# Study on liquid sloshing characteristics of a swaying rectangular tank with a rolling baffle

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## Abstract

This study addresses the sloshing characteristics of a liquid contained in a tank with a vertical baffle mounted at the bottom of the tank. Liquid sloshing characteristics are studied through an analytical solution procedure based on the linear velocity potential theory. The tank is forced to sway horizontally and periodically, while the baffle is fixed to the tank or rolling around a hinged point. The rectangular tank flow field is divided into a few sub-domains. The potentials are solved by a separate variable method, and the boundary conditions and matching requirements between adjacent sub-domains are used to determine the sole solution. The free surface elevations with no baffle or a low fixed baffle are compared with those in published data, and the correctness and reliability of the present method are verified. Then the baffle is forced to rotate around the bottom-mounted point. It is found that the baffle’s motion, including the magnitude and the phase together, can be adjusted to suppress the free surface elevation, and even the sloshing wave can be almost eliminated.

## Keywords

Forced waves Linear velocity potential theory Rolling Baffle Sloshing waves Swaying tank Wave elimination## 1 Introduction

Sloshing problems in liquid tank have to be considered for the design of cargo ships containing fluids such as OBO, LNG, LPG and VLCC. When the tank is forced to move, the free surface of the liquid in the tank deforms, fluid moves in large amplitudes, and a resultant impact pressure is exerted on the tank walls which may damage the local structure or influence the stability of the ship. Thus, sloshing problems have attracted significant attention. Liquid sloshing leads to high local pressure and large total force, both of which are important to consider in vessel design. The two most prominent causes of severe liquid movement in a cargo tank are the resonance that happens when the sway frequency of the tank is near to or equal to the natural frequency of the liquid movement, and the movement of liquid in a tank with little or no damping. In the case of resonance, the liquid movement in the tank becomes highly magnified. With regard to damping, a rectangular tank with baffles mounted at the bottom is the common engineering remedy applied to loaded liquid containers.

Liquid sloshing can be induced by tank motion. Scholars have carried out a great deal of research on tank sloshing, and most research methods are mature. They can be divided into three approaches, namely numerical, experimental and analytical methods.

There are two main numerical schemes for addressing sloshing problems: one is to apply potential flow theory [1, 2, 3], the other is to model the viscous Navier–Stokes equations [4, 5, 6]. Both methods are extensively used to study the weakening effect of the baffles. Popov et al. [7] studied forced turning and braking of the fluid within a tank, obtained the steady-state solutions and analyzed wave height, force and overturning moment by solving continuous Navier–Stokes equations numerically. Wu et al. [8] simulated sloshing waves in a 3D liquid tank by a finite element method based on potential flow theory. The finite element method was adopted by Pal [9] to study the fluid behaviors in a flexible thin-walled cylindrical tank. On the basis of the spatially averaged Navier–Stokes equations, Liu and Lin [10] employed the large eddy simulation approach to model sloshing in a tank with bottom mounted baffles. By using the boundary element method, Wang et al. [11] studied the internal sloshing flow of a tank freely moving in an incident wave by the boundary element method. Zhao et al. [12] investigated sloshing problems in partially filled membrane tank under forced excitations, and viscous effects were considered through validation from experimental tests. Chu et al. [13] focused on mitigating effects of the number and the height of baffles mounted at tank bottom by analyzing the data from large eddy simulation and experiments.

Large displacement of the liquid in a tank produces local high stress on the walls of the vessel. Under these circumstances, the fluid motion has strong nonlinear characteristics, which presents great obstacles to theoretical analysis and numerical calculation. Some assumptions need to be introduced into the analytical or numerical procedure to render the problems solvable, but such assumptions may lead to deviation from the accurate solution. In contrast, experimental studies do not need the artificial assumptions used in numerical or analytical methods, and from this aspect, they are more reliable. Celebi and Akyildiz [14] reported experimental results to show that a shear layer and dissipated energy caused by the viscous term would form in the presence of a baffle inside the tank. They also revealed that the motion of the tank had influence on the nonlinearity of the sloshing phenomena. To analyze the pressure on tank walls and the distorted free surface, a series of experiments were undertaken by Panigrahy et al. [15]. The sway motion of the tank was controlled by a shaking table. The sway frequency of the shaking table was changed, the fill level was altered, the experimental cases with and without baffles were repeated, and the pressure and the free surface elevation were studied.

