Two-dimensional Rectangular Tank Research Articles

Heterogeneity's impact on the sloshing motion in excited tanks

AbstractThis paper aims to investigate the effect of liquid heterogeneity on sloshing in a two-dimensional rectangular tank. The container is excited horizontally, and the density of the liquid at equilibrium can almost be considered a linear dependence of the depth. The linearized equations representing the sloshing phenomenon are solved numerically using ANSYS 2020 R2 software and analytically using the separation method of variables in conjunction with the Fourier analysis. On the other hand, a comparison study has been carried out between the obtained results and other works related to the same phenomenon. Overall, the results exhibit considerable effects of variation in the Heterogeneity Coefficient β on the free surface's motion and the pressure distribution within the liquid. Furthermore, the free surface of liquid heterogeneity is also affected by the variation of Excitation Frequency Ω and the Filling Rate h of the liquid in the tank.

Experimental study of liquid slamming in elastic rectangular tanks under the flip-through impact

This paper considers the hydroelasticity effect in liquid slamming. A series of experiments were designed and performed in a nearly two-dimensional (2D) rectangular tanks with four elastic materials. The effect of the Young's modulus on the kinematic and dynamic characteristics of fluid-structure interaction (FSI) during flip-through impact is investigated. Furthermore, the peak values of slamming pressure, impact duration, pressure impulse and structural displacement in different cases are discussed. The hydrodynamic force of fluid acting on the sidewall is defined and compared for different cases. The results show that the Young's modulus of the sidewall plays an important role in hydroelasticity induced by liquid slamming. And a strong hydroelastic behavior could be observed when the impact occurs, especially in the case with a small Young's modulus. In addition, the artificial neural network (ANN) method is adopted to build the hydroelastic response prediction model. The relationship between Young's modulus and peak structural displacement is predicted with the model. This study could help verify and calibrate the theoretical and numerical models of the FSI problems in the sloshing tank and provide guidance on the study of hydroelastic slamming for the flexible cargo containment system.

Effect of the drag coefficient on the performance of vertical porous baffles in a sloshing tank

Purpose The present work involves analytical and experimental investigation of sloshing in a two-dimensional rectangular tank including the effect of porous baffles to control and/or reduce the wave motion in the sloshing tank. The purpose of this study is to assess the analytical solutions of the drag coefficient effect on porous baffles performance to track free surface motion variation in the sloshing tank by comparison with experimental shake table tests under a range of sway excitation. Design/methodology/approach The linear second-order ordinary differential equations for liquid sloshing in the rectangular tank were solved using Newmark’s beta method and obtained the analytical solutions for liquid sloshing with dual vertical porous baffles of full submergence depths in a sway-oscillated rectangular tank following the methodology similar to Warnitchai and Pinkaew (1998) and Tait (2008). Findings The porous baffles significantly reduce wave elevation in the varying filled levels of the tank compared to the baffle-free tank under the range of excitation frequencies. It is observed that the Reynolds number-dependent drag coefficient for porous baffles in the tank can significantly reduce the sloshing elevations and is found to be effective to achieve higher damping compared to the porosity-dependent drag coefficient for porous baffles in the sloshing tank. The analytical model’s response to free surface elevation variations in the sloshing tank was compared with the experiment’s test results. The analytical results matched with shake table test results with a quantitative difference near the first resonant frequency. Research limitations/implications The scope of the study is limited to porous baffles performance under range sway motion and three different filling levels in the tank. The porous baffle performance includes Reynolds number dependent drag coefficient to explore the damping effect in the sloshing tank. Originality/value The porous baffles with low-level porosities in the sloshing tank have many engineering applications where the first resonant mode of sloshing in the tank is more important. The porous baffle drag coefficient is an important parameter to study the baffle’s damping effect in sloshing tanks. Hence, obtained analytical solution for liquid sloshing in the rectangular tank with Reynolds number as well as porosity-dependent drag coefficient (model 1) and porosity-dependent drag coefficient porous baffles (model 2) performance is discussed. The model’s test results were validated using a series of shake table sloshing experiments for three fill levels in the tank with sway motion at various excitation frequencies covering the first four sloshing resonant modes.

