Experimental findings of the suppression of rotary sloshing on the dynamic response of a liquid storage tank
Introduction
Liquid storage tanks are strategic structures that regulate and store diverse types of liquids, e.g. water, oil, and other chemicals. After strong earthquakes, water storage tanks contribute to the resilience of communities as a reservoir of a vital resource for human consumption and firefighting. However, post-earthquake observations have repeatedly demonstrated that sloshing of the water surface may cause the sinking of the floating roof, and produce instabilities in the tank wall (Jia and Ketter, 1989, Zama et al., 2012). Additionally, experimental observations of storage tanks under dynamic loads demonstrate that sloshing innately tends to transform into a chaotic motion, even for small vibrations of the tank (Abramson et al., 1966a). To provide solutions and avoid damage to the wall of storage tanks, the designer must understand the dynamic behaviour of the tank–water system, including the possible consequences of particular types of sloshing (Abramson et al., 1966a).
The motivation for this work stems from the procedure suggested by the guidelines for the seismic design of storage tanks, e.g. American Petroleum Institute (2007), Eurocode 8 (2006) and NZSEE (2009). In these guidelines, the contribution of sloshing to the tank response is pragmatically incorporated by performing a quasi-static analysis using the spring–mass model proposed by Housner (1957). The model assumes that storage tanks respond mainly in two different vibration modes, i.e. the impulsive and the convective mode. The impulsive mode corresponds to that part of the liquid that moves as if it is rigidly attached to the tank wall and excited by relatively high-frequency forces. The convective mode represents the sloshing of the liquid triggered by relatively low-frequency excitations. According to Malhotra et al. (2000), in most of the cases the first impulsive and the first convective modes are sufficient for a sufficiently accurate prediction of the dynamic response of tanks for design purposes. Consequently, the contribution of higher impulsive and convective modes are usually ignored.
However, the consequences in major earthquakes, observed by e.g. Hatayama (2008), Yazici and Cili (2008), Zama et al. (2012), and theoretical studies performed by e.g. Clough and Niwa (1979), Haroun (1980) and Manos and Clough (1982), have demonstrated that the seismic response of storage tanks is more complex than was initially believed. The complexity lies in the geometric and possible material nonlinear response of the structure. This involves, among other factors, the interaction between the fluid and the tank wall (fluid–structure interaction), see e.g. the works by Abramson et al. (1966b) and Ibrahim (2005), the tank geometry, the aspect ratio, i.e. the ratio of the tank radius to the liquid height , and the sloshing pattern (Chiba, 1993).
Three generic sloshing patterns have been identified in partially filled cylindrical storage tanks, (Ibrahim et al., 2001), i.e. planar, nonplanar, and chaotic, as shown in Fig. 1. The first pattern is associated with small oscillations where the free surface remains planar, see Fig. 1(a). The second pattern corresponds to higher sloshing heights than the first pattern, and the free surface is nonplanar, see Fig. 1(b). The third pattern refers to abrupt velocity changes with time that lead to chaotic motion, see Fig. 1(c). Royon-Lebeaud et al. (2007) studied both theoretically and experimentally the transition of the sloshing pattern from planar to nonplanar and from nonplanar to chaotic in down-sized cylindrical and rectangular tanks. However, their results showed that the sloshing pattern may be meaning-fully up-scaled to larger containers than employed in their study, if geometric and frequency similarities are fulfilled.
Within each of the modes of tank response, i.e. impulsive and convective, the sloshing pattern depends mainly on the characteristics of the excitation, e.g. duration, frequency content, and maximum acceleration. If the sloshing pattern remains planar, which indicates that high convective and impulsive modes of vibration are not activated, Housner’s spring–mass model may give relatively accurate results of the tank response.
Effective methods to keep the sloshing pattern planar include rigid and flexible baffles in the horizontal and vertical planes, floating roofs and mats. Shekari (2014) theoretically demonstrated that a ring-shaped baffle reduced the response of the tank–water system in terms of the maximum shear force and overturning moment of the tank. It is emphasised that the baffle should be located as near as possible to the liquid surface, which transforms the baffle into a type of a rigid lid. Garza and Abramson (1963) and Hosseini et al. (2017) also found experimentally that a suspended ring-shape baffle reduced the maximum sloshing height, but it did not affect the frequencies of the convective mode. Askari et al. (2011) analysed numerically the effect of a rigid empty cylinder in a cylindrical tank on the sloshing frequencies of the tank-fluid system. The internal cylinder is fixed at the top end and the lower part is partially submerged to suppress sloshing. It was concluded that if the lower part of the internal cylinder is submerged deeply enough, the free surface could be divided into two independent regions, resulting in a reduction of the sloshing amplitude
Floating roofs and mats modify the dynamic response of a tank–water system significantly. A floating roof reduces the sloshing height but increases the stress in the tank wall in conjunction with the provision of associated damping (Abramson and Ransleben, 1961). According to some numerical analyses performed by e.g. Golzar et al. (2012), Sakai et al. (1984), and Yoshida et al. (2010), the presence of a floating roof has little influence on the first sloshing frequency of the tank–water system. Consequently, current seismic design methods of storage tanks using only the first mode approach, e.g. those according to American Petroleum Institute (2007), Eurocode 8 (2006) and NZSEE (2009), the presence of a floating lid can be ignored. However, the studies mentioned above do not elucidate the effects of a mitigation of sloshing on the development of acceleration and strain in the tank wall.
