed, respectively the Reynolds Averaged Navier Stokes Equations (RANSE) and the Shallow Water Equations (SWE). The RANSE are solved using a modified form of the well established MAC method (SIMAC) able to treat both the free surface motion and the viscous stresses over the rigid walls accurately. The Shallow Water Equations are solved by means of a simple and powerful algorithm (CE-SE) able to deal with large impacting waves over the tank walls.Successively, in order to validate the mentioned algorithms and for a better understanding of the sloshing phenomenon, experimental tests have been carried out using a 0.5 m breadth rectangular tank in periodic roll motion.It has been shown that RANSE provide more accurate solutions than SWE for small or moderate amplitudes of excitation. In particular in this paper it is proved that the shallow water approximation can be efficiently adopted within liquid depth to tank breadth RATIO = 0.15, when examining the sloshing problem. By increasing the water level inside the tank, results by SWE show large qualitative and quantitative disagreement with experiments. Nevertheless, in the case of large amplitude excitation, when sprays and large breaking waves are expected, SWE provide a fairly good estimate of the sloshing induced waves.Finally a simple baffle configuration inside the tank has been considered. By the analysis of numerical results, it has been observed that the presence of a vertical baffle at the middle of the tank dramatically changes the sloshing response compared to the unbaffled configuration. It produces a jump-like effect, resulting in a weak magnification of the dynamic loads on the vertical walls out of resonance, and a strong reduction of the dynamic loads in the resonance condition.     

PSME model of parametric excitation of two-layer liquid in a tank

Internal waves driven by external excitation constitute important phenomena that are often encountered in environmental fluid mechanics. In this study, a pseudospectral σ-transformation model is used to simulate parametric excitation of stratified liquid in a two-layer rectangular tank. The σ-transformation maps the physical domain including the liquid free surface, the interface between the liquid layers, and the bed, onto a pair of fixed rectangular computational domains corresponding to the two layers. The governing equation and boundary conditions are discretised using Chebyshev collocation formulae. The numerical model is verified for two analytical sloshing problems: horizontal excitation of constant density liquid in a rectangular tank, and vertical excitation of stratified liquid in a rectangular tank. A detailed analysis is provided of liquid motions in a shallow water tank due to excitations in the horizontal and the vertical directions. Also, the effect of pycnocline on the wave motions and patterns is studied. It is found that wave regimes and patterns are considerably influenced by the pycnocline, especially when the excitation frequency is large. The present study demonstrates that a pseudospectral σ-transformation is capable to model non-linear sloshing waves in a two-layer rectangular tank.     

Investigation of the Effects of Baffles on the Shallow Water Sloshing in A Rectangular Tank Using A 2D Turbulent ISPH Method

Liquid sloshing is a common phenomenon in the liquid tanks transportation. Liquid waves lead to fluctuating forces on the tank wall. Uncontrolled fluctuations lead to large forces and momentums. Baffles can control these fluctuations. A numerical method, which has been widely used to model this phenomenon, is Smoothed Particle Hydrodynamics(SPH). The Lagrangian nature of this method makes it suitable for simulating free surface flows. In the present study, an accurate Incompressible Smoothed Particle Hydrodynamics(ISPH) method is developed and improved using the kernel gradient correction tensors, particle shifting algorithms, k–ε turbulence model, and free surface particle detectors. Comparisons with the experimental data approve the ability of the present algorithm for simulating shallow water sloshing. The main aim of this study is to investigate the effects of the vertical baffle on the damping of liquid sloshing. Results show that baffles number has a major role in sloshing fluctuation damping.     

弹性液舱内液体晃荡实验研究

通过物理模型实验,对水平简谐激励下弹性矩形液舱内液体晃荡问题进行了研究。实验中给出了不同液深和不同激励振幅下弹性液舱内液体最低阶固有频率。采用影像采集与分析系统获得液舱内自由液面的形状和高度,通过压力采集系统得到舱壁上的压力分布。实验中分析了不同液深、不同激励频率下弹性和刚性液舱内自由液面高度和晃荡压力的变化特性。比较了弹性和刚性液舱内不同位置处自由液面高度随外界激励频率的变化规律,以及舱壁上不同位置处的压力峰值随外界激励频率的变化规律。结果表明:非共振情况下,弹性和刚性实验结果及理论值三者的波高和压力较为接近。共振情况下,弹性和刚性波高基本相等,在近舱壁处二者明显大于理论值,而在远离舱壁处二者与理论值存在一定差别;弹性压力较刚性压力略小,但二者与理论值差别较大。     

