Free Surface Oscillations Research Articles

Studying Resonance Oscillations in the Eastern Part of the Posyet Bay

The field experiment with the installation of autonomous pressure gages in the Troitsa and Vityaz’ bays was held for studying the specific features of free surface oscillations in the Posyet Bay (the Sea of Japan). The interpretation and joint analysis of measurement data allowed obtaining the datasets of spatiotemporal parameters for free oscillations calculated using the spectral-difference model and for forced oscillations in the form of the response to the forcing with the periods of 10–50 minutes in the numerical model of shallow water with the difference approximation on the irregular triangular grid. The position of peaks on the model resonance curves and the values of frequencies of free oscillations are consistent with the position of clearly pronounced peaks of the energy spectrum from field data. This indicates the possibility of significant resonance enhancement of the amplitude of oscillations by periodic wind effects and incoming wave effects.

Numerical simulation of parametric liquid sloshing in a horizontally baffled rectangular container

Liquid sloshing is a problem of serious concern in partially filled tanks. Tank designers must ensure safe margins and develop methodologies to overcome a wide range of plausible situations related to transport, wind, earthquake loads etc. to assess the structural stability. In the present study, numerical simulations are carried out to investigate the sloshing dynamics of a partially filled rectangular container, subjected to vertical harmonic as well as seismic excitations. Unlike horizontal excitation, participation of higher modes is of prime concern in vertical (parametric) excitations. The present study numerically simulates and explores methodologies to control the slosh forces and free surface oscillations with the help of a baffle. Detailed numerical validations are carried out against other experimental and computational studies from the literature. Sloshing dynamics under imposed vertical harmonic excitations was investigated at its first and third modes. Based on a detailed study of transient wave profiles, force and pressure time histories, optimal baffle design was achieved. Optimal position of the baffle and its width are systematically obtained with reference to the quiescent free surface. The effectiveness of this baffle was tested against the well known Bhuj earthquake in India.

Tsunami and shelf resonance on the northern Chile coast

AbstractThis work presents the analysis of long waves resonance in two of the main cities along the northern coast of Chile, Arica, and Iquique, where a large tsunamigenic potential remains despite recent earthquakes. By combining a modal analysis solving the equation of free surface oscillations, with the analysis of background spectra derived from in situ measurements, the spatial and temporal structures of the modes are recovered. Comparison with spectra from three tsunamis of different characteristics shows that the modes found have been excited by past events. Moreover, the two locations show different response patterns. Arica is more sensitive to the characteristics of the tsunami source, whereas Iquique shows a smaller dependency and similar response for different tsunami events. Results are further compared with other methodologies with good agreement. These findings are relevant in characterizing the tsunami hazard in the area, and the methodology can be further extended to other regions along the Chilean coast.

An analytical solution for free liquid sloshing in a finite-length horizontal cylindrical container filled to an arbitrary depth

• Exact slosh analysis in an arbitrary-fill-depth finite horizontal cylindrical tank. • Full data for slosh frequencies as functions of fill depth and container length. • Extensive simulations exposing mode crossings among dissimilar frequency clusters. • Deep insights via benchmark general simulations and comparisons with other data. A general series-type theoretical formulation based on the linearized potential theory, the method of separation of variables, and the translational addition theorem for cylindrical Bessel functions is developed to study three-dimensional natural sloshing in a partially filled horizontally-mounted circular cylindrical tank of finite span. Assuming time-harmonic variations, the potential solutions associated with the Symmetric/Antisymmetric (S/A) modes of free liquid surface oscillations are first analytically expanded as series of bounded spatial functions with unknown modal coefficients. The impenetrability conditions of the rigid end-plates along with the free surface dynamic/kinematic boundary condition are then imposed. The zero-normal-velocity requirement of the lateral tank boundary is subsequently applied by innovative use of Graf's translational addition theorem for modified cylindrical Bessel functions. After truncation, four independent sets of homogeneous algebraic equations are obtained that are then numerically worked out for the natural sloshing eigen-frequencies and free surface oscillation mode shapes. Extensive numerical data include the first thirty six longitudinal/transverse Antisymmetric/Symmetric (AA, SA, AS, SS) dimensionless sloshing frequencies, for a wide range of liquid fill depths and container span to radius ratios. Also, the influence of fill depth on the free surface oscillation mode shapes is addressed through selected 2D images. Comprehensive numerical simulations illustrate the strong effects of container length and liquid fill depth on the calculated sloshing frequencies. It is revealed that the frequency branches with the same transverse mode number form a cluster that progressively merge together amid the tank fill-depth limits as the tank span ratio increases. On the other hand, when the tank length substantially decreases, the number of “frequency cross-overs” between various frequency clusters at certain liquid fill depths considerably increases. Moreover, primary advantages of proposed methodology in comparison to other approximate/numerical methods are explicitly pointed out, convergence of solution is tested, and accuracy/reliability of the results is demonstrated by comparisons with available data.

