Sloshing Modes Research Articles

Frequency analysis and control of sloshing coupled by elastic walls and foundation with smoothed particle hydrodynamics method

Broadband stochastic excitations from environment such as seismic activities, sea waves, wind gusts, and tremors can lead to the terrible damages of tall structures and buildings which usually have a low amount of vibration damping. Tuned liquid dampers (TLD) have been considered for around thirty years regarding their ability for passive suppression and absorb the vibration energy in such structures. In this paper, the fluid-structure interaction of a two-dimensional rectangular tank with elastic side walls and foundation is investigated by the smoothed particle hydrodynamics (SPH) method. First, the implementation of SPH for the frequency analysis of a rectangular container with flexible boundaries, partially filled with liquid, is described. The governing equations of the sloshing of an inviscid liquid in a two-dimensional Cartesian coordinate system coupled with the thin plate vibration on sides are linearized and solved by the SPH method. The sloshing and bulging modes of the system are derived and discussed in this paper. The results of the frequency spectrum of the sloshing tank coupled with the spring-damper system are analyzed and compared with the mechanical equivalent system. Results are validated by analytical benchmark and show the applicability as well as the simplicity of the SPH method for analytical vibration analysis of a fluid-structure coupled system. Then various control method including: Active mass damping (AMD) control designed based on proportional feedback control, the tuned liquid damper (TLD) control based on the analytical model, and a hybrid method of active tuned liquid damper (ATLD) are modeled and compared in a one-story structure. Results demonstrated that although the AMD is more efficient and stable in a specific range of feedback coefficient, the proposed ATLD is reliable in a wider range of control parameters.

Coupling Effects of Sloshing and Barge Motion in Variable Bathymetry

Abstract Two partially filled tanks fixed on rectangular barge floating in variable bathymetry is studied based on the linearized potential theory in the frequency domain. The internal sloshing motion is simulated by linearized superposition of natural sloshing modes. The motion of barge is solved by matching eigenfunctions expansion in the external fluid domain. The computational model is verified by experimental results. Coupling sway and roll motions of floating body with different filling levels are compared. The effects of both inclined bottom with different slopes and hump-shaped bottom with different heights are analyzed. Viewing from the results, coupling sway and roll motions of floating body are significantly affected by the shapes of bottoms.

On natural frequencies of three-dimensional moonpool of vessels in the fixed and free-floating conditions

The hydrodynamic interactions between the vessel motion and resonant wave response inside the three-dimensional moonpool are studied in the present paper. In particular, the shift of natural frequencies of a three-dimensional moonpool in the fixed and free-floating conditions is computed and examined systematically. Barges with different moonpool dimensions are analyzed to examine how the moonpool configurations affect the shift of resonant frequencies. Detailed analyses are performed for cases with and without recess. Numerical computations are performed for regular waves with wave headings θ=0∘, 45∘ and 90∘. Free-surface elevations inside the moonpool with respect to the incident wave frequencies are computed for both fixed and free-floating conditions. It is observed that natural frequencies of the moonpool in the free-floating condition are higher than those in the fixed condition. For cases without recess, the shift of resonant frequency is more prominent in piston mode resonance, in particular for shallow or long moonpools. In contrast, for cases with a recess, the shift of resonant frequency in the first sloshing mode is relatively larger than that in piston mode. In addition, the effects of motion RAOs on the free-surface elevation inside the moonpool in the free-floating condition are also examined. It is found that the moonpool resonance can induce a local hump in heave and pitch responses. Moreover, through deriving a modified frozen mode approximation model, we reveal that, comparing to the fixed condition, the vessel motion in the free-floating condition yields a reduction in the added mass due to the fluid underneath the moonpool so that the piston mode resonant frequency increases.