Numerical Simulation of Sloshing Using the MPS-FSI Method with Large Eddy Simulation

A numerical model has been developed to study sloshing of turbulent flow in a tank with elastic baffles. The Moving-Particle Semi-implicit method(MPS) is a kind of meshless Lagrangian calculation method. The large eddy simulation(LES) approach is employed to model the turbulence by using the Smagorinsky Sub-Particle Scale(SPS)closure model. This paper uses MPS-FSI method with LES to simulate the interaction between free surface flow and a thin elastic baffle in sloshing. Then, the numerical model is validated, and the numerical solution has good agreement with experimental data for sloshing in a tank with elastic baffles. Furthermore, under external excitations,the MPS is applied to viscous laminar flow and turbulent flow, with both the deformation of elastic baffles and the wave height of the free surface are compared with each other. Besides, the impact pressure with/without baffles and wave height of free surface are investigated and discussed in detail. Finally, preliminary simulations are carried out in the damage problem of elastic baffles, taking the advantage of the MPS-FSI method in computations of the fluid–structure interaction with large deformation.     

The air–water sloshing problem: Fundamental analysis and parametric studies on excitation and fill levels

The problem of liquid sloshing has gained recent attention with the proliferation of liquefied natural gas (LNG) carriers transporting liquids in partially filled tanks. Impact pressures caused by sloshing depend on the tank fill level, period and amplitude of oscillation of the tank. In this paper, we first present the rudiments of a linear potential theory for sloshing motions in a two-dimensional rectangular tank, due to small amplitude sway motions. Although this topic is fundamental, we clarify inconsistencies in the published literature and texts.Numerical investigations were carried out on the sloshing motions in a two-dimensional tank in the sway excitation. The fluid domain was modeled using a finite volume approximation, and the air–water interface was tracked using a volume-of-fluid (VOF) technique. Computational results for free surface elevation and impact pressure are found to be in good agreement with theory and published data. The fill levels were varied from 10% to 95%, and the excitation time periods were varied from 0.8 to 2.8 s for a constant sway amplitude of 0.25 m (peak–peak) at 1:30 scale. The results of the parametric study are compared with theoretical predictions and suggestions are made on incorporating sloshing effects in standard seakeeping analysis for LNG carriers.     

Boundary element analysis of liquid sloshing characteristics in axisymmetric tanks with various porous baffles

This paper aims to investigate the effects of the porous baffles on the suppression of sloshing for the tanks with axisymmetric geometries under lateral excitation. Based on the assumptions of inviscid, irrotational, incompressible liquid and small amplitude sloshing, an axisymmetric boundary element method (BEM) for 3D Laplace equation is derived by using the Green's theorem together with the weighted residual method. And a zoning method is employed to model fluid domain in the tanks with complex porous baffles. Meanwhile, the porous baffles are treated motioning together with the tanks, and the velocity across the porous baffle is assumed to be linearly proportional to the pressure gradient between each side of the porous baffle. And the mechanism of suppressing the sloshing response is mainly the energy dissipation of the fluid passing through the porous baffle. Moreover, the linear free surface boundary conditions are also used to solve the governing equations. Compared with other numerical methods, the most prominent advantage of the BEM in solving axisymmetric potential problem is that only the boundaries of half the cross-section instead of the entire problem domain should be discretized, which can cut down large amount of memory and time costs. The present method is verified by comparing the numerical results with the existing literatures, and excellent agreements are obtained. Meanwhile, the proposed models are applied to investigate the effects of the porous baffles on sloshing response in circular cylindrical, annular cylindrical and conical tanks. The effects of the porous baffle length, porous-effect parameter, installation angle and baffle height on the sloshing force, natural frequency and surface elevation are studied. Additionally, some typical sloshing pressure distributions, velocity potential contours and velocity fields are plotted. The results show that swirls at the tips of the baffles can be observed in many cases, and the top-mounted porous baffle makes more significant suppression effects on sloshing response than that of bottom-mounted porous baffle, while increasing the number of ring porous baffles can achieve better restraint effects on sloshing response. And increasing the baffle length of the horizontal wall-mounted ring porous baffle can significantly decrease the sloshing frequencies, as well as the first non-dimensional natural frequency decreases with decrease in porous-effect parameter of the coaxial porous baffle. In addition, remarkable effects on sloshing can be obtained when reasonable designed by selecting the optimal porous-effect parameter, installation angle and baffle height. And this paper can be a useful guide for the seismic design and analysis of many actual liquid storage tanks (such as the Advanced Passive PWR, large water cooling tower, etc.).     