Rotary sloshing induced by impeller action in unbaffled stirred vessels

Unbaffled stirred tanks are characterized by a highly swirling fluid motion and, unless they are completely filled and closed with a lid, a central whirlpool takes place, whose depth depends on the impeller rotational speed. In some cases, periodic rotating waves appear, regardless of the axial symmetry of the system. These waves may result into undesirable periodic stresses for the structure, the vessel and the process connections.In this paper, a comprehensive experimental study on the onset and the characterization of periodic free-surface oscillations in unbaffled stirred vessels is carried out. Different wave shapes (or oscillation modes) may be observed, depending on the operating conditions. The oscillation mode of the free liquid surface is experimentally evaluated by both visual observation and measurements of the local dynamic pressure time traces. The oscillation frequency is identified from the time traces by means of simple Fast Fourier Transform.The natural frequency of each oscillation mode is in excellent agreement with predictions obtained by the theory of sloshing in case of still liquids in cylinders, but the effect of impeller is here analyzed in detail.Several experiments using different tank geometries, filling ratios and liquid viscosities were carried out. In all cases, the observed frequency was found to increase linearly with the impeller speed, with a slope that depends on the impeller geometry and on the oscillation mode under analysis. A simple predictive model is developed and validated by comparison with experimental data.

Incompressible Smoothed Particle Hydrodynamics Modeling and Investigation of Fluid Mixing in a Rectangular Stirred Tank with Free Surface

In the present study, a robust modified Incompressible Smoothed Particle Hydrodynamics (ISPH) method is developed and applied to model a fluid mixer with a rotating stirrer and a tank body with free surface and horizontal oscillations. The mixer consists of a rectangular tank in which a stirrer rotates to mix the fluid. For the fluid in the tank, the free surface condition is considered. According to the Reynolds number, it is assumed that the mixing process is turbulent, and so a turbulent viscosity is defined. Although the smoothed particle hydrodynamics (SPH) method can result in some complexities, it is generally an easy and appropriate method for modeling mixing flow, free surface flow, and moving body problems, which will simultaneously be applied in the present study. The method is improved with kernel gradient corrective tensor, shifting particle algorithm, and turbulent viscosity, and it is validated against other well-known test cases and problems and is applied to model the mixing phenomenon. This study aims to increase the mixing rate and decrease the mixing time of the described mixer. To do so, the effect of the rotation of the stirrer on the mixing process is first investigated, after which the linear oscillation of the body is added and then the optimal shape of the tank is examined. The results show that the stirrer’s rotating speed, the linear oscillation of the tank body, and streamlining all have major effects on the mixing rate.

