Rigid Rectangular Tank Research Articles

Analysis of liquid sloshing frequencies in a partially filled 3D rectangular tank

This research investigates liquid sloshing in a 3D rigid rectangular tank. The impact of rigid baffle on sloshing frequencies has been studied. The mathematical modal has been developed using potential theory. The boundary value problem has an analytical solution in terms of velocity potential with undetermined frequency. We get a system of homogeneous algebraic equations using boundary and free surface conditions. The frequencies are calculated using the non-trivial solution condition. Frequencies of baffled tank are computed for various filling levels. The effects of filling level on frequencies are identified. ANSYS software is used to report the liquid domain and rigid baffle mode forms.

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.

Vibrations of rigid cantilevered plates in a rectangular tank filled with liquid

The article presents the developed analytical (exact) methods for solving problems of calculating the frequencies of natural vibrations of rigid and elastic partitions in a rigid rectangular tank filled with liquid. The results of solving these problems can be used in the design of hydraulic structures, as well as in shipbuilding when analysing the dynamics of structural elements of large oil tankers and gas carriers.

Experimental studies of suppressing effectiveness on sloshing with two perforated floating plates

In the present paper, model tests of suppressing sloshing fitted with two perforated floating plates are carried out. The study involves identification of system performance such as the suppression and the solidity ratio. Three different solidity ratios of perforated plates have been tried out as potential positive slosh damping devices. A series of painstaking experiments have been conducted in a rigid rectangular tank on six degrees of freedom motion platform under roll harmonic excitation. Comparison of the clean tank shows that the three types of perforated plates are all effective on damping the run-up and impact pressure along the bulkhead. The parametric study indicates that the perforated plate with the median solidity ratio is the most optimal one in suppressing sloshing among three configurations.

Sloshing response of partially filled rectangular tank under periodic horizontal ground motion.

The event caused due to the movement of the unrestricted free surface liquid in container due to any external excitation is known as sloshing. The problem of liquid sloshing phenomenon in stationary or in moving container is a great matter of concern for many researchers and engineers. The containers may range from a cup of milk, ponds, and lakes to fuel tanks of launching vehicles and cargo ships carrying variety of liquids such as oil, liquefied natural gas, and chemical fluids. As a result of sloshing there is spillage of liquid from the containers and it causes structural instability and structural damage. Due to these reasons, liquid retaining structures which are special in construction and in function from an engineering point of view must be constructed well to be resistant against oscillation of the liquid due to external excitations. The amount of liquid participating in the sloshing motion depends on the shape of tank, the liquid depth, internal objects, if any, orientation, duration, amplitude, and frequency contents of external excitations. This present study focuses on the forced vibration analysis of partially filled two-dimensional rigid rectangular tank numerically. A Finite element (FE) code in two dimensions is developed to understand the behavior of sloshing. This method is competent enough of evaluating both impulsive and convective response of tank-liquid system in terms of base shear and hydrodynamic pressure distribution along the walls of the containers.

Simulation of Sloshing in Rigid Rectangular Tank and a Typical Aircraft Drop Tank

In this research, the liquid sloshing behavior in a 3D rectangular tank was simulated and validated by applying peak acceleration load using computational fluid dynamics technique. The application of this sloshing phenomenon was carried out on a typical Aircraft Drop tank with and without baffle plates for 7 g peak acceleration. The structural integrity of the drop tank has been taken into consideration during the cruise flight condition. Further, an optimized design of a drop tank has been modeled. The comparison of computed results for 3D rectangular tank case with experimental results showed that the numerical technique is capable of simulating hydrostatic pressure loads exerted on tank walls. Similarly, the necessary sloshing loads in the form of hydrodynamic pressure generated on the tank walls have been estimated for different cases of a typical aircraft drop tank. The kinematic profiles of liquids were observed at different instances for various cases. Computational results indicated that there is a reduction in the peak pressure on aft side of the tank with the use of baffle plates.

Fluid damping in rectangular tank fitted with various internal objects – An experimental investigation

The potential of internal objects in changing the dynamic system characteristics of mobile liquid carrying rectangular containers is experimentally investigated in the present study. This study involves identification of system characteristics such as natural frequency and damping. Three different configurations of centrally installed internal objects; bottom-mounted vertical baffles, surface-piercing wall-mounted vertical baffles, and bottom-mounted submerged-blocks have been tried out as potential passive slosh damping devices. A series of painstaking experiments has been conducted in a rigid rectangular tank model on a shake table under lateral harmonic excitation. A frequency response of various internal object arrangements on free surface elevation has been studied. Having identified the system characteristics, the sloshing responses of liquid (water) to harmonic sinusoidal loading for different baffle configurations are investigated. The time variation of the free surface elevation to the baffle configuration and height have been highlighted. Sine sweep and Logarithmic decay method have been resorted to in the experimental discourse. The parametric study shows that the surface-piercing wall mounted baffles are the most effective one in slosh damping among the three configurations.