Numerical simulation of three-dimensional sloshing phenomena using a finite difference method with marker-density scheme

In the present study, three-dimensional sloshing phenomena occurring in liquid cargo tanks are numerically simulated. The Navier-Stokes equations and continuity equation are used for the governing equations, and solved with a finite difference method in a rectangular fixed staggered mesh system. The positions of free surface are defined by the Marker density method satisfying the free-surface boundary conditions, and the flows of the gas and liquid regions are simulated simultaneously. The irregular leg length and star method is employed on the cells near the free surface for the computations of pressure. The computation results are compared with other experimental results to verify the consistency of the present numerical method, and the agreements are reasonably good. Furthermore, the flow characteristics inside a partially filled liquid tank of a real sized ship oscillating regularly and irregularly are computed to verify the possibility of practical application of the present method.     

A set of canonical problems in sloshing. Part 0: Experimental setup and data processing

In a series of attempts to research and document relevant sloshing type phenomena, a series of experiments have been conducted. The aim of this paper is to describe the setup and data processing of such experiments. A sloshing tank is subjected to angular motion. As a result pressure registers are obtained at several locations, together with the motion data, torque and a collection of image and video information. The experimental rig and the data acquisition systems are described. Useful information for experimental sloshing research practitioners is provided. This information is related to the liquids used in the experiments, the dying techniques, tank building processes, synchronization of acquisition systems, etc. A new procedure for reconstructing experimental data, that takes into account experimental uncertainties, is presented. This procedure is based on a least squares spline approximation of the data. Based on a deterministic approach to the first sloshing wave impact event in a sloshing experiment, an uncertainty analysis procedure of the associated first pressure peak value is described.     

容器固有频率对液体晃荡的影响

针对矩形容器内液体晃荡问题,采用了时域高阶边界元方法建立自由水面满足完全非线性边界条件的数学模型。求解中采用混合欧拉-拉格朗日方法追踪流体瞬时水面,运用四阶龙格库塔方法更新下一时间步的波面和速度势。通过将计算得到的波面结果与实验数据、解析解和已发表结果对比,吻合良好,验证了本方法的准确性。进而采用谱分析方法分析了波面时间历程,得到容器各阶固有频率对液体晃荡的影响。研究发现,基频对液体晃荡的影响最大,且非线性越强,更高阶容器固有频率的影响越大。     

Experimental Studies on Sloshing Mitigation Using Dual Perforated Floating Plates in A Rectangular Tank

The performance of dual perforated floating plates in a rectangular tank is investigated based on the model tests under different external excitations for different filling rates.It is found that dual perforated floating plates in the tank can remarkably mitigate violent resonant sloshing responses compared with the clean tank,especially when the external excitation frequency is in the vicinity of the first-order resonant frequency.Next,the parametric studies based on different filling rates and external excitation amplitudes are performed for the first-order resonant frequencies.The presence of dual perforated floating plates seldom shifts the sloshing natural frequencies.Further,dual perforated floating plates change the sloshing modes from the standing-wave mode in the clean tank to the Utube mode,which can arise from the sloshing reduction to some extent.     

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.     

A finite element model of interaction between viscous free surface waves and submerged cylinders

This paper describes the simulation of the flow of a viscous incompressible Newtonian liquid with a free surface. The Navier–Stokes equations are formulated using a streamline upwind Petrov–Galerkin scheme, and solved on a Q-tree-based finite element mesh that adapts to the moving free surface of the liquid. Special attention is given to fitting the mesh correctly to the free surface and solid wall boundaries. Fully non-linear free surface boundary conditions are implemented. Test cases include sloshing free surface motions in a rectangular tank and progressive waves over submerged cylinders.     