Boundary element method in nonlinear sloshing analysis for shells of revolution under longitudinal excitations

Processes of liquid sloshing in rigid shells of revolution filled with an ideal incompressible liquid are studied. The shells are subjected to longitudinal excitations. The liquid motion in these containers is supposed to be irrotational. The problems of liquid sloshing are considered in weakly nonlinear formulations when the free surface elevation is small compared with the shell radius. The spectral boundary problem on natural sloshing modes and frequencies is solved using the in-house computational tool based on boundary element methods. The linear and constant interpolations of unknown functions inside boundary elements are involved. The modes obtained are considered as basic functions for analysis of forced liquid vibrations. The numerical procedure based on boundary element method and the multimodal approach is developed for numerical analysis of nonlinear sloshing effects in rigid shells of revolution under longitudinal excitations. The free-surface elevation and velocity potential are expanded into infinite series with obtained basic functions and unknown time-dependent coefficients. The problems of liquid vibrations are reduced to solving nonlinear systems of second order ordinary differential equations about these coefficients that in linear formulation turns to the system of uncoupled Mathieu equations. The parametric oscillations are considered both in linear and nonlinear statements.

Nonlinear mode coupling and internal resonance observed in a dusty plasma

In this paper, we report the first experimental observation of internal resonance in a dusty plasma, which shows the intrinsic nonlinearities of dust interactions in plasmas. When driving a system of vertically aligned dust particle pairs in the vertical direction, the horizontal motion is found to be excited during onset of internal resonance when the higher-frequency horizontal sloshing mode is nonlinearly coupled to the vertical breathing mode through the 1:2 commensurable relation. A theoretical model of the nonlinear interaction of dust particles in plasma is also provided and the results of the theoretical model are shown to match experimental observations.

Controlling the seismic response of structures under near-field earthquakes with fluid/structure interaction of cylindrical liquid tanks

In this paper, the effectiveness of using nonlinear fluid/structure interaction for controlling the seismic response of the structures under near-field earthquake is investigated. For this purpose an SDOF Structure is considered which is equipped with the circular cylindrical liquid tank. Considering its fundamental mode of vibration, the structure is an idealised model of multi-degree of freedom system. Under 72 horizontal near-field earthquake excitations, the dynamic response of this system is examined studying three liquid sloshing modes. The dynamic response is investigated in the neighbourhood of 1:1 resonance between first unsymmetrical sloshing mode and SDOF structural mode. For this matter, most of the pulse-like earthquakes identified by Baker from NGA-West2 database and some other well-known near-field earthquakes have been used as base excitation. The Energy flow from the structural mode to the first unsymmetrical liquid sloshing mode is explored, and the proficiency of the appended liquid tanks in reducing the dynamic response of the structural system under pulse-liked earthquake is shown. The results show appropriate response reduction using the fluid oscillations for the structure subjected to pulse-like excitations.

A two-dimensional numerical and experimental study of piston and sloshing resonance in moonpools with recess

The piston and first sloshing modes of two-dimensional moonpools with recess are investigated. Dedicated forced heave experiments are carried out. Different recess lengths are tested from $1/4$ to $1/2$ of the length of the moonpool at the mean waterline. A theoretical model to calculate the natural frequencies is developed based on linearized potential flow theory and eigenfunction expansion. Two numerical methods are implemented: a boundary element method (BEM) and a Navier–Stokes solver (CFD). Both the BEM and CFD have linearized free-surface and body-boundary conditions. As expected, the BEM over-predicts the moonpool response significantly, in particular at the first sloshing mode. The CFD is in general able to predict the maximum moonpool response adequately, both at the piston and first sloshing modes. Both numerical methods fail to predict the Duffing-type behaviour at the first sloshing mode, due to the linearized free-surface conditions. The Duffing behaviour is more pronounced for the largest recess. The main source of damping in the proximity of the first sloshing mode is discussed.

