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## Free Surface Sloshing Analysis

Sloshing is a free surface flow problem in a tank which is subjected to forced oscillation. Clarification of the sloshing phenomena is very important in the design of the tank. The violent sloshing creates localized high impact loads on the tank roof and walls which may damage the tank. Early simulations of the liquid sloshing problem have mostly been performed with waves of small steepness. The sloshing height was assumed to be too small so that the nonlinear boundary conditions may be neglected. Jacobsen  determined hydrodynamic pressures on a cylindrical tank. Graham and Rodriguez  gave a very thorough analysis of the impulsive and convective pressures in a rectangular container.

The most commonly applied idealization for estimating liquid response in seismically excited rigid, rectangular and cylindrical tanks was formulated by Housner  in 1957. He divided hydrodynamic pressures of contained liquid into two components: the impulsive pressure caused by the portion of the liquid accelerating with the tank and the convective pressure caused by the portion of the liquid sloshing in the tank. The convective component was modeled as a single degree of freedom oscillator. The study presented values for equivalent masses and their locations which duplicate forces and moments exerted by a liquid on a tank. Properties of the mechanical model can be computed from the geometry of the tank and the characteristics of the contained liquid. Housner's model (, ) is widely used to predict the maximum seismic response of tanks by means of a response spectrum characterizing the design earthquake (, ).

In 1989, Mclver  considered the two-dimensional sloshing of a fluid in a horizontal circular cylindrical container and the three-dimensional sloshing of a fluid in a spherical container. He used the linearized theory of water waves to determine the frequencies of free oscillations under gravity of an arbitrary amount of fluid in such tanks. Special coordinate systems were used and the problems were formulated in terms of integral equations which were solved numerically for the eigen values. Tables of the sloshing frequencies were presented for a range of fill-depths of the containers.

It was not until late that researchers started to investigate the nonlinear fluid sloshing problem. Numerical methods presented previously for the sloshing analysis can be roughly classified into three methods: the finite difference method, the boundary element method and the finite element method. In 1963, Hutton  and Kamatsu  obtained nonlinear frequency response curves of a liquid by using perturbation techniques. They also studied the stability of the surface sloshing. In 1980, Nakayama and Washizu (, ) modeled the nonlinear sloshing by finite element and boundary element methods and carried out numerical simulations of a two-dimensional liquid under horizontal and pitching periodic ground motions. In 1986, Ramaswamy  modeled the nonlinear sloshing of sinusoidally-excited liquids with viscous damping. In a following work , he used a Lagrangian-Eulerian finite element method to model the free surface fluid flow.

In 1987, Yamada  discussed the effect of nonlinear boundary conditions at the liquid surface on the sloshing heights in cylindrical tanks under horizontal and vertical ground motions. Based on a comparison with data obtained by linear analysis, he reported that this effect depends mainly on both the dimensionless sloshing height and the dimensionless liquid depth, as far as a small sloshing height is concerned. He proposed a simple formula for estimating this effect. On the basis of the large sloshing heights recorded during the 1983 Nihonkaichubu earthquake, he estimated the sloshing heights under nonlinear conditions to be about 10-25% larger than those calculated under linear conditions. Taking this into account, he stated that the response spectra of relatively long-period ground motions deduced from the recorded sloshing heights are nearly equal to the two-dimensional response spectra calculated from strong-motion seismograms. In the same year, Ibrahim  has conducted experimental investigations concerning the liquid sloshing in an excited tank. In 1988, Lepelletier  developed a nonlinear model to describe the fluid motion in a tank. He studied the liquid response behavior near resonance and the transient behavior of liquid in the sense of modal superposition when the liquid is subjected to steady-state periodic basin excitations. It was observed that shallow water waves predicted by linear theory becomes inadequate near resonance and that the wave shape is very sensitive to the frequency of excitation near resonance.

The fluid-structure interaction problem with the free surface sloshing was also investigated. Liu (, , ) presented a variational principle for fluid-structure interaction problems with sloshing that accounts for both seismic and body forces. He showed that various fluid-structure interaction formulations may be obtained from the developed functional. Bauer et al  analyzed the nonlinear hydroelastic vibrations of two cases of an infinitely long rectangular container of finite width and filled to a certain height with a compressible and non-viscous liquid. The first case was that in which the free surface was covered by a flexible membrane, exhibiting a nonlinear stress-strain relation and large oscillation amplitudes. This system exhibited a hard vibration. In the second case the hydroelastic vibrations of a liquid with a free surface, performing large amplitudes, and a nonlinear flexible membrane bottom has been treated. The influence of various system parameters on the coupled natural frequencies was investigated. The liquid exhibited a softening vibration characteristic, while the membrane shows a hardening effect, which with the increase of membrane prestrain could change to a soft vibration.

