Structural Analogy for Dynamic Analysis of Liquid Sloshing

The analogy between the wave equation of liquid and the Navier equations of structural elasticity is examined in detail. By introducing appropriate parameters, the structural counterpart of the liquid sloshing model can be easily built. Therefore, the dynamic analysis of liquid sloshing can be reduced to that of structural elasticity, and the existing FEM structural analysis computer programs can be applied to liquid sloshing analysis without any modification. The present method also reveals the internal relationship between liquid sloshing and structural vibration. The effectiveness and reliability of the method is illustrated by the numerical example.

A Numerical Study of Liquid Sloshing in a Two-dimensional Tank under External Excitations

In this research, liquid sloshing behavior in a 2-D rectangular tank was simulated using ANSYS-FLUENT software subject to single or multiple-coupled external excitations （such as sway coupled with roll, and sway and roll coupled with heave）. The volume of fluid （VOF） method was used to track the free surface of sloshing. External excitation was imposed through the motion of the tank by using the dynamic mesh technique. The study shows that if the tank is subjected to multiple coupled excitations and resonant excitation frequencies, liquid sloshing will become violent and sloshing loads, including impact on the top wall, will be intensified.

Coupling Frequency of the Liquid Sloshing in a Cylindrical Tank with a Flexible Baffle

The coupling oscillation of a liquid in a cylindrical tank with an elastic slosh baffle is investigated. Free surface conditions are considered in the study. The complexity of the coupled boundary-value problem for the liquid and elastic damping spacer results in significant analytical difficulties. Two different velocity potential functions are respectively used in the liquid domain above, or below the damping spacer. A coupled frequency equation is obtained by using the pair of velocity potential functions. The numerical and theoretical analysis show that the natural frequency changes according to the location and stiffness of the spacer. Results indicate that the frequency coupling between damping spacer and sloshing liquid is obvious near the free liquid surface. It is shown that the coupling frequency increases with the increase of damping baffle rigidity.

Numerical investigation of liquid sloshing in FLNG membrane tanks with various bottom slopes

To analyze the bottom slope's effect on the sloshing liquid in floating liquefied natural gas(FLNG)membrane tanks,a simulation model is built and applied to describe the liquid behavior in a sloshing container.The free surface motion is simulated by the volume-of-fluid method and the standard k-εturbulence model.Experimental data and numerical results from references are used to validate the accuracy of the proposed simulation model.To study the influence of the sloped bottom on the liquid sloshing,different slope sizes and filling ratios are numerically simulated at the lowest natural frequency.The results reveal that the natural frequency can be determined by the average peak values of hydrodynamic parameters.The natural frequency and pressure loading on the tank walls decrease with the increase in the slope size.The peak pressure on the wall decreases by 5.45 kPa with the increase in the slope ratio from 5%to 20%.However,the relationship between the peak pressure and slope ratio is more significant with lower filling rates.Liquid behavior is more stable and independent with the change of the slope structure at a high filling rate(60%).The results of numerical simulation and modeling are expected to provide reference data for the design and operation of the FLNG system.

Numerical simulation of liquid sloshing in a spherical tank by MPS method

This paper investigates the sloshing phenomena in a spherical liquid tank using the moving particle semi-implicit(MPS)method,a crucial study in fluid dynamics.Distinct from previous research focused on rectangular or LNG tanks,this work explores the unique motion patterns inherent to spherical geometries.The accuracy of our in-house MPS solver MLParticle-SJTU is validated against experimental data and finite volume method(FVM).And the MPS method reveals a closer alignment with experimental outcomes,which suggests that MPS method is particularly effective for modeling complex,non-linear fluid behaviors.Then the fluid’s response to excitation at its natural frequency is simulated,showcasing vigorous sloshing and rotational motion.Detailed analyses of the fluid motion are conducted by drawing streamline diagrams,velocity vector diagrams,and vorticity maps.The fluid’s motion response is explored using both time-domain and frequency-domain curves of the fluid centroid,as well as the sloshing force.