Quantification of Seismic Response of Partially Filled Rectangular Liquid Tank with Submerged Block

The seismic behavior of partially filled rigid rectangular tank with bottom-mounted submerged block is numerically simulated. Six different ground motions characterized on the basis of low-, intermediate-, and high-frequency contents are used to investigate the dynamic behavior of tank-liquid-submerged block system. A velocity potential-based Galerkin finite element model is developed for the analysis. The effect of submerged block on impulsive and convective response components of hydrodynamic behavior manifested in terms of base shear, overturning base moment, and pressure distribution along the tank wall as well as the block wall has been quantified.

Experiments on non-resonant sloshing in a rectangular tank with large amplitude lateral oscillation

Flow of two dimensional nonlinear sloshing in a rigid rectangular tank with free surface is considered. The flow is generated by oscillating the container in a lateral harmonic motion, i.e., d⁎=A⁎sin(2πf⁎t⁎) where d⁎ denotes the displacement of the container externally forced, A* the amplitude of displacement and f* the frequency. Thus, the maximum stroke, S*, of the container for a cycle, measured by a distance from the leftmost to the rightmost location on the container movement, is defined as S*=2A*. It has performed a sequence of experiments on a variety of S*- and f*-values to investigate large amplitude sloshing flows at off-resonant condition far from the system natural frequency, where large amplitude means that the stroke, S*, of the container movement is comparable with the breadth, L*, of the container, i.e., S*/L*∼O(1) and the excitation acceleration is also comparable with the gravity, i.e., π2(S*f*)2/g*∼O(1). Through PIV experiment, it shows that the flow physics on nonlinear off-resonant sloshing problem can be characterized into a combination of three peculiar sloshing motions: (1) standing wave motions which is similar with those of linear sloshing during run-down process, (2) run-up phenomenon like hydraulic jump along the vertical sidewall at the moment of turn-around of the container and (3) gradually propagating bore motion from one sidewall to the opposite wall which is similar with dam breaking problem.

A Boundary Element Investigation of Liquid Sloshing in Coupled Horizontal and Vertical Excitation

Sloshing flows in a two‐dimensional rigid rectangular tank under specified excitations in the coupled horizontal and vertical modes are simulated by using a higher‐order boundary element method (BEM). The liquid sloshing is formulated as an initial‐boundary‐value problem based on the fully nonlinear potential flow theory. And a semi‐mixed Eulerian‐Lagrangian technique combined with the 4th‐order Runge‐Kutta scheme is employed to advance the solutions in the time marching process. A smoothing technique is applied to the free surface at every several time steps to avoid the possible numerical instabilities. Numerical results obtained are compared with the available solutions to validate the developed model. The parametric studies are carried out to show the liquid sloshing effects in terms of the slosh frequencies and excitation amplitudes in horizontal and vertical modes, the second‐order resonance frequency, a bottom‐mounted vertical rigid baffle, free surface displacement, and hydrodynamic forces acting on the tank.

사각용기의 슬로싱 유동에 관한 연구

The two-dimensional sloshing problem in a rigid rectangular tank with a free surface is considered. The flow is generated by a container in harmonic motion in time along the horizontal axis, i.e., a container excited by u=Asin() where u denotes the container velocity imposed externally, A is the amplitude of the oscillation velocity, and f is the frequency of oscillation. Experimental apparatus is arranged to investigate the large-amplitude sloshing flows in off-resonant conditions, where the large amplitude means that A~O(1), and the distance, S, is comparable to the breadth, L, of the container, i.e., L/S~O(1). Comprehensive particle image velocimetry (PIV) data are obtained, which show that the flow physics of the nonlinear off-resonant sloshing problem can be characterized into three peculiar free surface motions: standing-wave motions similar to those of linear sloshing, a run-up phenomenon along the vertical sidewall at the moment of turn-over of the container, and gradually propagating bore motion from the sidewall to the interior fluid region, like a hydraulic jump.

Fluid Pressures on Unanchored Rigid Rectangular Tanks Under Action of Uplifting Acceleration