A two-dimensional semi-analytic solution on two-layered liquid sloshing in a rectangular tank with a horizontal elastic baffle

A semi-analytic study on two-layered liquid sloshing in a horizontal excited two-dimensional rectangular tank with a horizontal elastic baffle was performed. The present work is a further development of the semi-analytical technique for solving the single-layer liquid sloshing problem. The horizontal elastic baffle was located in the lower liquid or upper liquid. First, the complex liquid domain in a baffled tank was divided into several simple sub-domains to solve the analytic solution. The wet mode of a horizontal baffle was presented according to the Eulerian Bernoulli beam. The continuity boundary condition on the interfaces between two liquids and the virtual interfaces was given. Then, the formal solution was derived for each sub-domain and a horizontal baffle using the superposition principle and the method of separation of variables. Then, the total velocity potential subject to lateral excitation was summed over the container potential function and the liquid disturbance potential. The dynamic response equation for two-layered liquid sloshing was established. Multiple methods verified the semi-analytic solution's correctness and agreed well with other methods. Finally, the numerical analysis mainly shows that the density ratio becomes more significant for the coupled frequency when the baffle is in the upper liquid. A large baffle's width suppresses the liquid flow in the baffled tank effectively. Furthermore, adjusting the baffle and the layered liquid parameters can significantly suppress layered liquid sloshing. For the seismic response, the elevation energy is mainly located after 2.5 s.

A fast semi-analytic solution of liquid sloshing in a 2-D tank with dual elastic vertical baffles and walls

Liquid sloshing characteristics in a lateral excited two-dimensional (2-D) rectangular tank with dual elastic vertical baffles and walls have been investigated analytically. The elastic baffle or wall is regarded as an Eulerian Bernoulli beam. The complex liquid domain divides into six simple sub-domains according to the superposition principle. The formal solution of the potential function in each sub-domain is obtained using the separation of variables method. Then the formal baffle or wall deflection solution can be derived from the transverse vibration equation. The Eigen equation can be established according to the free surface, interface, and coupled vibration conditions. The coupled frequency can be obtained by solving the Eigen equation. The velocity potential under lateral excitation consists of the rigid container velocity potential and the disturbance potential. Then the coupled dynamic response equation is established using the free liquid surface equation and the coupled vibration equation. The correctness of the semi-analytic solution is verified against available data. Therefore, the numerical model of the present work can be computed quickly with semi-analytic solution and obtain the mechanism of liquid-structure interaction. Finally, the effect of baffle parameters and excitation frequency on liquid sloshing is discussed in detail.

A Semi-Analytical Solution of Liquid Response in a Two-Dimensional Rectangular Tank with a Vertical Rigid Baffle Under Pitching Excitation

The present work uses a semi-analytic mathematical model to study the dynamic response of the partially liquid-filled rectangular tank equipped with a vertical rigid baffle at the bottom subjected to pitching excitation. The velocity potential consists of the rigid container velocity potential and the disturbance potential, which contains a Stokes–Joukowski potential. The liquid domain is divided into four simple sub-domains. The formal solution of the Stokes–Joukowski potential in each sub-domain is solved using the superposition principle and the separation of variables method. The dynamic response equation is established by using the free liquid surface equation. The ADINA solution is compared with the present method solution, which verifies the correctness of the proposed method. Finally, the effect of baffle parameters on liquid sloshing for a baffled tank subjected to harmonic and seismic pitching excitations is discussed in detail.

Analysis of loads caused by waves on the deck near the free surface of the offshore platform using computational fluid dynamics

ABSTRACT The analysis of stresses imposed on structures by offshore environments is critical for their safety and smooth operation. Physical model testing for structural responses on offshore platforms is time-consuming and expensive. Computational fluid dynamics (CFD) can be used to predict hydrodynamic loads induced by fluid flow around an offshore platform. In CFD, model validation and verification are performed before model testing. A two-dimensional rectangular wave tank is developed using ANSYS Fluent for the validation, verification, and fluid-structure interaction modelling. Stokes waves are generated in the wave tank. The validation and verification of the wave tank are performed by comparing the free surface elevation of simulated waves with the ideal and experimental wave. After successful model testing, a rectangular deck is introduced in the tank to determine the wave-in-deck loads. It is concluded from the study that CFD results agree well with analytical and experimental results.

Haline Convection within a Fresh-Saline Water Interface in a Stratified Coastal Aquifer Induced by Tide

Sea-tide effects on the fresh-saline water interface (FSI) in a stratified coastal aquifer are examined through laboratory experiments. The physical model, a two-dimensional rectangular flow tank, is filled with layered aquifers and aquitards. The aquifers serve as the main entrances/exits of water to/from the system through significant horizontal flows, creating unstable conditions of heavier saline water above lighter freshwater for short periods of time. Several processes create mixing; this instability results in haline convection, creating downward fingering, stable rising of horizontal saltwater front, and unstable upward fingerings of flushing freshwater. The time lag between the sea tide fluctuations and the emergence of adequate fresh- and saltwater is higher in a stratified system compared to a homogeneous system. Furthermore, longer tide cycles lead to the enlargement of the FSI’s toe horizontal movement range. The combination of tidal forcing with a layering aquifer structure leads to a wider FSI by creating a significant salt- and freshwater mixing inside each layer, vertical flows between the layers, and saltwater bodies at isolated areas. Haline convection within the FSI might be the reason for the wider fresh-saline interfaces that are found in field studies.

