Sloshing Modes Research Articles

Experiments on sloshing mitigation using tuned oscillating baffles

We experimentally confirm the efficacy of submerged oscillating baffles for sloshing mitigation in a partially filled open container of water. As with the more typical applications of tuned mass dampers, when the natural frequency of the baffle is close to that of the first sloshing mode, the main resonance peak is split into two peaks of lower amplitude. The reduction in resonant sloshing amplitude is superior to that of a fixed baffle.

Three-dimensional numerical simulation of a cylindrical oscillating water column (OWC) device placed in a wave flume

A three-dimensional computational fluid dynamic (CFD) model is developed for simulating a cylindrical Oscillating Water Column (OWC) device for harvesting wave energy. The single-phase CFD model solves the Reynolds-Averaged Navier-Stokes (RANS) equations using the finite element method for simulating the wave motion and uses an aerodynamic model to simulate the flow through the air turbine. The model is implemented to simulate wave energy harvesting of a cylindrical OWC device. Through harmonic decomposition, it is found that the first harmonic wave surface elevation oscillates like a piston in the OWC chamber. However, both the amplitude and phase of the second harmonic wave surface elevation vary significantly along the wave direction in the OWC chamber, demonstrating a sloshing mode of the second harmonic. This study further proved the reduction of the capture width ratio caused by the transverse sloshing mode when the wavelength is the same as the wave flume width. The effect of the width of the wave flume on the peak CWR is found to be negligibly small when the wave flume width is three times the OWC diameter.

Modeling Torsional Stiffness and Damping of Liquid Slosh in Spacecraft Diaphragm Tanks

This study presents a pendulum-based empirical model of liquid slosh in spacecraft diaphragm tanks. Diaphragm torsional stiffness and effective damping are predicted within a range of lateral excitation frequency typical of launch or on-orbit operations. A pendulum-analog model with two primary slosh modes is extracted from diaphragm tank slosh data corresponding to steady-state triaxial force measurements at the support points of the tank under sinusoidal excitation. The approach leads to repeatable and accurate determination of pendulum model parameters. A gradient-based algorithm and a procedure for setting initial conditions for optimal parameter estimation are proposed to predict the liquid slosh forces and torques acting on the tank as frequency response functions of apparent mass and apparent mass moment. The pendulum models were assessed by comparing model-predicted force/torques with measurements, showing good agreement (1–5% prediction error in the time domain) under most test conditions (fill level, excitation amplitude, and frequency). An empirical model of the stiffness and damping is also proposed for the tested range of operational conditions. Because of the nonlinear nature of liquid slosh behavior in a diaphragm tank, both the stiffness and damping are local values for a given set of test conditions.

Experimental Study on Impulse-Induced Liquid Sloshing Modes and Frequencies in Rectangular Tanks

Storage tanks are extensively used containers in various industries for storing diverse liquids. The sloshing of liquid in these containers, often caused by earthquakes, is an emerging issue in the context of damage sustained by tanks. Thus, the seismic resilience of a tank is a critical performance criterion in tank design. Given the complexity of liquid sloshing, the fundamental characteristics of this phenomenon must be examined to gain insights into fundamental sloshing behavior. Liquid sloshing typically involves a combination of fundamental and higher sloshing modes, and the modes and frequencies of liquid sloshing are therefore crucial for assessing this complex behavior. In this study, a shaking table test was employed to analyze the characteristics of liquid sloshing modes. The shapes of the first three sloshing modes under unidirectional excitation were determined and used to validate their symmetry. The findings indicate that the first sloshing mode exerts a more pronounced influence on liquid sloshing during free oscillation induced by unidirectional and horizontal excitation. The modal contribution factor of the first sloshing mode escalates with an increase in the ratio of the liquid level to the length of the tank parallel to the shaking direction, eventually converging toward a constant value.