Experimental and Numerical Study of the Sloshing Modes of Liquid Storage Tanks with the Virtual Mass Method

Experimental and Numerical Study of the Sloshing Modes of Liquid Storage Tanks with the Virtual Mass Method

Nonlinear response of vertical paired structure in complex plasma

The effect of the ion wake downstream of grains immersed in a flowing plasma is observed experimentally through the apparent non-reciprocal grain–grain interactions. Here we extend amplitude-frequency response analysis to examine the nonlinear components of the interaction of a dust particle pair aligned with the ion flow. The particle pair is modeled as two forced coupled oscillators, and a new coordinate system is introduced in which the breathing and sloshing modes are linearly decoupled. Multiple-scale analysis is used to derive analytical expressions for the response in the vertical direction for each mode in this coordinate system. By fitting the analytical expressions to experimentally measured response curves, the nonlinear part of the wake-modified particle–particle interaction is determined and identified as the source for the coupling between the sloshing and breathing modes. It is found that the restoring force acting on the downstream particle is more linear, thus stabilizing the oscillations of the downstream grain.

An experimental investigation on the dynamics of liquid sloshing in a rectangular tank and its interaction with an internal vertical pole

In the present study we perform shake table experiments to study the fluid structure interaction effects between the sloshing liquid and the internal structure. A partially filled storage tank is systematically subjected to (i) horizontal (ii) vertical and (iii) seismic excitations. To start with, forced horizontal excitations are imposed by varying the amplitude of excitation ranging from 0.001 L to 0.0075 L, where, L is the tank length. As the amplitude of excitation is increased, sub-harmonic wave features were observed at the wave crests, which indicate the presence of higher slosh modes. The resulting resonant free surface oscillations are validated against available theoretical and computational studies. Furthermore, the tank was subjected to vertical harmonic excitations at nth slosh frequency (ωn), where, n = 1, 2, 3 and 4 covers first four sloshing modes. The excitation frequency (Ωn = ωn∕ωv) of 0.5 and the corresponding forcing amplitudes (κv) are chosen such that, the combinations were with in the unstable range of Matheiu's instability diagram. For higher modes of vertical excitation, the free surface oscillations were found to be better damped. A detailed study of flow visuals, free surface fluctuations, dynamic pressure data, power spectral densities, structural responses etc reveal the nature of coupling with the container wall and the circular cylinder. The shake table is further programmed to simulate the well known El Centro earthquake excitation, where the storage tank is imposed with horizontal as well as combined horizontal and vertical components of excitation (at 50% of actual amplitude) to investigate the fluid structure interaction effects. When the partially filled storage tank is subjected to seismic excitation, spiky jet like features were observed over the free surface.

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.

Influence of moonpool on the total resistance of a drillship by the effect of water motions inside the moonpool

Moonpools are openings right through the hull from continuous deck to bottom of the ship, allowing equipment or mini-submarines to be put into the water at a location on the vessel with minimum ship motion. Open moonpools in a drillship are causing additional resistance when the ship is in forward speed. It was shown that the water inside the moonpool started to oscillate at forward speed. The water mass in the moonpool is subjected to sloshing and piston modes. The vertical motion is piston mode and the longitudinal one is called as sloshing mode. This water particle motion inside the moonpool is mainly depended on the geometry, moonpool depth, and encountered wave frequency. Out of this, moonpool geometry is one of the key factors for the performance of the moonpool. The varying cross-section geometry is one of the practically possible and economically feasible solutions to reduce the oscillation to a considerable level is attempted in this paper. Also the resistance caused by the moonpool and the free surface generated around the hull is investigated with the use of computer simulation.

Revisiting the experiment of a free-surface resonance of a liquid in a vibration tank using a nonlinear fluid–structure computational model

Using an advanced nonlinear fluid–structure reduced-order computational model, this work revisits and explains a resonance of the free surface of water contained in a thin elastic cylindrical tank, which was experimentally exhibited by Lindholm, 1962 and Abramson, 1966. The proposed simulation model allows the experimental setup to be reproduced. The structure undergoes large displacements and large deformations (geometrical nonlinear effects of the structure) that play an important role on the liquid vibrations. The experimental setup is simulated using a large-scale numerical model of the elastic cylindrical tank partially filled with water that is considered as a compressible fluid and that takes into account surface tension and sloshing effects. The results show that for a frequency external excitation in the frequency band [500,2,500]Hz of the fluid–structure system, unexpected high-amplitude sloshing vibrations are observed in the frequency band [0,150]Hz. The observed phenomenon, which cannot be reproduced with a linear fluid–structure model, is explained by the transfer of the vibrational energy from the frequency band of excitation into a low-frequency band (and then exciting the first sloshing modes) by a non-direct coupling mechanism between the structural modes and the sloshing modes.