A Computational Study of High-Speed Microdroplet Impact onto a Smooth Solid Surface

Numerical solutions of high-speed microdroplet impact onto a smooth solid surface are computed, using the interFoam VoF solver of the OpenFOAM CFD package. Toward the solid surface, the liquid microdroplet is moving with an impinging gas flow, simulating the situation of ink droplets being deposited onto substrate with a collimated mist jet in the Optomec Aerosol Jet printing process. The computed values of maximum spread factor, for the range of parameters of practical interest to Aerosol Jet printing, were found in very good agreement with some of the correlation formulas proposed by previous authors in the literature. Combining formulas selected from different authors with appropriate modifications yields a maximum spread factor formula that can be used for first-order evaluations of deposited in droplet size during the Aerosol Jet technology development. The computational results also illustrate droplet impact dynamics with lamella shape evolution throughout the spreading, receding-relaxation, and wetting equilibrium phases, consistent with that observed and described by many previous authors. This suggests a scale-invariant nature of the basic droplet impact behavior such that experiments with larger droplets at the same nondimensional parameter values may be considered for studying microdroplet impact dynamics. Significant free surface oscillations can be observed when the droplet viscosity is relatively low. The border line between periodic free surface oscillations and aperiodic creeping to capillary equilibrium free surface shape appears at the value of Ohnesorge number around 0.25. Droplet bouncing after receding is prompted with large contact angles at solid surface (as consistent with findings reported in the literature), but can be suppressed by increasing the droplet viscosity.

Hydrodynamic analysis of three–unit arrays of floating annular oscillating–water–column wave energy converters

This paper deals with the hydrodynamic analysis of an array of Oscillating Water Column (OWC) devices, made up of coaxial cylinders, which are floating either independently or as a unit forming a floating platform. The platform is considered either free – floating or as TLP configuration connected to the sea bottom. Numerical results concerning the three boundary value problems, namely, the diffraction, the motion – and the pressure – dependent radiation ones are given. They have been obtained through an analytical solution method using matched axisymmetric eigenfunction expansion formulations. In all cases the interaction phenomena with neighbouring bodies have been taken properly into account using the physical idea of multiple scattering. Numerical results for the first – and the mean second – order wave forces, the hydrodynamic interaction coefficients along with pressure hydrodynamic parameters, inner air pressure and free–surface oscillation amplitude inside and outside of each device are parametrically evaluated and supplemented by experimental data.

Effects of Magnetic Pressure Constraint on Dynamic Deformation and Oscillation Behavior of Metal Free Surface

A new method was presented to constrain deformation and oscillations of liquid metal free surface by using a high frequency magnetic field. A magnetic field generator was designed to investigate its feasibility using numerical simulation and physical simulation. The results indicate the feasibility of controlling bulge deformation of liquid metal surface using magnetic pressure. Sunken deformation with a slight fluctuation occurs on the surface when magnetic pressure acts on the static liquid metal surface. The largest amplitude remains within ±0.8 mm even if current reaches 1400 A. In case of the bump-type deformation, the magnetic pressure strengthens gravity field by the superposition effect, dissipates the kinetic energy of metal flow impacting on free surface in advance, reduces the velocity of free surface, and decreases the bulge height. On the region without magnetic field, the liquid metal surface rises and tends to be flat because of the significant damping effect on surface fluctuation. The constraint strength of the magnetic pressure increases with the augment of current intensity. However, different heights of bulge deformation should have a corresponding reasonable coil current for achieving the best constraint effect.

Liquid sloshing in partly-filled laterally-excited cylindrical tanks equipped with multi baffles

Baffles are used effectively to reduce the sloshing response of liquid in the liquid storage containers. This study is aimed at analysis of transient lateral sloshing in a partially-filled cylindrical tank with multi baffles including floating circular baffle, wall-mounted ring baffle, floating ring baffle and their combination form, and those baffles with inclination using a coupled multimodal method and scaled boundary finite element method (SBFEM). Slosh frequencies and mode shapes are initially estimated using the extending semi-analytical SBFEM by applying the linearized free surface boundary condition using the zoning method, where the liquid domain is firstly divided into several simple sub-domains so that the liquid velocity potential in each sub-domain has continuous boundary conditions of class C1. As a key point, a new type of local co-ordinate system for SBFEM with axisymmetric geometry is presented for each sub-domain. Based on the multimodal method, significant improvement in computational time is achieved by reducing the generalized eigenvalue problem to a standard one involving only the velocity potentials on the two-dimensional half free-surface length. The generalized coordinates of the free-surface oscillations under a lateral excitation are then obtained from superposition of the natural slosh modes. The sloshing mass and lateral slosh force are also formulated in terms of the generalized coordinates and hydrodynamic coefficients. The validity of the model is examined through comparisons with available other solutions, and the results show that the present method has higher accuracy and efficiency with a very small number of degrees of freedom for the simulation of the complex sloshing problem partly-filled laterally-excited cylindrical tanks. It is also shown that consideration of only the first sloshing mass is adequate to represent the dynamic behavior of the liquid container quite accurately. The effects of liquid fill level, baffled arrangement and length of those baffles upon the sloshing masses and sloshing forces are discussed in detail.