If the sloshing pattern evolves from planar to nonplanar, the liquid surface may respond by generating a considerable sloshing height that can lead to a chaotic sloshing. Chaotic sloshing increases the magnitude of stress in the floating roof and the stress in the tank wall (Matsui, 2007, Matsui and Nagaya, 2013, Yamauchi et al., 2006). A nonplanar sloshing, that eventually transforms into the chaotic sloshing observed in circular and rectangular containers, is the catalyst leading to generic rotary sloshing. Rotary sloshing occurs when the frequency of the applied lateral harmonic force is close to the lowest frequency of the tank–water system (Abramson et al., 1966b, Berlot, 1959, Ibrahim et al., 2001). Dodge et al. (1965) and Hutton, 1964, Hutton, 1963 describe the characteristics of rotary sloshing and emphasise the causes. Collectively their theoretical results reveal there is a coupling between the motion of the fluid in directions both parallel and perpendicular to the excitation direction. The coupling effects were studied by Kana, 1989, Kana, 1987 and Miles (1984a), who concluded that rotary sloshing produces a cross-axis force in the direction horizontally perpendicular to the excitation that cannot be neglected for design purposes.
Opinions concerning the causes of rotary sloshing are divided. On the one hand, the rotation of the centre of gravity of the tank–water system may cause an angular moment by producing a rotation of the fluid particles (Berlot, 1959). On the other hand, high modes of vibration of the tank wall in the axial and circumferential directions, called breathing vibrations (Lindholm et al., 1962), may cause rotary sloshing (Graham, 1960). In the circumferential direction, the tank wall can vibrate in two asymmetric modes that are circumferentially 90 degrees out-of-phase with each other. Thus the circumferential mode, excited by a harmonic external force, contains what are called the driven mode and the companion mode, see e.g. (Dowell and Ventres, 1968, Evensen, 1967). The driven and the companion modes of the tank wall can occur simultaneously. This usually occurs when the frequency of the harmonic excitation is close to the natural frequency of a tank–water system and the combined vibrations appear like a travelling wave (Amabili, 2008). The effects of the travelling wave on the natural frequencies of empty and liquid-filled cylindrical tanks are numerically studied by e.g. Amabili (2008) and Morand and Ohayon (1995). However, it is not clear whether rotary sloshing is a cause or a consequence of the simultaneous occurrence of the two asymmetric modes.
Another possible cause of rotary sloshing is the phenomenon called secondary (internal) resonance. According to Faltinsen and Timokha (2009), when a tank–water system is excited by a harmonic load with the fundamental frequency of the system, the excited water will subsequently excite the system. The excited water can have a significant contribution to the response of the system. Numerical results provided by Matsui and Nagaya (2013) confirmed that the contribution of the second convective mode to the development of axial stress should never be ignored. Consequently, several higher modes are needed to adequately reflect the characteristics of rotary sloshing. However, why after a while a rotary sloshing changes the direction of the rotation, is still not clear.
To the authors’ best knowledge, no experimental data is published about the effects of rotary sloshing on the tank response and the occurrence of breathing vibrations. The objective of this research is to experimentally elucidate the effects of rotary sloshing on the dynamic response of a storage tank partially filled with water. The dynamic response monitored and interpreted includes the development of both axial and hoop strain in the tank wall, and the acceleration at the bottom, middle, and top of the tank. The influence of rotary sloshing on the tank breathing vibrations, which enhance the development of strain in the tank wall, is also investigated.
Additionally, a floating lid is utilised to suppress rotary sloshing. Results from the tank–water system response with and without the floating lid are compared. To evaluate the effectiveness of the floating lid the tank response is analysed by applying a spectrogram, which is a visual representation of the intensity of a variable over a spectrum of frequencies as a function of time. The experiments performed also provide data for the calibration of numerical models for analysing the behaviour of storage tanks when rotary sloshing occurs.
Section snippets
Characteristics of rotary sloshing
The alignment between natural frequencies of the tank–water system and the external force produces an extremely complex sloshing response (Amabili, 2008). Furthermore, solving the mathematical formulation of chaotic sloshing in a storage tank under dynamic loads can be very challenging. According to Ohayon and Schotté (2017), the fluid–structure interaction during rotary sloshing can be idealised as a problem of hydro-elastic sloshing. However, in a cylindrical tank, the vibration of a pendulum
Tank model
A low-density polyethylene (LDPE) cylindrical tank of height and radius was filled with tap water to form an aspect ratio of , i.e. the water level is . Fig. 5(a) shows the physical LDPE tank. The tank is roofless and unanchored, i.e. uplift may occur. Uplift is defined as a transient and partial separation of the base of the tank from the supporting foundation. The effects of uplift on the tank response have been experimentally assessed by e.g. Hernandez-Hernandez
Description of rotary sloshing
Rotary sloshing was observed in all experiments. Fig. 7 shows the history of a floating white ball that is used to track the time–history of the rotational displacement of a typical particle on the water surface. The ball is light and smooth, and this provides minimal resistance to rotation. The pattern shown in Fig. 7 occurs when the sine excitation has a frequency of . The phenomenon is described in the following four steps:
(1) The sloshing starts being planar, see Fig. 7(b), until the
Conclusions
A low-density polyethylene tank containing water was tested on a shake table using nine sine excitations with a range of frequencies between and . The main aim was to experimentally determine the effects of rotary sloshing on the development of hoop and axial strain and the acceleration of the tank wall. Additionally, the effects of the rotary sloshing on the occurrence of breathing vibrations, and response frequencies of the tank–water system were assessed. For some of the tests a
CRediT authorship contribution statement
Diego Hernandez-Hernandez: Conceptualization, Methodology, Writing - original draft, Data curation, Visualization, Formal analysis, Software, Investigation. Tam Larkin: Supervision, Writing - review & editing, Validation. Nawawi Chouw: Supervision, Writing - review & editing, Validation. Yann Banide: Data curation, Visualization, Investigation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The authors wish to thank the Mexican Government for awarding the first author the doctoral scholarship “CONACyT-SENER Hidrocarburos” for his PhD research at the University of Auckland.
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.
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