On the nonlinear moonpool responses in a drillship under regular heading waves

In this study, the nonlinear and viscous damping effects on the free-surface elevations of the recess-type moonpool inside a drillship are investigated. Based on a three-dimensional nonlinear potential flow (NPF3D) model, the nonlinear moonpool responses excited by regular heading waves are simulated in the time domain. To consider the vortex-shedding damping effects, induced by nonlinear moonpool responses, the pressure drop model of Chu et al. [Chu et al., “Effects of nonlinearity and viscous damping on the resonant responses in two-dimensional moonpools with a recess,” Appl. Ocean Res. 127, 103295 (2022)] is extended to three-dimensional and combined with NPF3D to form a viscous modified nonlinear potential flow model (referred to as NPF3D_V). The pressure drop model is composed of two parts in order to account for the energy loss from the first harmonic (piston-mode motions) and higher harmonics (sloshing-mode motions), respectively. The investigation focuses on the piston-mode resonance and secondary resonances of the first and second longitudinal sloshing modes. The response amplitude operators of the higher harmonics, by which the nonlinear effects are evaluated, are computed by the NPF3D_V model. It is found that the higher harmonics are noticeable at the excitation frequencies ωn0/m, where secondary resonances of the nth longitudinal sloshing mode are triggered. In addition, it is found that increasing the length of the recess can promote the nonlinear response of the moonpool significantly. For the moonpool with a long recess, the higher harmonics at secondary resonance are comparable to the first harmonics.

Sloshing reduction in microgravity: Thermocapillary-based control and passive baffles

In this work, we numerically analyze different thermocapillary-based strategies for active control of sloshing in microgravity. The fluid system considered is an open rectangular container holding a L×H=30 × 15 mm2 volume of liquid. A classical controller is implemented to reduce the natural sloshing motion of the system, which is characterized by its decay time τd. The controller produces an output signal ΔT that is applied anti-symmetrically at the lateral walls of the container, driving thermocapillary flow. Its performance is characterized via the functional P=(1−λ) τ̂+λ κ̂, which allows for a trade-off between the damping of unwanted sloshing modes and the cost of implementing the control, with normalized decay time τ̂ and cost κ̂, and modulated by λ∈[0, 1]. We optimize P for discrete values of λ and determine the optimal performance envelope. The results show that thermocapillary controllers are generally able to reduce τd by a 50% factor, with reasonable cost and controller output. A novel strategy combining thermocapillary controllers and passive baffles is further proposed and assessed. On their own, passive baffles can reduce the natural decay time of sloshing by an 80% factor at zero cost. In addition, they significantly alleviate the thermal requirements associated with the control, while providing additional improvements in τd. Finally, the aforementioned strategies are tested against a reboosting maneuver of the International Space Station, showing their potential for sloshing reduction in microgravity.

A new type of wave tank: prototype and proof of concept

The prototype and proof of concept of a new type of wave tank are described. If successful, this innovation would enable many wave tank experiments for only a few percent of current costs. Tank testing remains indispensable for new designs, and its high cost is a significant inhibitor of offshore innovation. To date, all wave tanks generate and propagate scaled but real water waves towards a physical model. This new type of tank instead reproduces the velocity field that such waves would have, but only near the model. In practice, this can be approximately achieved only for cases where the marine structure is much shorter than the wavelength, and its draft shallow compared to the wavelength and depth. In such cases the wavefield is time-varying but approximately spatially uniform near the structure. Therefore, it can be reproduced by moving a container of water like wave orbitals; and ensuring slosh modes are minimised. A prototype was built, and after initial challenges in controlling slosh excitation, a proof of concept was successfully achieved: a cylinder was excited by surge oscillations and its response appears consistent with rough calculations. Range of potential applications of the new type of wave tank, lessons learnt by prototyping, and next developments are discussed.

Viscous damping of steady-state resonant sloshing in a clean rectangular tank

A machine learning of the unknown a priori viscous damping, incorporated into the single-dominant nonlinear ‘inviscid’ modal theory by Faltinsen et al. (J. Fluid Mech., vol. 407, 2000, pp. 201–234) on resonant sloshing (the forcing frequency close to the lowest natural sloshing frequency) in a clean (no internal structures) rigid rectangular tank, is proposed. The learning procedure requires a set of measured phase lags between the harmonic horizontal tank excitation and the steady-state resonant wave response. A good consistency with experiments by Bäuerlein & Avila (J. Fluid Mech., vol. 925, 2021, A22) on the liquid-mass centre motions is shown. The latter confirms that the free-surface nonlinearity (causing an energy flow from the primary-excited to higher natural sloshing modes) and viscous damping of the higher natural sloshing modes matter, as well as that the damping rates can depend on the steady-state wave amplitude.

Learning the single-dominant modal system on resonant sloshing in a rectangular tank

A machine learning technique is proposed to derive the damping rate functions in the co-called single-dominant modal system describing the resonant liquid sloshing in a rectangular tank performing harmonic longitudinal motions. Its implementation is demonstrated for the steady-state (periodic) resonance waves when the modal system admits an analytical asymptotic solution depending on the introduced damping rate functions. Recent experiments by Bäuerlein & Avila (2021) are employed to show that the viscous damping of the higher natural sloshing modes matters, and the damping functions depend on the wave amplitude.