Усталені резонансні коливання рідини у вертикальному циліндричному баці при орбітальних збуреннях

The nonlinear Narimanov-Moiseev multimodal equations are used to study the swirling-type resonant sloshing in a circular base container occurring due to an orbital (rotary) tank motion in the horizontal plane with the forcing frequency close to the lowest natural sloshing frequency. These equations are equipped with linear damping terms associated with the logarithmic decrements of the natural sloshing modes. The surface tension is neglected. An asymptotic steady-state solution is constructed and the response amplitude curves are analyzed to prove their hard-spring type behavior for the finite liquid depth (the mean liquid depth-to-the-radius ratio h>1). For the orbital forcing only swirling occurs. This behavior type is supported by the existing experimental data. Phase lags, which are piecewise functions along the continuous amplitude response curves in the undamped case, become of the non-constant character when the damping matters. The wave elevations at the vertical wall are satisfactory predicted except for a frequency range where the model test observations reported wave breaking and/or mean rotational flows.

Jump rope vortex in liquid metal convection

Understanding large-scale circulations (LSCs) in turbulent convective systems is important for the study of stars, planets, and in many industrial applications. The canonical model of the LSC is quasi-planar with additional horizontal sloshing and torsional modes [Brown E, Ahlers G (2009) J Fluid Mech 638:383-400; Funfschilling D, Ahlers G (2004) Phys Rev Lett 92:194502; Xi HD et al. (2009) Phys Rev Lett 102:044503; Zhou Q et al. (2009) J Fluid Mech 630:367-390]. Using liquid gallium as the working fluid, we show, via coupled laboratory-numerical experiments in tanks with aspect ratios greater than unity ([Formula: see text]), that the LSC takes instead the form of a "jump rope vortex," a strongly 3D mode that periodically orbits around the tank following a motion much like a jump rope on a playground. Further experiments show that this jump rope flow also exists in more viscous fluids such as water, albeit with a far smaller signal. Thus, this jump rope mode is an essential component of the turbulent convection that underlies our observations of natural systems.

Investigation on sloshing response of water rectangular tanks under horizontal and vertical near fault seismic excitations

In this study, sloshing response of liquid is investigated in partially filled rectangular tanks subjected to earthquake ground motions. To verify the numerical modeling, shaking-table tests were carried out in sub-scale tanks for different excitation amplitudes, frequencies and fill water levels. Using the calibrated models, nonlinear numerical simulations were performed for practical tanks under different horizontal and vertical near fault excitations. The results express that the liquid pressure induced on tank under vertical seismic loading can be increased especially in near-field earthquakes, however, its effect on sloshing wave height is negligible. Also the results for different tanks under both horizontal and vertical earthquakes show that free surface vertical displacement depends on the spectral response acceleration at the first fundamental period of sloshing behavior. Moreover, the induced pressure is strongly influenced by the earthquake peak ground acceleration and also by the spectrum response acceleration at the first sloshing mode.

An oil sloshing study: adaptive fixed-mesh ALE analysis and comparison with experiments

We report in this work a numerical analysis of the sloshing of a squared tank partially filled with a domestic vegetable oil. The tank is subject to controlled motions with a shake table. The free-surface evolution is captured using ultrasonic sensors and an image capturing method. Only confirmed data within the error range is reported. Filling depth, imposed amplitude and frequency effects on the sloshing wave pattern are specifically evaluated. The experiments also reveal the nonlinear wave behavior. The numerical model is based on a stabilized finite element method of the variational multi-scale type. The free-surface is captured using a level set technique developed to be used with adaptive meshes in Arbitrary Lagrangian–Eulerian framework. The numerical results are compared with the experiments for different sloshing conditions near the first sloshing mode. The simulations satisfactorily match the experiments, providing a reliable tool for the analysis of this kind of problems.