Review on the Progress and Issues in Liquid Tank Sloshing of Ships

With the development of large liquid cargo ships,liquid tank sloshing has gradually become a hot research topic in the area of shipping and ocean Engineering.Liquid tank sloshing,characterized by strong nonlinearity and randomness,not only affects the stability of the ship but also generates a huge impact force on the wall of the tank.To further investigate liquid tank sloshing,a comprehensive review is given on the research process of the most focused subjects of liquid sloshing.Summarizing the existing research will help to identify issues in the current field and provide useful references.The methods for investigating sloshing,the research progress and the situations worldwide are discussed.The advantages and defects of experiments and numerical simulations are also explored.The problems which need to be explored in the future are subsequently proposed.

Influence of baffle position on liquid sloshing during braking and turning of a tank truck

The influence of baffle position on liquid sloshing during the braking and turning of a tank truck was studied using a volume of fluid (VOF) model. The forces,their positions and weight distribution during braking and the forces and rolling moment during turning were calculated. The reliability of the calculation method was validated by comparisons with experimental results. The results showed that during braking,liquid splashes in the tank and the maximum forces and G (the ratio of weight acting on the front axle to the rear axle) are large when A (the ratio of the arch area above the baffle to the area of cross section)≤0.1. When A≥0.2,as the position of the baffle is lowered,the maximum of Fx (the force in direction x) first decreases then increases,and the maximum of Fy (the force in direction y) and G increase. During turning,liquid splashes in the tank and the maximum forces and M (the rolling moment) are large when D (the ratio of the arch area above the baffle to the area of cross section)≤0.2. When D≥0.3,as the position of the baffle is lowered,the maximums of Fy,Fz (the force in direction z) and M increase.

Numerical simulations of multi-layer-liquid sloshing by multiphase MPS method

In this paper,a multiphase moving particle semi-implicit(MPS)method is developed by introduction of various multiphase models into the original MPS method for single-phase flows,and is then applied to the numerical simulations of two-layer-liquid and three-layer-liquid sloshing in a 2-D rectangular tank under different external excitations.To validate the accuracy of the present multiphase MPS method,the qualitative and quantitative results,including wave profiles,interface elevations and impact pressures,are compared with experimental data and other numerical results,which show a good agreement.In particular,the evolutions of free surface and phase interfaces are compared and an obvious discrepancy is found,indicating that the phase interfaces may suffer from complex deformations while the free surface keeps relatively quiet.Moreover,the different modes of phase interface are observed under different excitation frequencies.Then,the violent sloshing of a three-layer-liquid system is simulated and the phenomenon of internal wave breaking is well captured,which mainly benefitted from the Lagrangian basis of MPS method and its advantage in multiphase flows with complex interfaces.Finally,the three-layer-liquid sloshing with a vertical baffle are simulated to investigate the suppressing effect of baffle on multi-layer-liquid sloshing.

Laboratory investigation of the hydroelastic effect on liquid sloshing in rectangular tanks

A sloshing experiment is conducted to study the hydroelastic effect in an elastic tank. For this purpose, a translational harmonic excitation is applied to a 2-D rectangular tank model. The lowest-order natural frequencies of the liquid in the tank are determined through the sweep test. The wave elevation and the sloshing pressure are obtained by changing the excitation frequency and the liquid depth. Then the characteristics and the variation of the elevation and the pressure are discussed. The results are compared with the experimental results and the theoretical calculations in a rigid tank. Our analysis indicates that, in the non- resonant cases, the elastic results, the rigid experimental results and the theoretical values are all close to each other. In contrast, under the resonant condition, the elastic experimental result is slightly smaller than the rigid one. Also, the theoretical values are smaller than the experimental results at the resonant frequency.

Viscous liquid sloshing damping in cylindrical container using a volume of fluid method

Liquid sloshing is a kind of very complicated free surface flow and exists widely in many fields.In order to calculate liquid sloshing damping precisely a volume of fluid method based on finite volume scheme is used to simulate free surface flows in partly filled cylindrical containers.A numerical method is pre-sented to simulate the movement of the free surface flow,in which a piecewise linear interface con-struction scheme and an unsplit Lagrangian advection scheme instead of Eulerian advection scheme are used.The damping performance of liquid sloshing in cylindrical containers under fundamental sloshing mode is investigated.There are four factors determining the surface-wave damping:free surface,boundary-layer,interior fluid and contact line.In order to study different contributions from these four factors to whole damping,several examples are simulated.No-slip and slip wall boundary conditions on both side wall and bottom wall of the cylindrical containers are studied to compare with the published results obtained by solving Stokes equations.In the present method the first three main factors can be considered.The simulation results show that the boundary-layer damping contribution increases while the interior fluid damping contribution decreases with increase of Reynolds number.