Thermocapillary-driven dynamics of a free surface in microgravity: Control of sloshing

Numerical simulations are used to analyze the dynamics of a free surface excited by thermal modulations at the lateral boundaries that generate a time-dependent thermocapillary flow. Fluid parameters are selected to be representative of 5 cSt silicone oil. Following the work of Gligor et al. [“Thermocapillary-driven dynamics of a free surface in microgravity: Response to steady and oscillatory thermal excitation,” Phys. Fluids 34, 042116 (2022)], the response of the free surface to oscillatory thermal excitation is characterized by the displacement of the contact points, and a frequency sweep is used to obtain a Bode-type diagram that reveals a resonance peak in the vicinity of the first sloshing mode. The ability of the thermocapillary flow to excite this sloshing mode suggests a control strategy that uses thermal modulations to dampen sloshing motion. After the response of the isothermal surface to a generic pulse-like inertial perturbation is measured, a classical proportional integral derivative control is implemented and the effect of its gains is considered separately. The efficacy of the controller is characterized by the decay time of the contact point oscillations and by a cost function. The effect of possible delays in the control loop is accounted for. Finally, a controller with a derivative gain is selected and used to dampen the motion induced by a reboosting maneuver of the International Space Station.

Numerical study of coupled slosh modes in a 3D vessel subjected to multi-directional excitations

Present study explores liquid sloshing in a three-dimensional (3D) rectangular tank subjected to surge and coupled surge, sway and heave excitations. A number of applications of practical interest in nuclear systems such as, spent fuel storage tanks, fast reactor vessels, pool type coolant etc will be benefited from such analysis. Effect of harmonic excitation is investigated emphasizing finite liquid depth when the internal resonance condition 1:1 is satisfied between the natural frequencies of dominant modes (1,0), (0,1) and (0,2). The numerical model was validated against available in–house experimental shake table studies. Temporal snapshots of velocity magnitudes and free surface time history at different locations are presented when the tank is excited under two different loading conditions. The amplification of diagonal slosh modes were observed under planar excitation. This analysis assumes significance, as the attendant hydrodynamic forces could eventually cause fatigue failure of the tank structure. Non-linear square-like, swirling and wave breaking features were observed, when the tank was subjected to earthquake excitation.

An upright bottomless vertical cylinder with baffles floating in waves

Damping of the surge and pitch motions, as well as the first lateral sloshing mode in a rigid free-floating upright circular dock with bilge boxes and open bottom is investigated. Model tests are carried out on a 0.80 m diameter model in regular waves with wave periods near the highest natural sloshing period, and the internal free-surface elevation and model’s rigid body motions are measured. Perforated and solid annular baffles of relatively small widths are also installed inside the dock at various submergences. The experimental results are compared to a semi-analytical approach, where a three-dimensional domain decomposition method based on linear potential flow theory is adopted to calculate the hydrodynamic coefficients and exciting forces in heave, surge and pitch. A reduced natural sloshing frequency, as well as a damping ratio estimated from the energy dissipated due to flow separation from the baffles, are introduced in the free-surface boundary conditions to model the effects of the baffle. It shows good agreement with experimental data when the ratio between the draft of the baffle and the internal radius of the cylinder is dB/a=0.27, and tends to under-predict the damping ratio for shallower drafts, most likely due to free-surface interactions. The solid baffle damps the sloshing response most efficiently, reducing the amplitude at the resonant peak by more than 56%.

Natural frequencies and modal shapes of three-dimensional moonpool with recess in infinite-depth and finite-depth waters

In this study new extensions of the theoretical models developed by Molin (2001) and Molin et al. (2018) are proposed to solve the resonance problem for three-dimensional circular and rectangular moonpools with recesses. Eigenfunction expansions are adopted to describe the velocity potentials in all the subdomains. Then, the velocity potentials and normal velocities are matched at the common boundaries such that the eigenvalue problems are formulated and solved, yielding the natural frequencies and associated modal shapes of the free surface. Both the models for infinite and finite water depths are derived for circular moonpools with recesses. Applications are also made for the three-dimensional rectangular moonpools with recesses. In addition, frozen-mode approximation is derived, which yields simple formulas for prediction of the natural frequencies for piston-mode resonances. The proposed models in this study are validated by comparing the obtained results with the experimental data and the results using other numerical models. Moreover, for the first sloshing mode in rectangular moonpool, simple approximation model is proposed based on the assumption that the half of the wavelength of a standing wave is the same as the moonpool length. The validity of the model is examined by comparing the solutions with the results by the diffraction–radiation code WAMIT.