Although uplift motion of flat-bottom cylindrical shell tanks has been considered to contribute toward various damages to the tanks, the mechanics were not fully understood. As well as uplift displacement of the tanks, fluid pressure accompanying the uplift motion of the tanks may play an important role in the cause of the damage. An accurate estimate of the fluid pressure induced by the uplift motion of the tanks is indispensable in protecting the tanks against destructive earthquakes. As a first step of a series of research, this study mathematically derives the fluid pressure on a rigid rectangular tank with a unit depth accompanying angular acceleration, which acts on a pivoting bottom edge. The rectangular tank employed herein is equivalent to a thin slice of the central vertical cross section of a rigid flat-bottom cylindrical shell tank. Assuming a perfect fluid and velocity potential, a continuity equation is given by the Laplace equation in Cartesian coordinates. The fluid velocities accompanying the motions of the walls and bottom plate constitute the boundary conditions. Since this problem is set as a parabolic partial differential equation of the Neumann problem, the velocity potential is solved with the Fourier-cosine expansion. The derivative of the velocity potential with respect to time gives the fluid pressure at an arbitrary point inside the tank. A mathematical solution for evaluating the fluid pressure accompanying the angular acceleration acting on the pivoting bottom edge of the tank is given by an explicit function of a dimensional variable of the tank, but with the Fourier series. The proposed mathematical solution well converges with a few first terms of the Fourier series. Values of the fluid pressure computed by the explicit finite element (FE) analysis well agrees with those by the proposed mathematical solution. For the designers’ convenience, diagrams that depict the fluid pressures normalized by the maximum tangential acceleration given by the product of the angular acceleration and diagonals of the tank are also presented. Consequently, the mathematical solution given by the Fourier series converges easily and provides accurate evaluation of the fluid pressures on a rigid rectangular tank accompanying the angular acceleration acting on the pivoting bottom edge. Irregularity in the fluid pressure distribution increases as the tank becomes taller.

Fluid Pressures on Unanchored Rigid Flat-Bottom Cylindrical Tanks Under Action of Uplifting Acceleration

To protect flat-bottom cylindrical tanks against severe damage from uplift motion, accurate evaluation of accompanying fluid pressures is indispensable. This paper presents a mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank in the same manner as the procedure outlined and discussed previously by the authors (Taniguchi, T., and Ando, Y., 2010, “Fluid Pressures on Unanchored Rigid Rectangular Tanks Under Action of Uplifting Acceleration,” ASME J. Pressure Vessel Technol., 132(1), p. 011801). With perfect fluid and velocity potential assumed, the Laplace equation in cylindrical coordinates gives a continuity equation, while fluid velocity imparted by the displacement (and its time derivatives) of the shell and bottom plate of the tank defines boundary conditions. The velocity potential is solved with the Fourier–Bessel expansion, and its derivative, with respect to time, gives the fluid pressure at an arbitrary point inside the tank. In practice, designers have to calculate the fluid pressure on the tank whose perimeter of the bottom plate lifts off the ground like a crescent in plan view. However, the asymmetric boundary condition given by the fluid velocity imparted by the deformation of the crescent-like uplift region at the bottom cannot be expressed properly in cylindrical coordinates. This paper examines applicability of a slice model, which is a rigid rectangular tank with a unit depth vertically sliced out of a rigid flat-bottom cylindrical tank with a certain deviation from (in parallel to) the center line of the tank. A mathematical solution for evaluating the fluid pressure on a rigid flat-bottom cylindrical tank accompanying the angular acceleration acting on the pivoting bottom edge of the tank is given by an explicit function of a dimensional variable of the tank, but with Fourier series. It well converges with a few first terms of the Fourier series and accurately calculates the values of the fluid pressure on the tank. In addition, the slice model approximates well the values of the fluid pressure on the shell of a rigid flat-bottom cylindrical tank for any points deviated from the center line. For the designers’ convenience, diagrams that depict the fluid pressures normalized by the maximum tangential acceleration given by the product of the angular acceleration and diagonals of the tank are also presented. The proposed mathematical and graphical methods are cost effective and aid in the design of the flat-bottom cylindrical tanks that allow the uplifting of the bottom plate.

ALE and Fluid Structure Interaction for Sloshing Analysis

Liquid containment tanks are, generally, subjected to large deformations under severe earthquake conditions due to coupling forces between tank and the contained liquid. The accurate description of these forces is vital in order to diminish or eliminate the potential risk of tank failure during an earthquake. Yet, analytical formulations derived for the seismic analysis of liquid storage tanks are not capable to capture the complex fluid-structure effects since they include many assumptions and simplifications not only for the behavior of fluid and structure but also for the external excitation. On the other hand, an appropriate numerical method allows us to cope with large displacements of free surface of the fluid, high deformations of the structure and correctly predicts the hydrodynamic forces due to the high-speed impacts of sloshing liquid on a tank wall and roof. For this purpose, a new coupling algorithm based on the penalty formulation of finite element method which computes the coupling forces at the fluidstructure interface is developed in this paper. This algorithm is constructed on a two superimposed mesh systems which are a fixed or moving ALE mesh for fluid and a deformable Lagrangian mesh for structure. The fluid is represented by Navier-Stokes equations and coupled system is solved using an explicit time integration scheme. In order to verify the analysis capability of coupling algorithm for tank problems, numerical method is applied for the analyses of a rigid rectangular tank under harmonic excitation and a flexible cylindrical tank subjected to earthquake motion and numerical results are compared with existing analytical and experimental results. Strong correlation between reference solution and numerical results is obtained in terms of sloshing wave height.