Coupling between resonant sloshing and lateral motions of a two-dimensional rectangular tank

Abstract

Frequency analysis and control of sloshing coupled by elastic walls and foundation with smoothed particle hydrodynamics method

Broadband stochastic excitations from environment such as seismic activities, sea waves, wind gusts, and tremors can lead to the terrible damages of tall structures and buildings which usually have a low amount of vibration damping. Tuned liquid dampers (TLD) have been considered for around thirty years regarding their ability for passive suppression and absorb the vibration energy in such structures. In this paper, the fluid-structure interaction of a two-dimensional rectangular tank with elastic side walls and foundation is investigated by the smoothed particle hydrodynamics (SPH) method. First, the implementation of SPH for the frequency analysis of a rectangular container with flexible boundaries, partially filled with liquid, is described. The governing equations of the sloshing of an inviscid liquid in a two-dimensional Cartesian coordinate system coupled with the thin plate vibration on sides are linearized and solved by the SPH method. The sloshing and bulging modes of the system are derived and discussed in this paper. The results of the frequency spectrum of the sloshing tank coupled with the spring-damper system are analyzed and compared with the mechanical equivalent system. Results are validated by analytical benchmark and show the applicability as well as the simplicity of the SPH method for analytical vibration analysis of a fluid-structure coupled system. Then various control method including: Active mass damping (AMD) control designed based on proportional feedback control, the tuned liquid damper (TLD) control based on the analytical model, and a hybrid method of active tuned liquid damper (ATLD) are modeled and compared in a one-story structure. Results demonstrated that although the AMD is more efficient and stable in a specific range of feedback coefficient, the proposed ATLD is reliable in a wider range of control parameters.

A formula for calculating fundamental natural frequency of partially-filled tanks

By thoroughly considering the fluid-structure interaction, this paper aims to analytically analyze the free vibration of liquid sloshing in partially-filled two-dimensional flexible rectangular tanks. It is assumed that the fluid is compressible, irrotational and inviscid. Additionally, a closed-form formula is derived for estimating the fundamental natural frequency of this system by performing comprehensive curve fitting. Through various numerical examples, the correctness of authors’ exact solutions is corroborated, and the effects of different parameters on sloshing natural frequencies and mode shapes of the aforesaid fluid-structure system are assessed.

Liquid sloshing in a two-dimensional rectangular tank: A numerical investigation with a T-shaped baffle

In this study, the liquid sloshing in a closed, partially filled, T-shaped baffled and unbaffled two-dimensional rectangular tank was numerically investigated. The tank was rotated with two different rotation angles of 4° and 8°, and a fixed angular velocity of 3.3 rad/s, which was determined by taking the natural frequency of the tank into consideration, for the filling depths of 50% and 75%. The baffle heights as well as the rotation angles and filling depths were systematically varied and the effect of these parameters on the hydrodynamic loads on the tank wall and free surface elevations was examined. The calculations were carried out in two different ways, by means of laminar and turbulent viscous flow solvers, which use the finite differences and finite volume discretization techniques in conjunction with the “volume of fluid” technique, respectively. The general flow topology, free surface deformations, and vorticity distributions are provided and discussed in detail. It was found that the baffle was fully effective in pressure and wave damping when its height was greater than 80% of the liquid level. • Laminar and turbulent flow computations were performed for the simulation of liquid sloshing in a two-dimensional rectangular tank with a T-shaped baffle. • The effect of the baffle height, liquid depth and rotational angles on the pressure levels acting on the side wall were demonstrated. • The topological features of the flow, free surface deformations and vorticity distributions at constant sloshing phases were presented.

Numerical Simulation of Sloshing Motion in a Rectangular Tank using Differential Quadrature Method

The numerical simulation of sloshing waves for Laplace equation with nonlinear free surface boundary condition in a two-dimensional (2D) rectangular tank is performed using the Differential Quadrature Method (DQM). Application of the DQM to the Laplace equation and the nonlinear free surface boundary condition gives two sets of Ordinary Differential Equations (ODEs) in time. These two sets of ODEs are coupled with each other and can be expressed as a system of nonlinear ODEs which can be further discretized in time using various time integration schemes. The resultant system of nonlinear algebraic equations can then be solved using various iterative methods. In this study, the backward difference time integration scheme (of order six) in conjunction with the Newton-Raphson method is used to solve the resultant system of nonlinear ODEs. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison study with the available solutions in the literature is performed. Numerical results reveal that the DQM can be used as an effective tool for handling nonlinear sloshing problems.