Secondary resonance of liquid sloshing in square-base tanks undergoing the circular orbit motion

The transient wave sloshing in the square-base tank horizontally shaken in a circular orbit is numerically studied. The liquid sloshing is simulated by the boundary element method (BEM) based on the fully-nonlinear potential-flow theory. The tank is firstly excited at the first odd natural sloshing frequency. Resonant swirling waves are observed travelling along the tank sides, when the even sloshing mode is aroused. Then, the tank is excited at half of the first even natural frequency. Techniques of the FFT filter and wavelet analysis are applied to distinct the wave components from the wave elevation histories, through which the occurrence of the secondary resonance is identified. During the secondary resonance, three typical wave motion patterns are observed, i.e. swirling waves, standing waves and double-peak travelling waves. Effects of the excitation amplitude and the liquid depth on the secondary resonance are investigated. Further, the secondary resonance by oscillating the tank at the difference of the first even and first odd natural frequency is studied. The first odd sloshing mode is found to contribute to the dominance of the even mode wave component during the secondary resonance.

Damped steady-state resonant sloshing in a circular base container

To describe the damped resonant sloshing in a circular base container, the nonlinear modal equations by Faltinsen et al (2016 J. Fluid Mech. 804 608–45) are equipped with linear damping terms associated with the logarithmic decrements of the natural sloshing modes. The damping coefficients express a cumulative effect of diverse dissipative phenomena. The surface tension is neglected, the container performs a prescribed periodic sway/surge/pitch/roll motion, the forcing frequency is close to the lowest natural sloshing frequency, and the mean liquid depth—to the tank radius ratio h ≳ 1.2. An asymptotic steady-state solution of the modal equations is derived; its stability is analysed by the linear Lyapunov method. The dominant amplitudes and the phase-lags of the two primary excited natural sloshing modes are governed by four (secular) nonlinear algebraic equations whose structure is the same as if the container were to perform an elliptic orbital horizontal translatory motion. The steady-state response curves are studied versus the semi-axes ratio of the horizontal elliptic orbit; a line segment (horizontal longitudinal) and a circle (rotary forcing) are two limiting cases. For the longitudinal forcing, planar (standing) and swirling steady-state waves are possible, otherwise, only swirling occurs. A focus is on the phase-lags, which are piecewise functions along the continuous amplitude response curves in the undamped case, but they become of the non-constant character when the damping matters. A comparison is done with measurements of the phase-lag by Royon-Lebeaud et al (2007 J. Fluid Mech. 577 467–94) (longitudinal forcing) to show that, if the damping rates are associated with the boundary layer at the wetted tank surface and the bulk viscosity, a satisfactory agreement is established with lower wave amplitudes but the cumulative damping effect must be larger to fit the experiments with increasing amplitudes. For elliptic forcing, stable swirling can be co- or counter-directed with the forcing direction. However, damping makes the counter-directed swirling impossible as the elliptic forcing orbit tends to a circle.

Liquid Induced Vibrations of Truncated Elastic Conical Shells with Elastic and Rigid Bottoms

A method of estimating natural modes and frequencies of vibrations for elastic shells of revolution conveying a liquid is proposed. The vibration modes of the liquid-filled elastic shells are presented as linear combinations of their own vibration modes without liquid. The explicit expression for fluid pressure is defined using Bernoulli’s integral and potential theory suppositions. Non-penetration, kinematic, and dynamic boundary conditions are applied at the shell walls and on a free liquid surface, respectively. The solution of the hydro-elasticity problem is found out using an effective technique based on coupled finite and boundary element methods. Computational vibration analysis of elastic truncated conical shells with different fixation conditions is accomplished. Sloshing and elastic walls frequencies and modes of liquid-filled truncated conical tanks are estimated. Both rigid and elastic bottoms of shells are considered. Some examples of numerical estimations are provided to testify the efficiency of the developed method