Resonant Sloshing Research Articles

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.

Steady-State Resonant Sloshing in an Upright Cylindrical Container Performing a Circular Orbital Motion

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. 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). This behavior type is supported by the existing experimental data. The wave elevations at the vertical wall are satisfactorily predicted except for a frequency range where the model test observations reported wave breaking and/or mean rotational flows.

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.

Nonlinear sloshing in rectangular tanks under forced excitation

A numerical code is developed based on potential flow theory to investigate nonlinear sloshing in rectangular Liquefied Natural Gas (LNG) tanks under forced excitation. Using this code, internal free-surface elevation and sloshing loads on liquid tanks can be obtained both in time domain and frequency domain. In the mathematical model, acceleration potential is solved in the calculation of pressure on tanks and the artificial damping model is adopted to account for energy dissipation during sloshing. The Boundary Element Method (BEM) is used to solve boundary value problems of both velocity potential and acceleration potential. Numerical calculation results are compared with published results to determine the efficiency and accuracy of the numerical code. Sloshing properties in partially filled rectangular and membrane tank under translational and rotational excitations are investigated. It is found that sloshing under horizontal and rotational excitations share similar properties. The first resonant mode and excitation frequency are the dominant response frequencies. Resonant sloshing will be excited when vertical excitation lies in the instability region. For liquid tank under rotational excitation, sloshing responses including amplitude and phase are sensitive to the location of the center of rotation. Moreover, experimental tests were conducted to analyze viscous effects on sloshing and to validate the feasibility of artificial damping models. The results show that the artificial damping model with modifying wall boundary conditions has better applicability in simulating sloshing under different fill levels and excitations.

Numerical analysis of the flow field in a sloshing tank with a horizontal perforated plate

Liquid sloshing is a type of free surface flow inside a partially filled water tank. Sloshing exerts a significant effect on the safety of liquid transport systems; in particular, it may cause large hydrodynamic loads when the frequency of the tank motion is close to the natural frequency of the tank. Perforated plates have recently been used to suppress the violent movement of liquids in a sloshing tank at resonant conditions. In this study, a numerical model based on OpenFOAM (Open Source Field Operation and Manipulation), an open source computed fluid dynamic code, is used to investigate resonant sloshing in a swaying tank with a submerged horizontal perforated plate. The numerical results of the free surface elevations are first verified using experimental data, and then the flow characteristics around the perforated plate and the fluid velocity distribution in the entire tank are examined using numerical examples. The results clearly show differences in sloshing motions under first-order and third-order resonant frequencies. This study provides a better understanding of the energy dissipation mechanism of a horizontal perforated plate in a swaying tank.

Tracking free surface and estimating sloshing force using image processing

Ultrasonic level sensors are commonly used to measure the motion of the free surface in fluid sloshing. They are used to measure the elevation of the free surface at a single point. The sloshing forces are generally measured with load sensors, which require two sets of measurements, with and without the fluid in the tank. This paper develops a method, which tracks the free surface motion during sloshing with a camera and uses the captured images to estimate the forces due to sloshing in a rectangular tank. One of the major assumptions is that the displacement input which causes sloshing is one dimensional and the resulting sloshing motion is two dimensional. For the method to correctly estimate the sloshing forces along the displacement input direction, sloshing should be around the resonant sloshing frequency. This new method can track the motion of the complete free surface rather than a single point. It estimates the sloshing forces using image processing and potential flow theory, without the need for a load cell measurement. Free surface shapes and sloshing force estimates obtained by image processing are compared with those measured by the sensors. Good agreement is observed for low amplitude sloshing around fundamental resonance frequency.