Unravelling the large-scale circulation modes in turbulent Rayleigh-Bénard convection (a) Contribution to the Focus Issue Turbulent Thermal Convection edited by Mahendra Verma and Jörg Schumacher.

The large-scale circulation (LSC) is the most fundamental turbulent coherent flow structure in Rayleigh-Bénard convection. Further, LSCs provide the foundation upon which superstructures, the largest observable features in convective systems, are formed. In confined cylindrical geometries with diameter-to-height aspect ratios of , LSC dynamics are known to be governed by a quasi-two-dimensional, coupled horizontal sloshing and torsional (ST) oscillatory mode. In contrast, in cylinders, a three-dimensional jump rope vortex (JRV) motion dominates the LSC dynamics. Here, we use dynamic mode decomposition (DMD) on direct numerical simulation data of liquid metal to show that both types of modes co-exist in and cylinders but with opposite dynamical importance. Furthermore, with this analysis, we demonstrate that ST oscillations originate from a tilted elliptical mean flow superposed with a symmetric higher-order mode, which is connected to the four rolls in the plane perpendicular to the LSC in tanks.

Gravity–capillary multi-component wave patterns generated by a single-frequency wave-maker oscillation and subsequent resonances

Cross-waves are standing waves with crests perpendicular to a wave-maker; they are subharmonic waves excited by parametric instability. The modulational and chaotic behaviours of nonlinear cross-waves have been studied widely since the 1970s. Most of the previous work has focused on gravity waves where surface tension can be neglected. In this work we study cross-waves that are highly dependent on surface tension as well as gravity. By oscillating a planar wave-maker either vertically or horizontally with frequencies of 25 Hz through 40 Hz at one end of a rectangular basin, two-dimensional multi-component surface patterns are realized. Using the free-surface synthetic Schlieren technique to measure the surface elevations, multi-dimensional Fourier transforms are utilized to track the evolutionary spectrum of the water surface in both the temporal and spatial domains. Wavelet transforms are implemented to show the development of the various frequency components. Three-wave resonances with and without first subharmonics are observed for small nonlinearity. Three-dimensional oblique propagating cross-waves are generated at higher nonlinearity; unlike most previous cross-wave experiments, this staggered pattern propagates far downstream. Experimental evidence shows that two oblique propagating waves form a two-dimensional short-crested pattern, and that the lateral component of the waves develops into parametric sloshing modes corresponding to the width of the tank. Two regimes of nonlinear wave patterns, resonant triads and oblique propagating cross-waves, are delineated.

Challenges and procedures for experiments with steady and unsteady model velocities in a water towing tank

Challenges for steady and unsteady model motion in a large water towing tank and procedures to overcome them are the focus of the presented work. Some challenges are attributed to experiments conducted in water, whereas others are uniquely ascribed to a towing tank facility. Data convergence and outlier detection are studied based on the phase averaged pressure in order to ensure proper data quality. Pressure measurements are performed with non-surface mounted sensors. Therefore, inertia effects are detrimental when the attached tubing is not fully de-aired. A procedure for de-airing the pressure sensor cavity and its tubing is described. An iterative approach is developed that compensates for nonlinear distortion of the model’s velocity profile. Further, vibration effects are examined by distinguishing mechanical and flow-induced frequencies that scale with the instantaneous model velocity. Sloshing waves are excited, which are a function of the water basin size. The first sloshing mode defines the required sensor offset time in between test cases when prevailing sloshing waves have not fully decayed. This appropriate selection of sensor offset time reduces data scatter and enables a reasonable waiting time in between test cases. A skim plate installed just below the water surface offers a potential solution to alleviate surface wave effects over the model.

The Sloshing and Diapycnal Meridional Overturning Circulations in the Indian Ocean

AbstractThe meridional overturning circulation (MOC) seasonality in the Indian Ocean is investigated with the ocean state estimate product ECCO v4r3. The vertical movements of water parcels are predominantly due to the heaving of the isopycnals all over the basin except off the western coast. Aided by the linear propagation equation of long baroclinic Rossby waves, the driving factor determining the strength of the seasonal MOC in the Indian Ocean is identified as the zonally integrated Ekman pumping anomaly, rather than the Ekman transport concluded in earlier studies. A new concept of sloshing MOC is proposed, and its difference with the classic Eulerian MOC leads to the so-called diapycnal MOC. The striking resemblance of the Eulerian and sloshing MOCs implies the seasonal variation of the Eulerian MOC in the Indian Ocean is a sloshing mode. The shallow overturning cells manifest themselves in the diapycnal MOC as the most remarkable structure. New perspectives on the upwelling branch of the shallow overturn in the Indian Ocean are offered based on diapycnal vertical velocity. The discrepancy among the observation-based estimates on the bottom inflow across 32°S of the basin is interpreted with the seasonal sloshing mode. Consequently, the “missing mixing” in the deep Indian Ocean is attributed to the overestimated diapycnal volume fluxes. Decomposition of meridional heat transport (MHT) into sloshing and diapycnal components clearly shows the dominant mechanism of MHT in the Indian Ocean in various seasons.