In both experimental studies and the numerical simulations, one major difficulty is tracking the deformation of the free liquid surface. From the viewpoint of an actual project, quantifying the free surface elevation in the tank is not necessary, as the most important concerns are understanding how large the natural frequencies of the tank are, and taking effective measures to avoid resonance. The resonance that occurs in a liquid cargo tank and effectiveness of mechanisms for suppressing liquid sloshing can be well characterized by analytical methods. Compared with the experimental method and the numerical method, the analytical method can easily give the relationship between physical quantities, and has the advantages of fast calculation speed and small error. Abramson [16] applied linear potential flow theories to analyze the liquid motion in cylindrical and spherical tanks. Evans and Mciver [17] studied the influence of a vertical baffle on the resonant frequency of the fluid in a rectangular container by linear wave theory. The accuracy of the simple approximation is evaluated by comparing it with the exact solution based on eigenfunction expansions. It was found that a surface-piercing baffle can significantly change the resonant frequencies, while the effect of a bottom-mounted baffle is usually negligible. Wu [18] employed velocity potential theory to study the second-order resonance of sloshing. Zhou [19] deduced the analytical solution of the wave radiated by a bottom-opened rectangular body floating in water of finite depth. Zhang et al. [20] found that floating foams can weaken both the sloshing amplitude and the hydrodynamic pressure through observations from experiments and an analytical potential-flow solution.

## 2 Governing equation and boundary conditions

A two-dimensional rectangular tank model is shown in Fig. 1. The tank is undergoing horizontal oscillation with displacement *X*(t), the total length of the rectangular tank is 2*l*, the height from the free liquid surface to the bottom of the tank is taken as *h*, and a rigid baffle with a height of \(h_{1}\) is fixed to the bottom of the tank. The distance from the rigid baffle to the right wall is *a*, and to the left is *b*. In the coordinate system, the *z*-axis is vertical and the *x*-axis is horizontal and is coincident with the baffle line.

*denotes the translational velocity of the tank, \({\varvec{\Lambda }}\) represents the rotational speed, and*

**U****r**is the position vector to the hinged point of the baffle. According to the Bernoulli equation, the dynamic free surface condition can be expressed as [18, 26]

*g*is the gravity acceleration. Equation (6) is transformed by Taylor expansion, ignoring the higher order term, and the linearized free surface condition is obtained

## 3 Sloshing of a swaying liquid tank with a fixed baffle

### 3.1 Expressions for potential in the fixed baffle case

### 3.2 Solution for the unknown coefficients in the fixed baffle case

## 4 Sloshing of a swaying liquid tank with a rolling baffle

### 4.1 Expression for the potential in the rolling baffle case

### 4.2 Solution for the unknown coefficients in the rolling baffle case

## 5 Analytical results

### 5.1 Convergence study with the number of the series

*N*is performed here. The size and parameters of the tank are \(a=b=l=h=0.7\,\hbox {m}\). The motion can be expressed as \(X = -X_{0} \cos \Omega _{0} t\) and the external excitation frequency is \(\Omega _0 =1\,\hbox {rad/s}\). The baffle’s height \(h_{1}/h\) is taken as 0.3, 0.5, 0.7, 0.9, respectively. The results with \(N=15\) and \(N=50\) are presented in Fig. 2, which shows that they are in very good agreement. This means \(N=15\) can provide sufficiently accurate results; therefore, in the following discussion, the potential expressions are truncated with 15 series, unless otherwise stated. When \(h_1 /h\) increases from 0.3 to 0.9, the characteristic frequency \(\omega _1\) in Eq. (24) decreases from 4.2 to 1.85, and they all are much larger than the excitation frequency \(\Omega _0 =1\,\hbox {rad/s}\) and thus no resonance happens. In range of \(\omega _1 /\Omega _0\) changing from 4.2 to 1.85, the influence of the characteristic frequency \(\omega _1\) on the value of free surface elevation is relatively smaller.