In 1989, Barron and Chng  studied fluid sloshing problems in circular containers by both theoretical and experimental methods. A circular cylindrical container with various levels of fluid attached to a low frequency suspension was analyzed by means of the method of asymptotic expansion. Experimental studies in the form of resonance and transient vibration tests have been conducted on a test rig. A theoretical analysis was applied to a mathematical model of the test rig. The equations of motion were simulated by a digital simulation method, for both linear and nonlinear conditions. Results showed parametric resonance effects of the fluid wave height and a horizontal fluid force wave forms. Kobayashi et al  conducted experimental and analytical study to determine the liquid natural frequencies and the resultant slosh forces in horizontal cylindrical tanks. They presented a study of the liquid sloshing response for small and large wave heights. In the former case, they presented an effective calculation method of the longitudinal slosh response by substituting an equivalent rectangular tank for a horizontal cylindrical tank. In the latter case, impulsive sloshing forces were observed for longitudinal excitation when the slosh liquid hit the top of the tank. They reported that the measured slosh forces including the impulsive forces were larger than the calculated ones.

Nash et al  treated large amplitude surface waves by expanding liquid elevations above mean free surface and velocity potentials in terms of a power series involving a dimensionless parameter which is a function of the peak amplitude of the tank excitation as well as the tank radius. Minowa  performed a similar study and concluded that the deformation of side walls produced the heaving type sloshing mode. He tried some nonlinear and linear sloshing analyses to reach a better understanding of the high sloshing waves. Popov et al  presented a numerical solution of the nonlinear liquid motion in a horizontal cylindrical container with circular cross section. They discretized and solved the 2-D governing differential equations in an Eulerian mesh. Li et al  obtained the critical stability conditions for large amplitude water wave oscillations.

In 1991, Krasnopol'skaya et al  studied the possible modes of vibration of the free surface of a liquid in a rigid container. They presented results of a study on interaction of regular and chaotic vibrations of the free surface of a liquid inside a rigid cylindrical shell, excited by an electric motor. In 1992, Kurihara et al  investigated experimentally the sloshing impact on a roofed tank. They proposed a formula to predict the impact pressures caused on the roofed liquid tanks due to sloshing. Hwang et al  employed the panel method, which was based on the boundary integral technique, to investigate the three dimensional sloshing problem. In 1993, Veletsos  investigated the sloshing action of layered liquids in rigid cylindrical and long rectangular tanks. The analysis was formulated for systems with N superimposed layers of different thickness and densities. In the same year, Isaacson  studied the earthquake-induced sloshing in rigid circular tanks.

In addition to these investigations, shaking table tests were conducted by Okamoto et al (, , ) to simulate the two-dimensional liquid sloshing behavior. They also performed numerical simulations for the sinusoidally-excited liquids by employing the Lagrangian-Eulerian finite element method. They considered the large amplitude 2-D sloshing wave in a multi-sloped wall tank with roof. In this investigation, the developed analysis accommodates double-valued free surface function. Prediction of the transient response of liquid subjected to seismic-type ground excitation has not been attempted until recently due to the complex nature of the physical and numerical behavior of liquid under prolonged transient and steady state base shaking. Haroun and Chen (, ) addressed the sloshing phenomenon in seismically-excited rectangular liquid storage tanks. They employed a semi-analytical method to study the effect of large amplitude sloshing on two-dimensional tanks under arbitrary horizontal excitations. The nonlinearity of both the kinematic and the dynamic conditions was considered. Because the location of the free surface is unknown, a numerical scheme was developed to transform a two dimensional uniform rectangular grid into boundary conforming curvilinear grid with prescribed arbitrary boundaries. A set of governing nonlinear equations was obtained and solved in the uniform rectangular mesh by employing a finite difference scheme. Three-dimensional large amplitude sloshing in rectangular tanks has also been attempted. Su (, ) has performed numerical simulation of the three-dimensional large amplitude liquid sloshing in rectangular containers subjected to vertical excitation.    Next: Fluid-Structure Interaction Up: Contribution of Research on Previous: Contribution of Research on
A. Zeiny
2000-09-06