Resonant three-dimensional nonlinear sloshing in a square-base basin. Part 4. Oblique forcing and linear viscous damping

Faltinsenet al.(J. Fluid Mech., vol. 487, 2003, pp. 1–42) (henceforth, Part 1) examined an undamped nonlinear resonant steady-state sloshing in a square-base tank by developing an approximate (asymptotic) Narimanov–Moiseev-type multimodal theory. The focus was on longitudinal and diagonal harmonic tank excitations. Neglecting the linear viscous boundary-layer damping was justified for model tanks with breadths of the order of metres. However, nonlinear sloshing in clean tanks of smaller size (count in centimetres) may be affected by damping in finite depth conditions. Qualitative and quantitative properties of the damped resonant steady-state sloshing in a square-base tank are now studied by using the modal theory from Part 1 equipped with the linear damping terms. The tank harmonically oscillates along an arbitrary horizontal (oblique) direction. An analytical asymptotic steady-state undamped solution is derived and the corresponding response curves are analysed versus the forcing direction. When the tank width$=$breadth$=$ $L\sim 10$ cm, the surface tension effect on the free-surface dynamics can be neglected but the linear viscous damping should be included into the Narimanov–Moiseev nonlinear asymptotic modal theory. We analytically show that the steady-state damped sloshing possesses a series of distinguishing features so that, e.g. the square-like standing wave regime fully disappears and becomes replaced by swirling. Typical response curves of the damped steady-state resonant sloshing are studied for the liquid depth-to-width ratio exceeding 0.5. The computational results of the steady-state resonant response amplitudes are in a satisfactory agreement with observations and measurements by Ikedaet al.(J. Fluid Mech., vol. 700, 2012, pp. 304–328), which were conducted with a relatively small laboratory container.

Sloshing reduction in a swaying rectangular tank by an horizontal porous baffle

Sloshing occurs in a tank containing liquid with free surface. It may result in resonant liquid motions causing carrier-vessel instability or structural damage of tank walls. The horizontal porous baffle is capable of dissipating the sloshing energy and reducing the sloshing pressures acting on tank walls. In the context of linear potential theory, the MEEM (matched eigenfunction expansion method) is applied to obtain the analytic solution for the sloshing with porous horizontal baffle. A BEM (boundary element method) with the porous boundary conditions is also independently developed for double checking and the application to more general cases. Two baffle positions at the center and at both walls of a rectangular tank are considered for various porosities, lengths, and submergence depths. The theoretical prediction is then compared with a series of experiments conducted by authors using harmonically oscillated rectangular tank with various baffle porosities and submergence depths. The measured data reasonably correlate with the predicted values. It is found that horizontal porous baffles installed at both tank walls significantly suppress violent resonant sloshing responses compared to one installed at the tank center. Through parametric study using the developed numerical tools, the optimal porosity, length, and submergence depth can also be determined.

Resonant sloshing in an upright annular tank

Resonant sloshing in an upright annular tank is studied by using a new nonlinear modal theory, which is complete within the framework of the Narimanov–Moiseev asymptotics. The applicability is justified for a fairly deep liquid (the liquid-depth-to-outer-tank-radius ratio $1.5\lesssim h=\bar{h}/\bar{r}_{2}$) and away from the non-dimensional inner radii $r_{1}=\bar{r}_{1}/\bar{r}_{2}=0.08546$, 0.17618, 0.27826, 0.31323, 0.31855, 0.43444, 0.46015, 0.48434, 0.68655, 0.70118. The theory is used to describe steady-state (stable and unstable) resonant waves due to a harmonic excitation with the forcing frequency close to the lowest natural sloshing frequency. We show that the surge-sway-pitch-roll excitation is always of either longitudinal or elliptic type. Existing experimental results on the horizontally excited steady-state wave regimes in an upright circular tank ($r_{1}=0$) are utilised for validation. Inserting an inner pole with the radii $r_{1}\approx 0.25$ and 0.35 ($1.5\lesssim h$) causes that no stable swirling and/or irregular waves exist. The response curves for an elliptic-type excitation are examined versus the minor-axis forcing-amplitude component. Stable swirling is then expected being co- and counter-directed to the angular forcing direction. Passage to the rotary (circular) excitation keeps the co-directed swirling stable for all resonant forcing frequencies but the stable counter-directed swirling disappears.

Nonlinear simulation of resonant sloshing in wedged tanks using boundary element method

Nonlinear sloshing problems in wedged tanks are simulated based on the paralleled boundary element method. An improved semi-Lagrangian procedure is introduced for the free-surface updating near non-vertical tank walls. Two types of resonances are mainly considered. One is the classical resonance, which is induced by a lateral excitation on the tank. Featured sloshing phenomena in wedged tanks are observed compared with those in rectangular tanks. The other type is the Faraday waves due to the forced tank oscillation perpendicular to the undisturbed free-surface plane. Effects of the excitation frequency, acceleration amplitude, instability regions and initial free-surface disturbance on behaviours of the resonant sloshing in wedged tanks are investigated.