Shallow-water sloshing motions in rectangular tank in general motions based on Boussinesq-type equations

Based on the highly accurate Boussinesq-type equations in terms of velocity potential, the shallow-water sloshing in a two-dimensional rectangular tank is studied. The rectangular tank in harmonic sway, heave and roll motions with small excitation amplitudes is considered. The total velocity potential is divided into two parts: the particular solution and the remaining part to be determined by the Boussinesq-type equations. The Stokes-Joukowski potential is adopted in the particular solution for the roll excitation motion. The comparisons of the numerical results indicate that the shallow-water sloshing motions in a rectangular tank can be predicted well based on the Boussinesq-type equations.

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.

Global force and moment in rectangular tanks through a modal method for wave sloshing

Basing on a modal description of the sloshing phenomenon, formulas for the global force and moment acting on two-dimensional rectangular tanks are proposed. These are extensively validated through comparison with experimental data for roll motions at different angles of excitement. Moreover, to extend the applicability of the modal method to the most violent breaking cases, a diffusive variant of the scheme is proposed. This relies on the use of a proper diffusive term in the continuity equation and allows for the overcoming of some numerical issues related to the sloshing dynamics in very shallow waters. Finally, a qualitative description of the interaction between diffusion, dispersion and nonlinearities has been proposed for the present modal scheme, along with a physical interpretation of the diffusive term.

The coupling analysis of tank motion and sloshing by a fully nonlinear decoupling method

In this paper, the three degrees-of-freedom motion of a two-dimensional rectangular liquid tank under wave action is simulated by the boundary element method in time domain. The coupling effects between tank motion and internal sloshing flow are investigated in partially filled conditions. The fourth-order Runge–Kutta method is adopted to update the wave shape and velocity potential on the free surface. The fully nonlinear mutual dependence of the incident wave, tank motion and internal sloshing flow is decoupled through an auxiliary function method, by which the liquid tank acceleration can be obtained directly without knowing the pressure distribution. The corresponding validation of numerical model is carried out and indicates that the accuracy of the present method is satisfactory to evaluate the dynamic responses of tank and sloshing motion. The corresponding response amplitude operators of tank motions for various wave frequencies, amplitudes and filling conditions are obtained, and the nonlinear coupling effects of sloshing flow on the tank responses are analyzed. It is found that the coupling effects have significant influence on sway and roll motion while have little impact on heave motion. The most important coupling effects on roll motion are the split of peak. In addition, due to the nonlinearity of sloshing flow, the roll motion amplitude is not linearly proportional to wave amplitude.

Analytical solution for free vibration of flexible 2D rectangular tanks

This paper provides a brief literature review on dynamic behavior of liquid storage containers. Moreover, the free vibration of two-dimensional deformable rectangular tanks fully-filled with a compressible fluid is analytically investigated, and the exact solutions are derived. In this study, the fluid-structure interaction is rigorously considered, and the fluid is assumed to be irrotational, inviscid and compressible. Furthermore, a closed-form expression is introduced for evaluating the fundamental frequency of the flexible 2D rectangular tanks filled with compressible liquid. In addition, an approximate formula is suggested to determine the corresponding pressure distribution on tank walls. Finally, numerical examples are performed to verify the presented closed-form relations. To study the effect of various parameters on natural frequencies and mode shapes of the fluid-structure system, a sensitivity analysis is also conducted.

Numerical model of sloshing in rectangular tank based on Boussinesq-type equations

The highly accurate Boussinesq-type equations in terms of velocity potential are adopted for the simulation of sloshing phenomena in a two-dimensional rectangular tank. The fully nonlinear free surface conditions are used and linearized energy dissipation term is added into the dynamic free surface condition. The total velocity potential is divided into two parts: the particular solution and the rest which is calculated by the Boussinesq-type model. Different filling levels of liquid in the tank are considered for the validation of the numerical model, including shallow water, intermediate water depth and finite water depth cases. The linear theory based on the superposition of eigen-modes corresponding to natural sloshing frequencies is introduced. Compared with results calculated by linear theory and results from literatures, the numerical model based on Boussinesq-type equations presents good performances in predicting the sloshing motions in rectangular tank with small excitation amplitude.

Flow Visualization of Sloshing in an Accelerated Two-Dimensional Rectangular Tank

In this paper, sloshing in a 2-Dimensonal (2-D) square liquid tank subjected to horizontal excitation is investigated by means of Computational Fluid Dynamics (CFD) technique. The tank with/out the vertical and horizontal baffles located at each side walls of the tank is moved on positive (+) x-axis for the excitation of 4.5 m 2 /s for each case studies. After series of simulations results obtained for each tank configuration are compared and flow is visualized for the tanks 50% filled with the fresh water.