Coupled analysis of nonlinear sloshing and ship motions

A coupled numerical model considering nonlinear sloshing flows and the linear ship motions has been developed based on a boundary element method. Hydrodynamic performances of a tank containing internal fluid under regular wave excitations in sway are investigated by the present time-domain simulation model and comparative model tests. The numerical model features well the hydrodynamic performance of a tank and its internal sloshing flows obtained from the experiments. In particular, the numerical simulations of the strong nonlinear sloshing flows at the natural frequency have been validated. The influence of the excitation wave height and wave frequency on ship motions and internal sloshing has been investigated. The magnitude of the internal sloshing increases nonlinearly as the wave excitation increases. It is observed that the asymmetry of the internal sloshing relative to still water surface becomes more pronounced at higher wave excitation. The internal sloshing-induced wave elevation is found to be amplitude-modulated. The frequency of the amplitude modulation envelope is determined by the difference between the incident wave frequency and the natural frequency of the internal sloshing. Furthermore, the coupling mechanism between ship motions and internal sloshing is discussed.     

Finite element analysis of two-dimensional nonlinear sloshing problems in random excitations

A two-dimensional nonlinear random sloshing problem is analyzed by the fully nonlinear wave velocity potential theory based on the finite element method. A rectangular container filled with liquid subjected to specified horizontal random oscillations is studied. Both wave elevation and hydrodynamic force are obtained. The spectra of random waves and forces have also been investigated, and the effects of the peak frequencies and spectral width of the specified spectrum used for the generation of the random oscillations are discussed. It is found that the energy mainly concentrates at the natural frequencies of the container and is dominant at the ith order natural frequency when the peak frequency is close to the ith order natural frequency. Some results are compared between the fully nonlinear solutions, the linear solutions and the linear plus second-order solutions.     

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。     

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.     

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。     

Experimental Study on A Sloshing Mitigation Concept Using Floating Layers of Solid Foam Elements

A sloshing mitigation concept taking advantage of floating layers of solid foam elements is proposed in the present study. Physical experiments are carried out in a liquid tank to investigate the hydrodynamic mechanism of this concept. Effects of the foam-layer thickness, excitation amplitude, and excitation frequency on the sloshing properties are analyzed in detail. It is found that the floating layers of solid foam elements do not evidently affect the fundamental natural sloshing frequency of the liquid tank evidently among the considered cases. At the resonant condition, the maximum wave height and dynamic pressure are greatly reduced as the foam-layer thickness increases. Higher-order pressure components on the tank side gradually vanish with the increase of the foam-layer thickness. Cases with different excitation amplitudes are also analyzed. The phenomenon is observed when the wave breaking in the tank can be suppressed by solid foam elements.     

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.     

Experimental Study on Sloshing Characteristics in the Elastic Tank Based on Morlet Wavelet Transform

Hydroelastic effect of sloshing is studied through an experimental investigation. Different excitation frequencies are considered with low-fill-depth and large amplitude. Morlet wavelet transform is introduced to analyze the free surface elevations and sloshing pressures. It focuses on variations and distributions of the wavelet energy in elastic tanks. The evolutions of theoretical and experimental wavelet spectra are discussed and the corresponding Fourier spectrums are compared. Afterwards, average values of the wavelet spectra are extracted to do a quantitative study at various points. From the wavelet analysis, sloshing energies are mainly distributed around the external excitation frequency and expanded to high frequencies under violent condition. In resonance, experimental wavelet energy of the elevation in elastic tanks is obviously less than that in the rigid one; for sloshing pressures, the elastic wavelet energy close to the rigid one and conspicuous impulse is observed. It recommends engineers to concern the primary natural frequency and impulsive peak pressures.     

Parametric Vibration Analysis of Submerged Floating Tunnel Tension Legs

According to the characteristics of submerged floating tunnel anchored by tension legs,simplifying the tube as point mass and assuming that the tension leg is a nonlinear beam model hinged at both ends,the nonlinear vibration equation of the tension leg is derived.The equation is solved by the Galerkin method and Runge Kutta method.Subsequently,numerical analysis of typical submerged floating tunnel tension leg is carried out.It is shown that,the parametric vibration response of the submerged floating tunnel tension leg is related to the amplitude and frequency of the end excitation.Without considering axial resonance and transverse resonance,it is reasonable that higher order modes are abandoned and only the first three modes are considered.The axial resonance amplitude of the second or third order mode is equivalent to the first order mode axial resonance amplitude,which should not be ignored.     

Numerical Simulation of Waves Generated by Seafloor Movements

Waves generated by vertical seafloor movements are simulated by use of a fully nonlinear two-dimensional numerical wave tank. In the souree region, the seafloor lifts to a designated height by a generation function. The numerical tests show that the linear theory is only valid for estimating the wave behaviors induced by the seafloor movements with a small amplitude, and the fully nonlinear numerical model should be adopted in the simulation of the wave generation by the large amplitude seafloor movements. Without the background surface waves, many numerical tests on the stable maximum elevations η0^max are carried out by beth the linear theory and the fully nonlinear model. The results of two models are compared and analyzed. For the fully nonlinear model, the influences of the amplitudes and the horizontal lengths on η^max are stronger than that of the characteristic duration times. Furthermore, results reveal that there are significant differences be- tween the linear theory and the fully nonlinear model. When the influences of the background surface waves are considered, the corresponding numerical analyses reveal that with the fully nonlinear model the η0^max near-linearly varies with the wave amplitudes of the surface waves, and the η0^max has significant dependences on the wave lengths and the wave phases of the surface waves. In addition, the differences between the linear theory and the fully nonlinear model are still obvious, and these differences are significantly affected by the wave parameters of the background surface waves, such as the wave amplitude, the wave length and the wave phase.     