The dynamics of the floodwater and the damaged ship in waves

The interaction between the dynamics of the floodwater and the damaged ship in waves is investigated by applying an integrated method, which couples a seakeeping solver and a Navier-Stokes solver. To reveal the effects of the water flooding and the sloshing on the damaged ship behaviour, the motion of a Ro-Ro ferry in regular beam seas is simulated, including the ship under the intact condition with and without internal water and the ship under a damaged condition. It is found that the shift of the natural roll frequency of a damaged ship and the decrease of its roll response are mainly due to the water sloshing inside the compartment. The effect of the resonant sloshing leads to the presence of a ship's second peak response at higher frequencies and it is significantly reduced by the water flooding through the damaged opening. The influence of the flooding and the sloshing on the ship behaviour is small with a further increase of the wave frequency.

Experimental study on sloshing in a tank with an inner horizontal perforated plate

Sloshing in a liquid tank may result in ship instability or structural damage. Inner structures are often used to restrain liquid sloshing and prevent tank damage. To increase energy dissipation and reduce the forces acting on structures, a horizontal perforated plate was designed and incorporated into a rectangular liquid tank in this study. Experimental studies were conducted, and a tank with an inner submerged horizontal perforated plate was excited under different amplitudes and frequencies. The free surface elevations on the side-walls and the resonant frequencies were carefully examined. The experimental results indicate that the horizontal perforated plate can significantly restrain violent resonant sloshing in the tank under horizontal excitation.

浮体を用いたLNG タンクのスロッシング防止に関する研究

Recent developments with offshore terminals and floating liquefied natural gas (FLNG) production, storage and offloading applications have included the new requirement that the storage tanks be able to handle unrestricted filling with LNG. If membrane-type tanks are used in such new systems, the occurrence of sloshing in the half-load condition becomes one of the most important design issues. In this paper, the authors propose a simple anti-sloshing floating device to be installed inside a liquid cargo tank. By setting the device on the free surface of the liquid cargo, the original free surface is divided into two or more sub-surfaces, and resonant sloshing of the liquid can be avoided by changing the natural frequency of its motion. This floating device can follow the liquid height change in the tank during loading and unloading operations of the liquid cargo, a feature that is especially preferable in FLNG applications. The authors examine the performance of the proposed anti-sloshing device by a series of model experiments and numerical simulations, which indicate that the device's performance is very promising.

An effect of vertical sloshing on a fluid pressure and a surface displacement in a tuned magnetic fluid damper

A tuned fluid damper (TLD) is a dynamic absorber which uses hyd rodynamic force of sloshing in a container to dampen oscillations of structures. The TLD using a magnetic fluid as working fluid is called a tuned magnetic fluid damper (TMFD). Sloshing of the magnetic fluid is controlled by the ma gnetic field on the TMFD. It is important to get the state of the sloshing, thus the change of the fluid level and the fluid press ure of the magnetic fluid is studied in several magnetic fields . The change of them is affected by the resonant sloshing of vertical component and not by the resonance of the main (horizontal) sloshing.

Analytical modeling of liquid sloshing in a two-dimensional rectangular tank with a slat screen

Potential-flow theory is employed with linear free-surface conditions, multimodal method, and a screen-averaged pressure-drop condition to derive an analytical modal model describing the two-dimensional resonant liquid motions in a rectangular tank with a vertical slat-type screen in the tank middle. The tank is horizontally excited in a frequency range covering the two lowest natural sloshing frequencies. The model consists of a system of linear ordinary differential [modal] equations responsible for liquid sloshing in compartments, as well as a nonlinear ordinary differential equation describing the liquid flow between the compartments. New experimental model tests on steady-state wave elevations near the tank wall are reported for the solidity ratios 0.328 ≤ Sn ≤ 0.963 where Sn is the ratio between the solid area and the full area of the screen. The experiments generally support the applicability of the model. The discrepancy can be explained by the free-surface nonlinearity. The screen acts as a damping mechanism for low and intermediate solidity ratios, but it causes an increase in the lowest resonant sloshing frequency at higher solidity ratios as if the screen had been replaced by an unperforated wall.

