Liquid sloshing in a swaying/rolling rectangular tank with a flexible porous elastic baffle

Highlights

• The anti-sloshing performance of a bottom-hinged and top-moored vertical porous elastic baffle was investigated.

• The analytical model based on the matched eigenfunction expansion method along with the Green function was developed.

• The porous effect of the baffle was considered by using Darcy’s law.

• The liquid sloshing and sloshing-induced forces were assessed with the baffle’s geometry and structural parameters.

Abstract

The flexible porous elastic baffle was proposed as an anti-slosh device to reduce the liquid sloshing in a swaying/rolling rectangular tank. The analytical model was developed to assess the effectiveness of the porous elastic baffle on liquid sloshing. A one-dimensional beam model, along with Darcy's model, was taken to implement the characteristics of porous elastic baffle. The matched eigenfunction expansion method (MEEM) with the Green function for the liquid sloshing interaction with the porous elastic baffle was employed. Numerical results were presented that illustrate the effects of the baffle's geometry (porosity, height), and structural parameters (flexural rigidity, stiffness of mooring line) on the liquid sloshing. Through parametric studies, the flexural rigidity, porosity, and mooring line's stiffness have to be adjusted properly to enhance the sloshing-reduction effect. The flexible porous elastic baffle showed the potential as a useful anti-slosh device within a limited application range compared with a rigid baffle.

Introduction

Sloshing is associated with an uncontrollable violent motion of free surface inside the liquid tank subjected to external excitations. The sloshing-induced loads may damage the tank; thus, the liquid sloshing should be reduced to avoid the liquid carrier instability and structural failure. Among anti-slosh devices, the vertical and horizontal baffles have been known as an effective tool for suppressing liquid sloshing. Abramson [1] first experimentally and theoretically studied the baffles to suppress liquid sloshing in the tank. Since then, many researchers have been greatly interested in various baffle configurations, including horizontal, vertical, ring, bottom-mounted, and surface-piercing baffles as well as porous (slotted, perforated) baffles. Their researches showed that appropriate baffle's dimensions, shapes, numbers, and arrangements can effectively reduce the liquid sloshing. Although such baffles effectively damp the sloshing energy, they impose a substantial weight on the liquid carrier. Recently, the flexible structure has also gained much interest in many engineering fields. Especially, the introduction of the flexible elastic material to the baffle is of some merits, such as light, inexpensive, and rapidly deployable.

There have been several previous investigations, both theoretical and experimental, of flexible wave barriers. The multi-mode deflection of the vertical tensioned membrane barrier can also be explored to widen the effective frequency range for wave attenuation ([2]). Aoki et al. [3], and Williams et al. [4,5] investigated the vertical flexible tensioned elastic wave barrier consisting of a compliant, beam-like structure, anchored to the seabed and kept under uniform tension by a small buoyancy chamber at the tip of the barrier. Additional stiffness was provided by mooring lines attached to the buoyancy chamber. The efficiency of the flexible tensioned membrane barrier can be improved further by adding structural porosity. Wang and Ren [6] introduced the flexible porous structure, which has several advantages, for example, it can reduce the wave loads, and the unwanted reflected and transmitted waves by dissipating the wave energy within the porous structure. The flow through the flexible porous structure was assumed to obey Darcy's law. Darcy's law demonstrates that the fluid velocity passing through the porous barrier is linearly proportional to the pressure drop, and has been successfully applied to the thin-walled porous structure by several authors ([[7], [8], [9]]). Williams [10] investigated theoretically the hydrodynamic properties of a floating flexible breakwater consisting of a membrane structure attached to a small float restrained by moorings using the boundary integral method. The wave reflection and transmission were observed to be sensitive to the membrane length, initial tension, and mooring line stiffness. Cho et al. [11] studied the scattering of oblique waves by a tensioned vertical flexible membrane structure hinged at the seafloor and attached to a rigid cylindrical buoy at its top. Cho et al. [12] extended the oblique wave interaction with a dual vertical flexible membrane wave barrier hinged at the sea bottom using the matched eigenfunction expansion method (MEEM) and boundary element method (BEM). Further, Kumar et al. [13] carried out an analysis to investigate the scattering of water waves by a vertical porous elastic barrier pinned both at the free surface and the seabed in a two-layer fluid of finite depth. Karmakar et al. [14] analyzed the multiple moored vertical surface-piercing elastic barriers as an effective breakwater in the presence of porosity using the method of least square approximation and the wide-spacing approximation. Koley et al. [15] used the Fredholm integral equations technique to study the oblique wave scattering by the vertical flexible porous elastic. Koley and Sahoo [16] studied the scattering of obliquely incident surface gravity waves by the vertical porous flexible membrane in water of finite depth analytically using the MEEM and numerically using coupled boundary element-finite difference method. (BEM, FDM).

The wave interaction with the flexible elastic baffle in an oscillating sloshing tank has been addressed in only a few studies. Furthermore, most researches have focused on the effectiveness of a flexible membrane constraining the free-surface displacement for the reduction of liquid sloshing rather than the anti-slosh baffles. Bauer [17,18] and Bauer and Eidel [19] analytically evaluated the natural frequencies of liquid in a partially filled rectangular and upright cylindrical tanks with a flexible membrane or plate constraining the free surface. Eigenfunctions and eigenvalues of the coupled liquid–elastic structure were obtained from the equation of an elastic membrane or plate through the compatibility condition at the surface of contact of liquid and elastic structure. These studies showed that the coupled slosh-membrane or slosh-elastic plate frequencies are higher than those of the unconstrained liquid alone. Kana and Dodge [20] experimentally investigated the anti-slosh effectiveness of an inextensible flexible bladder constraining the free-surface in upright and inverted tanks. The study verified substantial effects of a flexible bladder on the free-surface motion and the slosh frequencies and proposed a mechanical-equivalent model to describe the slosh dynamics of liquid coupled with the flexible bladder. Bohun and Trotsenko [21] proposed an analytical solution for the hydroelastic problem of free liquid oscillations in a circular cylinder container with an arbitrary axisymmetric bottom and an elastic membrane or plate constraining the free-surface. Parasil and Watnabe [22] investigated the interactions between a two-dimensional rubber-like membrane and the liquid in a rectangular tank. An Arbitrary Lagrangian-Eulerian (ALE) method was used to define the moving boundaries.

In this study, the liquid sloshing excited by a swaying/rolling rectangular tank equipped with the flexible porous elastic baffle has been investigated in the context of two-dimensional linear potential flow theory. The MEEM was employed to obtain the analytic solutions for the liquid sloshing. The elastic baffle is modeled by a compliant beam structure of uniform flexural rigidity, fixed at the tank bottom, and free at the upper end kept by a mooring line. The porous effect of the baffle was considered by using Darcy's law. The dynamic behavior of the flexible porous elastic baffle was described by an appropriate Green function. Numerical results were presented for illustrating the effects of the baffle's geometry (porosity, height), and structural parameters (flexural rigidity, stiffness of mooring line) on the liquid sloshing and sloshing-induced forces on the tank and baffle. Small-scale physical model tests with the porous elastic baffle of infinity rigidity (porous rigid baffle) were carried out to validate the present analytical model.