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。     

Finite-element computation of wave impact load due to a violent sloshing

A nonlinear sloshing problem is numerically simulated. During excessive sloshing, the sloshing-induced impact load can cause a critical damage on the tank structure. Recently the problem becomes an important research topic in LNG (Liquefied Natural Gas) Tanker and FPSO (Floating Production Storage Offloading) design. In this study, the wave impact load on the structure is obtained numerically by imposing the exact nonlinear free surface conditions and compared with that predicted by Morison's formula. As a theoretical model, a three-dimensional free surface flow in a tank is formulated in the scope of potential flow theory with the exact nonlinear free-surface conditions. A finite-element method based on Hamilton's principle is employed as a numerical method. The problem is treated as an initial-value problem. The nonlinear problem is numerically solved through an iterative scheme at each time step.     

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.     

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.).     

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.     

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.     

Experimental hydrodynamic investigation of a fixed offshore Oscillating Water Column device

Oscillating Water Column (OWC) is one of the pioneer devices in harnessing wave energy; however, it is not fully commercialized perhaps due to the complicated hydrodynamic behavior. Previous studies are significantly devoted to OWC devices located in nearshore and coastal regions where incident wave energy would experience dissipation more than offshore. In this paper, a 1:15 scaled fixed offshore OWC model is tested in a large towing tank of National Iranian Marine Laboratory. Wave spectrum shape effect on the efficiency of the OWC model is addressed. Moreover, the paper investigates the effects of the geometric and hydrodynamic factors on OWC device efficiency and uncovers new points in nonlinear interaction occurring inside the chamber; i.e. sloshing. The results indicate that shape of the spectrum inside the chamber is affected by the type of incident wave spectrum, especially for long waves. Pierson–Moskowitz spectrum leaded to higher efficiency rather than JONSWAP spectrum at longer incident wave periods. According to efficiency analysis, increasing wave height may lead to air leakage from the chamber followed by vortex generation, which is a reason for decreasing the efficiency of the OWC device. Furthermore, no shift in the resonant period of the OWC model, due to wave height increase, was observed at the opening ratios equal or smaller than 1.28%. Spectral analysis of water fluctuation inside the OWC chamber illustrates two modes of sloshing. The first mode can be seen at short period waves while the second mode is visible at long period waves. The sloshing modes approximately vanish by increasing draft value.     

三维完全非线性波浪水槽的数值模拟

用有限元求解拉普拉斯方程,建立了三维完全非线性数值波浪水槽.跟踪流体自由表面的方法为满足完全非线性自由表面条件的半拉格朗日法,对离散单元采用20节点的六面体二次等参数单元.并把数值计算结果与水面初始升高产生箱体内流体运动解析解和二阶斯托克斯波理论解进行了对比,结果表明该模型是稳定的、守恒的,能精确模拟非线性波浪的产生和传播.     

The effects of LNG-tank sloshing on the global motions of LNG carriers

The coupling and interactions between ship motion and inner-tank sloshing are investigated by a time-domain simulation scheme. For the time-domain simulation, the hydrodynamic coefficients and wave forces are obtained by a potential-thoery-based three-dimensional (3D) diffraction/radiation panel program in frequency domain. Then, the corresponding simulations of motions in time domain are carried out using convolution integral. The liquid sloshing in a tank is simulated in time domain by a Navier–Stokes solver. A finite difference method with SURF scheme is applied for the direct simulation of liquid sloshing. The computed sloshing force and moment are then applied as external excitations to the ship motion. The calculated ship motion is in turn inputted as the excitation for liquid sloshing, which is repeated for the ensuing time steps. For comparison, we independently developed a coupling scheme in the frequency domain using a sloshing code based on the linear potential theory. The hydrodynamic coefficients of the inner tanks are also obtained by a 3D panel program. The developed schemes are applied to a barge-type FPSO hull equipped with two partially filled tanks. The time-domain simulation results show similar trend when compared with MARIN's experimental results. The most pronounced coupling effects are the shift or split of peak-motion frequencies. It is also found that the pattern of coupling effects between vessel motion and liquid sloshing appreciably changes with filling level. The independent frequency-domain coupled analysis also shows the observed phenomena.     

一类相对转动时滞非线性动力系统的稳定性分析

建立了具有时变刚度、非线性阻尼和谐波激励的一类相对转动时滞非线性动力系统的动力学方程.采用多尺度法推导出时滞动力系统的分岔响应方程,运用奇异性理论研究系统结构稳定性,得到主共振稳态响应方程的转迁集以及不同参数下分岔曲线的拓扑结构.应用Hopf分岔理论讨论了时滞动力系统动态稳定性,给出了系统产生极限环的条件,最后用数值模拟的方法研究了时滞参数对系统极限环幅值的影响.