Oscillating characteristic of free surface from stability to instability of thermocapillary convection for high Prandtl number fluids

An investigation on oscillating characteristic of free surface in a liquid bridge for high Pr number fluids under gravity has been conducted numerically. Against the former studies, free surface is treated as a deformable surface in the direct numerical simulation (DNS) of liquid bridge by using a mass conserving level set method for the first time. The results show that, the moving track of vortex centers appears the shape of axisymmetric hook in the stable stage of thermocapillary convection, and the fluctuation of surface velocity is closely relative to the motion of cell flow toward the center and cold disk. In the process of transforming from the stable stage to oscillatory stage, the moving track of vortex centers of cell flow shows the asymmetry and step characteristic. The coupling effects of the temperature, velocity and free surface oscillations constitute complete mechanism of thermocapillary convection oscillations. The temperature firstly oscillates at the hot corner, and transfer direction of temperature oscillation is from the hot corner to inner. The velocity of surface flow is larger than that of internal flow, and the surface flow is supplied by the bulk return flow with the disturbance information of the velocity. The velocity oscillation lags behind the temperature oscillation at the hot corner. The closer to the intermediate height, the larger amplitudes of temperature and velocity are. The oscillation law of velocity and free surface at the intermediate height are the same, and there are two obvious oscillation bifurcations.

Dynamic Analysis of Partially Filled Non-circular Cylindrical Shells with Liquid Sloshing

The paper investigates the dynamic behavior of thin-walled reservoirs containing an ideal liquid taking into account the effects of hydroelastic interaction and sloshing. A mathematical statement of the problem is based on the principle of virtual displacements, which accounts for the pre-stressed non-deformed state of the shell caused by various force factors. The behavior of compressible liquid is described by linearized Euler equations, which are transformed by the Bubnov–Galerkin method. The dynamics of partially filled circular and elliptical cylindrical reservoirs are investigated numerically using a finite element procedure. It has been shown that allowing for sloshing considerably reduces eigenfrequencies of vibrations of the examined systems but has inessential effect on the displacements of the structure under non-stationary loads. Based on the modal analysis we present a classification of eigenmodes of free surface oscillations of a liquid in vertical tanks. It has been found that due to consideration of the sloshing effect, the frequency spectrum of the system can split into two parts in the case when the vibration frequencies of liquid differ from the vibration frequencies of an empty shell. Moreover, under harmonic excitation consideration of liquid sloshing leads to a more complicated amplitude-frequency curve characterized by displacement jumps.

Prediction of fluid dynamic instabilities of low liquid height-to-tank diameter ratio stirred tanks

In this work, original experimental data on the fluid flow instabilities associated with the oscillation of the liquid free surface in low liquid height-to-tank diameter ratio stirred tanks are presented. The local flow features of this type of tanks, which are commonly adopted as (bio)-reactors in important industrial productions such as drugs and bio-energy, are investigated by Particle Image Velocimetry. The local mean flow field and the turbulent characteristics are found to depend critically on the periodic variations of the liquid free surface, whose features vary as a function of the fill ratio. The frequency analysis of the dynamic pressure time series collected by a Pitot system shows that the frequency of the flow instabilities is mainly affected by the geometrical characteristics of the system. Therefore, the observed instabilities do not arise from the same origin of the so-called macro-instabilities previously identified in stirred vessels. A novel interpretation based on the liquid sloshing dynamics in cylindrical vessel is suggested.