Two Phase Linear Quadratic Gaussian Controller Design for Launch Vehicle Autopilot

The achievable payload of a launch vehicle is sensitive to the mass of the terminal stage which injects the payload into orbit, as well as to tracking errors introduced by the autopilot. The system is usually characterized by lightly damped slosh modes with uncertain parameters, whose stabilization is critical, while maintaining the overall performance of the system. Passive slosh stabilization involves introduction of mechanical baffles into the fluid tanks, which impacts the achievable payload mass. Optimal tracking performance along with active slosh stabilization is hence an ideal combination of autopilot design objectives. While optimal control methods are attractive for such systems, the lack of physical insight in the design procedure often limits the usefulness of these methods for launch vehicle autopilot design. A major challenge in stabilizing lightly damped slosh is the necessity to cater to very large mode magnitudes and phase swings exhibited by the system over the range of parameter perturbations. The significant contribution of this paper is the development of a two phase Linear Quadratic Gaussian approach, utilizing the physical insight into the system performance for tuning the weights in the performance index, to satisfy the time domain as well as frequency domain objectives. A robust design which exhibits superior time domain as well as frequency domain performance as compared to the benchmark classical controller is obtained, with near perfect phase stabilization of the lightly damped slosh dynamics.

A variational domain decomposition scheme for the natural sloshing modes in the baffled tanks

A variational domain-decomposition scheme is presented to obtain the analytically approximate natural sloshing modes in the rigid baffled tanks. The scheme consists of three steps: (i) dividing the unperturbed complicated liquid domain into several simple sub-domains by introducing artificial interfaces, and auxiliary artificial non-dimensional densities of fluids in these sub-domains are also introduced; (ii) obtaining spectral matrix problem by combing the approximate variational formulation of the approximate natural sloshing problem with the Trefftz method, eigenvalues of the spectral matrix problem are computed and plotted; (iii) extracting the pursued analytically approximate natural sloshing modes from the eigenfunctions given by the spectral matrix problem with artificial non-dimensional densities sufficiently close to 1. The fluid velocity potential in each sub-domain is expressed as a linear combination of a set of harmonic functions with unknown weight coefficients. A variational principle for natural sloshing of multi-layer immiscible fluid with multi free surface is developed and modified to be employed in step (ii). The extraction of the pursued analytically approximate natural sloshing modes is realized by classifying the obtained spectral parameters into two classes, one of which is dominated by the free-surface wave effect, whereas the other one is dominated by the effect of the artificial interfaces between adjacent sub-domains. To verify the present scheme, the results obtained are compared with results published or given by finite element analysis.

Tuned Sloshing Dampers With Large Rectangular Core Penetrations

Abstract Space restrictions at the top of tall buildings may necessitate using tuned sloshing dampers (TSD) tanks with large rectangular penetrations to accommodate the structural core of the tower. A finite element model is employed to predict the natural sloshing frequencies and mode shapes of liquid sloshing in a rectangular tank with a rectangular core. Equivalent mechanical properties are determined to predict the sloshing response. Frequency response predictions of wave heights, sloshing forces, and energy-dissipation per cycle agree with results from shake table testing conducted on a rectangular tank with a rectangular core. Energy dissipation due to flow around the core adds considerable damping to the liquid and is proportional to the response velocity-squared. Nonlinear coupling among sloshing modes results in multiple peaks in the frequency response plots near the fundamental resonant frequency. An interior core with a broad dimension in one direction substantially reduces the fundamental sloshing frequency and equivalent mechanical mass in the perpendicular direction; however, the fundamental sloshing frequency and equivalent mechanical mass in the parallel direction are only influenced marginally. Large rectangular cores reduce the proportion of the total water mass that is effective in controlling tower motion. A TSD with a rectangular penetrating core may enable a TSD option to be considered for the control of a tall building in cases where a traditional rectangular TSD is infeasible.