THE EQUIVALENT MECHANICAL MODELS OF LIQUID SLOSHING IN A SPHERICAL TANK WITH SPACER UNDER LOW GRAVITY

The dynamics of liquid sloshing in a spherical tank with spacer under low gravity was investigated. By extending the characteristic functions, the frequencies of liquid free sloshing, reaction force and moments on the wall of tank were obtained. By mechanical equivalent principle, the equivalent models of spring vibrator damper of liquid sloshing were given. The numerical results obtained show when a spacer is inserted in the tank, the sloshing frequency of liquid and the sloshing mass of liquid will decrease.

DYNAMIC CHARACTERISTICS OF LIQUID SLOSHING IN A RECTANGULAR TANK UNDER LOW GRAVITY

In this paper, the dynamics of liquid sloshing in a rectangular tank under low gravity was investigated. The basic frequencies and the velocity potentials of sloshing of liquid were obtained by expansion of the Fourier series. The characteristics of force and moment of the liquid acting on a rectangular tank were analyzed. The equivalent mechanical models of spring-vibrator-damper of sloshing of liquid were established with mechanical equivalent principle and numerical results were given. The numerical results show that the equivalent sloshing mass increases with the increase of gravity, and that the basic frequency and the equivalent sloshing mass increase with the increase of the liquid depth.

Numerical Simulation on Partially Liquid-Filled Sloshing with Baffle Under Different Density Ratios by the CLSVOF/IB Method

The density and viscosity ratios on partially liquid-filled sloshing with baffle have been investigated numerically in this study.As the key to success in the present simulation,the Coupled Level Set and the Volume of Fluid(CLSVOF)method and the Immersed Boundary(IB)method are used to capture gas/liquid and fluid/structure interfaces,respectively.Within the CLSVOF method,surface normal in weighting factors is calculated by the level set function,resulting in a more accurate solution.Furthermore,the Tangent of Hyperbola for INterface Capturing(THINC)coupled with the Weighted Linear Interface Calculation(WLIC)scheme is used for capturing moving interface.As a standard practice,we first validate the code by comparing it with experimental results of liquid sloshing,which involves large deformation of interface.In addition to the validation study of the present method,the problems of liquid sloshing with baffle are investigated to understand kinematics and dynamics behaviors under different density and viscosity ratios.

FORCES AND MOMENTS OF THE LIQUID FINITE AMPLITUDE SLOSHING IN A LIQUID-SOLID COUPLED SYSTEM

Nonlinear coupling dynamics between a spring-mass system and a finite amplitude sloshing system with liquid in a cylindrical tank is investigated. Based on a group of nonlinear coupling equations of six degrees of freedoms, analytical formulae of forces and moments of the liquid large amplitude sloshing were obtained. Nonlinearity of the forces and moments of the sloshing was induced by integrating on final configuration of liquid sloshing and the nonlinear terms in the liquid pressure formula. The symmetry between the formula of Ox and Oy direction proves that the derivation is correct. According to the coupled mechanism, the formulae are available in other liquid-solid coupled systems. Simulations and corresponding experimental results arecompared. It is shown that the forces and moments formulae by integrating on the final sloshing configuration are more reasonable. The omitted high-dimensional modal bases and high-order nonlinear terms and the complexity of sloshing damping are main sources of errors.

DYNAMIC CHARACTERISTICS OF LIQUID SLOSHING IN A TRANSVERSELY VIBRATING SPHERICAL TANK WITH A SPACER UNDER LOW GRAVITY

The dynamic characteristics of liquid sloshing in a transversely vibrating spherical tank with spacer under low gravity were investigated. By expanding the characteristic functions, the frequencies and velocity potentials of liquid free-sloshing were obtained. The dynamic equations and boundary conditions of liquid sloshing in the traverse vibrational spherical tank with spacer under low gravity were derived. By modifying the velocity potentials of liquid free-sloshing, the velocity potentials of liquid sloshing in the traverse vibrating spherical tank with a spacer were obtained. Furthermore, the forces and the moments acting on tank wall were given. The numerical results show when a spacer is inserted in the tank, the sloshing frequencies of liquid and the forces acting on tank wall will decrease, but the moment of force to the centre of the tank which is caused by the force of liquid acting on the spacer will occur.