Analysis of a resonance liquid bridge oscillation on board of the International Space Station

We study the singular event which took place when conducting an experiment with a liquid bridge aboard the International Space Station. The liquid bridge vibrated unexpectedly for several tens of seconds with an oscillation amplitude larger than 15% of its radius. At first glance, the analysis of the mass force measured by the accelerometer during the oscillation did not show any significant perturbation. However, our study reveals the existence of two small-amplitude vibrations of the experimental setup with practically the resonance frequency of the first lateral mode. These vibrations occurred a few tens of seconds before the liquid bridge oscillation reached its maximum amplitude, produced a mass force with a magnitude of the order of 10−5g. The numerical integration of the non-linear Navier–Stokes equations reproduces remarkably well the free surface oscillations measured in the experiments. It allows us to reconstruct the three-dimensional liquid bridge motion which took place in the experiment. The present study illustrates the sensitivity of liquid bridges in a microgravity environment, where tiny perturbations may produce significant vibrations which survive over long periods of time.

The production of viscoelastic capillary jets with gaseous flow focusing

The production of viscoelastic capillary jets with gaseous flow focusing is studied experimentally. In this technique, the liquid is injected at a constant flow rate through a feeding capillary located in front of the discharge orifice. A gas stream coflows with the jet across the orifice driven by a constant pressure drop. The gas stream sucks and drags the liquid, reducing the jet’s diameter well below the orifice diameter. Because of the rheological nature of the liquid, this focusing phenomenon differs from the Newtonian one in several regards. For given values of the polymer concentration, the injected flow rate, and the applied pressure drop, there is an interval of the capillary-to-orifice distance for which the jetting regime is reached. Outside that interval, the jet either suffers from the pull-out instability or breaks up before reaching the discharge orifice. Significant free surface oscillations can be observed in most of the jetting realizations. This oscillatory behavior is caused by a transient die swell effect which continuously appears right at the capillary exit. Ejection interrupts because the jet bulges to such an extent that the free surface pinches. Because of the stabilizing effect of the polymeric contribution to the axial stress, micrometer filaments with lengths up to 1 cm and Weber numbers on the order of 10−4 can be produced in front of the discharge orifice. The shear viscous stresses exerted on the emitted jet by the gas stream beyond the discharge orifice prevent the macromolecule recoiling. The resulting extensional viscosity inhibits the break-up process, and thus very long jets are produced.

Oscillation dynamics of free vortex surface in uncovered unbaffled stirred vessels

The main feature of unbaffled stirred tanks is the highly swirling liquid motion, which leads to the formation of a central vortex on the liquid free surface, when the vessel is operated without top-cover (Uncovered Unbaffled Stirred Tanks, UUST). One of the main drawbacks of such vessels, that limits their industrial applicability, is the possible onset of low-frequency sloshing of the free surface. In this work, original data on oscillation dynamics in UUST are presented. In particular, data focus on the oscillation amplitude as well as on their frequency. Data were obtained by means of a novel experimental technique based on digital image analysis. The effect of impeller geometry (Rushton turbine, pitched blade turbine, Lightnin A310 propeller) and that of scale-up were finally investigated. It was found that the natural oscillation frequencies only depend on vessel scale and geometry, while impeller type plays a role in the way the free surface oscillation frequencies change with impeller speed.

Free surface oscillations of liquid hydrogen in microgravity conditions

Experiments were performed to investigate the reorientation behavior and axial sloshing of liquid parahydrogen in a partly filled right circular cylinder due to a step reduction of gravity. Different temperature gradients along the cylinder wall in vertical direction were imposed to examine the influence of a wall superheat on the free surface. Experiments were conducted in the drop tower at the University of Bremen which provides a microgravity time of 4.7s and a compensated gravity environment of 10−6g0 (acceleration due to gravity). The thermal preparation of the experiments allowed to create defined wall temperature gradients and a stratified or homogenous liquid temperature distribution. Several sensors along the cylinder wall and in the vapor region monitored the temperature. The pressure inside the experiment was recorded and visual access was enabled by an endoscope. The experiments showed that the wall superheat has an influence on the free surface as well as on the temperature and pressure evolutions.