Forced stationary gas‐structure interaction vibrations in a rectangular parallelepiped tank

AbstractA closed rigid rectangular parallelepiped tank is filled with gas and a part of one of its walls is a thin linearly elastic rectangular plate. The problem about the determination of the forced stationary vibrations of the obtained gas–structure interaction system under the action of a source is considered. The method of the crossed strips of Warburton and the Bubnov–Galerkin method are used to create a method of investigation of the dynamic behavior of this gas–structure interaction system. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Non‐linear sloshing in partially liquid filled containers with baffles

AbstractA finite element method is used for computing the non‐linear sloshing response of liquid in a two‐dimensional rigid rectangular tank with rigid baffles. The potential formulation is considered for the liquid domain and a mixed Eulerian–Langrangian scheme is adopted. The solution is obtained by the Galerkin method. The fourth‐order Runge–Kutta method is employed to advance the solution in the time domain. A regridding technique is applied to the free surface of the liquid, which effectively eliminates the numerical instabilities without the use of artificial smoothing. Through the comparison with the available results for the rectangular tank without baffle, the validity of the present formulation is checked and then extended to the solution of tanks with rigid baffles. The effects of baffle parameters such as position, dimension and numbers on the non‐linear sloshing response are examined. The present numerical solution procedure is also applied to the non‐linear sloshing problems in a circular cylindrical container with annular baffle. Copyright © 2006 John Wiley & Sons, Ltd.

Nonstationary fluid‐structure interaction vibrations in a parallelepipedtank

AbstractA rigid rectangular parallelepiped tank is filled with a compressible and inviscid fluid as a part of one of its walls is a thin linearly elastic rectangular plate. The problem about the determination of the nonstationary vibrations of the received fluidstructure interaction system is considered. The Laplace transforms, the method of Bubnov‐Galerkin and the method of the crossed strips of G. Warburton are amalgamated to create some fast convergent method to investigate the dynamic behavior of the fluid‐structure interaction system under consideration in the cases of arbitrary supporting conditions of the plate. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Behaviour of Lagrangian triangular mixed fluid finite elements

The behaviour of mixed fluid finite elements, formulated based on the Lagrangian frame of reference, is investigated to understand the effects of locking due to incompressibility and irrotational constraints. For this purpose, both linear and quadratic mixed triangular fluid elements are formulated. It is found that there exists a close relationship between the penalty finite element approach that uses reduced/selective numerical integration to alleviate locking, and the mixed finite element approach. That is, performing reduced/ selective integration in the penalty approach amounts to reducing the order of pressure interpolation in the mixed finite element approach for obtaining similar results. A number of numerical experiments are performed to determine the optimum degree of interpolation of both the mean pressure and the rotational pressure in order that the twin constraints are satisfied exactly. For this purpose, the benchmark solution of the rigid rectangular tank is used. It is found that, irrespective of the degree of mean and the rotational pressure interpolation, the linear triangle mesh, with or without central bubble function (incompatible mode), locks when both the constraints are enforced simultaneously. However, for quadratic triangle, linear interpolation of the mean pressure and constant rotational pressure ensures exact satisfaction of the constraints and the mesh does not lock. Based on the results obtained from the numerical experiments, a number of important conclusions are arrived at.

Numerical Analysis of Sloshing with Large Deformation of Elastic Walls and Free Surfaces Using MPS method.

Moving particle semi-implicit (MPS) method was developed for analyzing incomplessible flow with particle models. In the present study, a numerical model for thin walls is developed and the MPS method is extended to analyze fluid-structure interactions with large deformation of free surfaces and walls. First, sloshing in a rigid rectangular tank is calculated. A finite amplitude wave occurs in a shallow tank, while a standing wave appears in a deep tank. Results are in good agreement with the experiments by Hayama et al., though the calculated sloshing amplitude is smaller than the experimental one in the deep tank. This is attributed to unknown dissipation mechnisms in the present method. Next, sloshing in a tank with elastic walls is calculated. The calculated resonance peak is shifted to a lower frequency due to large deformation of the walls.

ピッチング励振を受ける直方体容器内液面のパラメトリック振動

Three-dimensional nonlinear liquid motion in a rigid rectangular tank partially filled with liquid, in response to pitching excitation, is investigated. In the case that symmetric modes are dominant, the nonlinear ordinary differential equations governing the liquid surface oscillation are derived by applying Galerkin's method to the basic equations. It is confirmed that vertical excitation causes parametric excitation. In addition, it is noted that pitching excitation also causes parametric excitation when the pitching axis does not intersect the vertical line passing the center of the tank bottom. An experiment was conducted using a model tank. Good agreement was found between the theoretical and experimental results.