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.

Numerical Simulation of Sloshing with Large Deforming Free Surface by MPS-LES Method

Moving particle semi-implicit （MPS） method is a fully Lagrangian particle method which can easily solve problems with violent free surface. Although it has demonstrated its advantage in ocean engineering applications, it still has some defects to be improved. In this paper, MPS method is extended to the large eddy simulation （LES） by coupling with a sub-particle-scale （SPS） turbulence model. The SPS turbulence model turns into the Reynolds stress terms in the filtered momentum equation, and the Smagorinsky model is introduced to describe the Reynolds stress terms. Although MPS method has the advantage in the simulation of the free surface flow, a lot of non-free surface particles are treated as free surface particles in the original MPS model. In this paper, we use a new free surface tracing method and the key point is ＂neighbor particle＂. In this new method, the zone around each particle is divided into eight parts, and the particle will be treated as a free surface particle as long as there are no ＂neighbor particles＂ in any two parts of the zone. As the number density parameter judging method has a high efficiency for the free surface particles tracing, we combine it with the neighbor detected method. First, we select out the particles which may be mistreated with high probabilities by using the number density parameter judging method. And then we deal with these particles with the neighbor detected method. By doing this, the new mixed free surface tracing method can reduce the mistreatment problem efficiently. The serious pressure fluctuation is an obvious defect in MPS method, and therefore an area-time average technique is used in this paper to remove the pressure fluctuation with a quite good result. With these improvements, the modified MPS-LES method is applied to simulate liquid sloshing problems with large deforming free surface. Results show that the modified MPS-LES method can simulate the large deforming free surface easily. It can not only capture the large impact pressure accurately on rolling tank wall but also can generate all physical phenomena successfully. The good agreement between numerical and experimental results proves that the modified MPS-LES method is a good CFD methodology in free surface flow simulations.

Finite Element Analysis to Two-Dimensional Nonlinear Sloshing Problems

A two-dimensional nonlinear sloshing problem is analyzed by means of the fully nonlinear theory and time domain second order theory of water waves. Liquid sloshing in a rectangular container Subjected to a horizontal excitation is simulated by the finite element method. Comparisons between the two theories are made based on their numerical results. It is found that good agreement is obtained for the case of small amplitude oscillation and obvious differences occur for large amplitude excitation. Even though, the second order solution can still exhibit typical nonlinear features of nonlinear wave and can be used instead of the fully nonlinear theory.

Nonlinear coupled dynamics of liquid-filled spherical container in microgravity

Nonlinear coupled dynamics of a liquid-filled spherical container in microgravity are investigated. The governing equations of the low-gravity liquid sloshing in a convex axisymmetrical container subjected to lateral excitation is obtained by the variational principle and solved with a modal analysis method. The variational formulas are transformed into a frequency equation in the form of a standard eigenvalue problem by the Galerkin method, in which admissible functions for the velocity potential and the liquid flee surface displacement are determined analytically in terms of the Gaussian hypergeometric series. The coupled dynamic equations of the liquid-filled container are derived using the Lagrange＇s method and are numerically solved. The time histories of the modal solutions are obtained in numerical simulations.

The Two-dimensional Study of the Interaction between Liquid loshing and Elastic Structures

Sloshing phenomenon in the liquid cargo carriers has caught the attention of researchers as the interaction between the sloshing waves and structure is one of the key point and difficulty in the study of sloshing.In this paper,we captured the free surface with a volume of fluid(VOF) method and then calculated the motions and responses of the structure by adopting the Reynolds-averaged Navier-Stokes(RANS) equations for the whole fluid domain.With the use of user defined functions(UDF) in Fluent,the interaction between fluid and structure was then simulated.As a reasonable simplification,the authors studied the response of a single cantilever in a tank under sloshing loads;Further study should pay more attention to the mechanisms of interaction between sloshing waves and elastic structures.