Complete two-dimensional analysis of sea-wave-induced fully non-linear sloshing fluid in a rigid floating tank

A time-independent finite-difference method and a fifth-order Runge–Kutta–Felhberg scheme were used to analyze the dynamic responses of sea-wave-induced fully non-linear sloshing fluid in a floating tank. The interaction effect between the fully non-linear sloshing fluid and the floating tank associated with coupled surge, heave and pitch motions of the tank are analyzed for the first time in the present pilot study. For the analysis of fluid motion in the tank, the coordinate system is moving (translating and rotating) with tank motion. The time-dependent water surface of the sloshing fluid is transformed to a horizontal plane and the flow field is mapped on to a rectangular region. The Euler equations as well as the fully non-linear kinematic free surface condition were used in the analysis of the sloshing fluid. The strip theory for linearized harmonic sea-wave loading was adopted to evaluate the regular encounter wave force. In addition, the dynamic coefficients used in the dynamic equations of tank motion were also derived based on strip theory and a harmonic motion of the tank. The characteristics of free and forced tank motions with and without the sloshing effect are studied. By the damping effect, the response of free oscillation will damp out and that of forced oscillation will approach a steady state. Without sea-wave action, the contribution of the sloshing load would enlarge the angular response of tank motion as well as the rise of free surface and the sloshing effect will delay the damping effect on angular displacement. On the contrary, under sea-wave action, the sloshing effect will decrease the dynamic response of tank motion and rise of free surface. The interaction, sloshing and coupling effects are found to be significant and should be considered in the analysis and design of floating tanks.     

Experimental and numerical investigation of sloshing using different free surface capturing methods

We investigated the use of numerical methods to predict liquid sloshing phenomena in a moving tank and compared our results to model test measurements. The numerical techniques for the free surface, based on the so-called finite Volume-of-Fluid (VoF) approach, comprised an incompressible VoF method, an incompressible coupled Level-Set and Volume-of-Fluid (clsVoF) method, and a compressible VoF method. We assessed the capability of these three numerical methods to achieve suitable numerical predictions of sloshing phenomena, specifically, air pockets and bubbles on the free surface inside a test tank. To observe the described sloshing phenomena, we simulated tank motions leading to well defined single impact wave motions. We performed repeated physical tests for validation purposes. Computed velocity and pressure time histories were compared to experimental data we obtained from Particle Image Velocimetry (PIV) and pressure sensor measurement. Grid sensitivity and turbulence model studies were performed. We demonstrated that the compressible VoF method was the most suitable method to obtain accurate predictions of sloshing phenomena.     

Two phase modal analysis of nonlinear sloshing in a rectangular container

Sloshing, or liquid free surface oscillation, in containers has many important applications in a variety of engineering fields. The modal method can be used to solve linear sloshing problems and is the most efficient reduced order method that has been used during the previous decade. In the present article, the modal method is used to solve a nonlinear sloshing problem. The method is based on a potential flow solution that implements a two-phase analysis on sloshing in a rectangular container. According to this method, the solution to the mass conservation equation, with a nonpenetration condition at the tank walls, results in velocity potential expansion; this is similar to the mode shapes used in modal method. The kinematic and dynamic boundary conditions create a set of two-space-dimensional differential equations with respect to time. The numerical solution of this set of differential equations, in the time domain, predicts the time response of interfacial oscillations. Modal method solutions for the time response of container sloshing due to lateral harmonic oscillations show a good agreement with experimental and numerical results reported in the literature.     

A modal approach to second-order analysis of sloshing using boundary element method

This paper aims at developing a modal approach for the non-linear analysis of sloshing in an arbitrary-shape tank under both horizontal and vertical excitations. For this purpose, the perturbation technique is employed and the potential flow is adopted as the liquid sloshing model. The first- and second-order kinematic and dynamic boundary conditions of the liquid-free surface are used along with a boundary element model which is formulated in terms of the velocity potential of the liquid-free surface. The boundary element model is used to determine the natural mode shapes of sloshing and their corresponding frequencies. Using the modal analysis technique, a non-linear model is presented for the calculation of the first- and second-order potential which can be used to obtain a reduced-order model for the sloshing dynamics. The results of the presented model are verified with the analytical solution for the second-order analysis of sloshing in a rectangular tank and very good results were obtained. Also, the second-order sloshing in some other example tanks with complex bed shapes is studied. The second-order resonance conditions of liquid sloshing in the example tanks are investigated and some conclusions are drawn.