Mechanism and flow measurement of AC electrowetting propulsion on free surface

A free surface in contact with a floating object can be vertically oscillated by applying an alternating current electrowetting-on-dielectric (AC EWOD). The oscillation of the free surface generates a propelling force on the centimeter-sized floating object. This paper describes a propulsion mechanism in free-surface oscillation along with its experimental results. Flow visualizations, wave patterns measured by the free-surface synthetic schlieren method, and PIV measurements show that the oscillation generates a capillary Stokes drift on the water surface and two counter-rotating spiral underwater vortices, leading to an ejecting flow (streaming flow) normal to the wall of the boat. The momentum of the ejecting flow produces a reaction force on the wall and ultimately propels the floating boat. The propulsion speed of the boat highly depends on the amplitude, frequency, and shape of the AC EWOD signal. Curve fittings based on the Stokes drift well match the experimental measurements of propulsion speed. The width of the EWOD electrode also has significant effects on the boat speed.

Gravity wave instability in a turbulent free-surface Taylor–Couette flow: experiments and comparison with an amplitude equation with additive noise

We present an experimental and theoretical study on the gravity-wave instability developing in a highly turbulent free-surface Taylor–Couette flow, for which only the inner cylinder rotates. Above a critical rotation speed, from an axisymmetric turbulent base state a non-axisymmetric fluctuating gravity-wave state develops, with an m = 1 azimuthal wave number. The bifurcation is discontinuous and presents hysteresis. In contrast to previously reported work (Mujica N and Lathrop D 2006 J. Fluid Mech. 51 49–62), here we compare our experimental results with a universal model based on a quintic subcritical amplitude equation with additive noise. In general, the model describes correctly the mean free-surface oscillation amplitude and its fluctuations, although differences exist in the bistability region width and the free-surface fluctuations in the gravity wave state. These differences are due to the finite time measurements and non-linear effects, respectively. Indeed, we show that longer measurement times allow the system to transit in either direction (from or to the base state), which results in the shrinking of the bistability region. For very long measurement times, and in a very narrow range of rotation rates, the system presents a series of random reversals between both states. Finally, by removing the mean wave and flow oscillations in the measured free-surface and bulk pressure signals, we demonstrate that their dynamic fluctuations depend on the system state.

Electromagnetic flow control of a bifurcated jet in a rectangular cavity

The effect of Lorentz forcing on self-sustained oscillations of turbulent jets (Re=3.1×103) issuing from a submerged bifurcated nozzle into a thin rectangular liquid filled cavity was investigated using free surface visualization and time-resolved particle image velocimetry (PIV). A Lorentz force is produced by applying an electrical current across the width of the cavity in conjunction with a magnetic field. As a working fluid a saline solution is used. The Lorentz force can be directed downward (FL<0) or upward (FL>0), to weaken or strengthen the self-sustained jet oscillations. The low frequency self-sustained jet oscillations induce a free surface oscillation. When FL<0 the amplitude of the free surface oscillation is reduced by a factor of 6 and when FL>0 the free surface oscillation amplitude is enhanced by a factor of 1.5.A large fraction of the turbulence kinetic energy k=12ui′ui′‾ is due to the self-sustained jet oscillations. A triple decomposition of the instantaneous velocity was used to divide the turbulence kinetic energy into a part originating from the self-sustained jet oscillation kosc and a part originating from the higher frequency turbulent fluctuations kturb. It follows that the Lorentz force does not influence kturb in the measurement plane, but the distribution of kosc can be altered significantly. The amount of energy contained in the self-sustained oscillation is three times lower when FL<0 compared to the situation with FL>0.