The effect of fluid compressibility on the simulation of sloshing impacts

Resonant and near-resonant sway-induced sloshing flow in a rectangular container is used to compare various combinations of compressibility models for air and water. The numerical model is implemented in a commercial RANS computational fluid dynamics (CFD) code. A criterion based on wave propagation is developed to assess the importance of including fluid compressibility. For sloshing flows with low levels of fluid impact, this can be simulated with incompressible fluid models for both air and water. When modelling sloshing at low-filling levels with a travelling wave, which generates large air bubble entrainment, the choice of fluid compressibility model is shown to have a significant influence on pressure magnitude and frequency of oscillation required for structural assessment. Further comparisons with theoretical models show that a full thermal energy compressibility model is also required.

MODAL MODELLING OF THE NONLINEAR RESONANT FLUID SLOSHING IN A RECTANGULAR TANK II: SECONDARY RESONANCE

In Part I of this work [Math. Mod. Meth. Appl. Sci.15 (2005) 1431–1458], we studied periodic (steady-state) solutions of an asymptotic nonlinear modal system which describes two-dimensional resonant sloshing in a rectangular tank. The system was derived by Faltinsen et al. (2000) under the assumption that the primary excited (lowest) natural mode gives the largest contribution to the wave patterns. We found that this assumption is not true in a certain frequency domain due to internal (secondary) resonance leading to an amplification of the second mode. This frequency domain can also be identified for the critical depth-to-breath ratio h = 0.3368 … which was discussed in Part I. In Part II, this secondary resonance is modelled by a double-dominant modal system by Faltinsen and Timokha (2001). A comparative analysis with the results from Part I is presented. The emphasis is placed on the case of the mentioned critical ratio when a double turning point arises in the branching diagram. The appearance of the double turning point explains why classical laboratory experiments by Fultz (1962) underestimate the value of the critical depth.

Resonant hydraulic sloshing in a tank of variable depth

The forced resonant oscillations of a fluid in a tank of variable depth are considered within the hydraulic approximation. It is shown that for certain bottom topographies a continuous periodic output dominated by the first normal mode is possible. This contrasts with the case of a tank of constant depth, where hydraulic jumps are a feature of the motion. The amplitude and frequency of the output are connected by a cubic equation. The fluid response can act like that of a hard or soft spring, depending on the bottom topography. There is also a critical bottom topography that yields a higher order response amplitude.

Two-dimensional resonant piston-like sloshing in a moonpool

This paper presents combined theoretical and experimental studies of the two-dimensional piston-like steady-state motions of a fluid in a moonpool formed by two rectangular hulls (e.g. a dual pontoon or catamaran). Vertical harmonic excitation of the partly submerged structure in calm water is assumed. A high-precision analytically oriented linear-potential-flow method, which captures the singular behaviour of the velocity potential at the corner points of the rectangular structure, is developed. The linear steady-state results are compared with new experimental data and show generally satisfactory agreement. The influence of vortex shedding has been evaluated by using the local discrete-vortex method of Graham (1980). It was shown to be small. Thus, the discrepancy between the theory and experiment may be related to the free-surface nonlinearity.

Sloshing in a vertical circular cylindrical tank with an annular baffle. Part 2. Nonlinear resonant waves

Weakly nonlinear resonant sloshing in a circular cylindrical baffled tank with a fairly deep fluid depth (depth/radius ratio ≥ 1) is examined by using an asymptotic modal method, which is based on the Moiseev asymptotic ordering. The method generates a nonlinear asymptotic modal system coupling the time-dependent displacements of the linear natural modes. Emphasis is placed on quantifying the effective frequency domains of the steady-state resonant waves occurring due to lateral harmonic excitations, versus the size and the location of the baffle. The forthcoming Part 3 will focus on the vorticity stress at the sharp baffle edge and related generalisations of the present nonlinear modal system.