### 5.2 Verification of the results

### 5.3 Influence of baffle height on wave elevation

### 5.4 Influence of baffle height on fundamental frequency

### 5.5 Sloshing of a swaying liquid tank with a rolling baffle

The height of the baffle cannot be easily changed after tank construction is completed. We can achieve the purpose of stabilizing the tank by adjusting the forced movement of the baffle. As linear systems satisfy the superposition principle, the movement of a tank can be decomposed into two kinds of motion. The first is the whole tank moving with velocity \(U(t)=U_{0} \sin \Omega _{0} t\), which has been described in detail in Sect. 5.1 for a fixed baffle relative to the tank. Its translational equation can be set as \(X(t) = -X_{0} \cos \Omega _{0} t\). The second is to point (\(0, \hbox {-}\) h) as the rotating center and make a slight roll at angular velocity \(\varpi (t) = \varpi _{0} \sin (\Omega _{\mathrm{p}} t + \varphi )\) under the external excitation frequency \(\Omega _{\mathrm{p}}\). The symbol \(\varphi \) stands for the initial phase. Suppose the equation of the roll angle with time is \(\theta (\hbox {t}) = - \theta _{0} {\text {cos}}(\Omega _{p} t+ \varphi )\). Here, \(\theta _0\) represents the maximum roll angle that can be reached by the baffle’s movement, \(\theta (t)=0\) corresponds to the vertical position and it is positive that the baffle rotates counter-clockwise.

This section mainly concentrates on how to control the movement of the baffle in order to suppress the rise of the free surface in the tank and avoid the resonance that occurs the tank does translational motion and the baffle is forced to roll. We adjust the initial phase of baffle movement so that wave height \(\eta _1\) and wave height \(\eta _2\) are opposite, and adjust the maximum roll angle of the baffle’s movement so that the amplitude of \(\eta _2\) is almost equal to that of \(\eta _1\). Finally, the purpose of elimination of waves can be achieved.

## 6 Conclusions

A baffle was mounted at the tank bottom to inhibit sloshing; it divides the liquid tank into three regions which were solved analytically based on linear potential theory. The potential and its directional directive between different regions were continuous. Firstly, the baffle was fixed to the tank and the tank was in *sway* motion. The analytical method was verified by comparing the present result with published data. When the excitation frequency is close to the first natural frequency of the unbaffled tank, resonance will happen. Increase of the baffle’s height can inhibit the resonance, as it can make the natural frequency of the tank smaller. Then the baffle was set to roll around a hinged point. According to the superposition principle of linear systems, the movement of the liquid tank can be divided into the sway motion of the liquid tank and the roll motion of the baffle. The disturbed sloshing waves produced by two kinds of motion are summed up to obtain the total wave surface. By changing the initial phase of the forced motion of the baffle and the maximum roll angle of the baffle, the elevation of the sloshing waves generated by sway of the tank and roll of the baffle can be adjusted to be equal, and the phase to be opposite. Through this adjustment, the purpose of wave elimination can be achieved whether the frequency of sway motion is close to or far away from the resonant frequency. Under linear theory, the maximum free surface elevation is prone to emerge at two sides of the tank. Wave elimination method in the present paper is effective both for two sides as well as for the interior part of the tank.

## Notes

### Acknowledgements

This work is supported by National Key R&D Program of China (2018YFFC0310500) and the National Natural Science Foundation of China (Grant No. 51679045). This work is also supported by Lloyd’s Register Foundation (LRF) through the joint center involving University College London, Shanghai Jiaotong University and Harbin Engineering University, to which the authors are most grateful. LRF supports the advancement of engineering-related education, and funds research and development that enhances safety of life at sea, on land and in the air.

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