JIANG Mei-rong (蔣梅榮), REN Bing (任冰), WANG Guo-yu (王國玉), WANG Yong-xue (王永學(xué))

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China, E-mail: meirongjiang@live.cn

Laboratory investigation of the hydroelastic effect on liquid sloshing in rectangular tanks*

JIANG Mei-rong (蔣梅榮), REN Bing (任冰), WANG Guo-yu (王國玉), WANG Yong-xue (王永學(xué))

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China, E-mail: meirongjiang@live.cn

(Received July 1, 2013, Revised October 8, 2013)

A sloshing experiment is conducted to study the hydroelastic effect in an elastic tank. For this purpose, a translational harmonic excitation is applied to a 2-D rectangular tank model. The lowest-order natural frequencies of the liquid in the tank are determined through the sweep test. The wave elevation and the sloshing pressure are obtained by changing the excitation frequency and the liquid depth. Then the characteristics and the variation of the elevation and the pressure are discussed. The results are compared with the experimental results and the theoretical calculations in a rigid tank. Our analysis indicates that, in the nonresonant cases, the elastic results, the rigid experimental results and the theoretical values are all close to each other. In contrast, under the resonant condition, the elastic experimental result is slightly smaller than the rigid one. Also, the theoretical values are smaller than the experimental results at the resonant frequency.

liquid sloshing, elastic tank, wave elevation, sloshing pressure

Introduction

Sloshing must be considered for almost any moving vehicle or structure containing a liquid with a free surface and it can be the result of the resonant excitation of the tank liquid. Since the 1950s, the liquid sloshing in tanks has received a great deal of attention in the fields of the aerospace applications, the naval architectures, the ocean engineering and the civil engineering. Recently, the risk of leakage and the related security problems in the liquefied natural gas (LNG) carriers become increasingly an important issue due to the huge demands on the LNG terminals and the transportation systems all over the world[1].

Experiments are considered to be the most reliable method to reveal sloshing mechanisms, and the results also serve as the validation data for numerical simulations. A rather comprehensive review and discussion of the analytical and experimental researches was made, mainly focusing on the axially symmetric fuel tanks, as the starting point of the later successive researches in several areas[2].

There was a considerable amount of work on the 3-D liquid sloshing in the rigid tanks. The results of several research programs investigating sloshing in LNG carriers were delineated, and a three-parameter Weibull distribution was proposed to describe the peak pressure probability distribution[3]. Model tests were reported in which the sloshing forces on the instrumented structural members and the sloshing pressures were measured in shiplike tanks and shiplike internal structures[4]. The 2-D transient sloshing was modeled in the test and compared with the free surface elevation with the multidimensional modal method[5]. A time-resolved particle image velocimetry technique was applied in order to characterize the details of the fluid dynamics, and the wave loads were computed by integrating the experimental pressure distributions[6]. The pressures along the boundaries of the 2-D tanks of different geometries were measured, synchronized with the fast acquisition of images of the flow at the moments and locations of the impacts[7].

Meanwhile, several studies were carried out on the 3-D effect of the sloshing in rigid tanks. A series of model tests on the transient sloshing were condu-cted, to classify all 3-D nonlinear wave motions[8]. The pressure distribution and the 3-D effect under the pitch[9]and surge excitations[10]were studied in baffled tanks. Sloshing in a cylindrical tank with various fill levels and ring baffles was experimentally investigated, and it was found that ring baffle arrangements were very effective in reducing the sloshing loads[11]. In addition, the coupling effect between the sloshing in rigid tanks and the ship motion was also studied[12,13].

Liquid sloshing was studied extensively, but mostly focusing on the rigid tanks. The actual tank is elastic and will respond to the sloshing loads[14-16]. For example, in the LNG carriers, the large size bulkheads of the membrane-type tanks are typical elastic structures. The fluid–structure interaction could play an important role in the determination of the sloshing impact load due to the elasticity of the membrane-type containment system. Meanwhile, the safety of the structure can be strongly influenced by the dynamical response, such as the structural strain or movement. Elastic tanks were not extensively studied. Therefore, it is desirable to investigate the sloshing in elastic tanks and the related hydroelastic problems. A large scale test was conducted to analyze the hydroelastic effect[17], and a significant influence of the stiffness on the pressure pulse was observed, which shows that the elastic integrated pressure pulse is about 10% lower than the stiff one but that the peak pressure distributions of the two are very similar. The elasticity of the tank structure was found to have a significant influence on the height and shape of the impact pressure peak in the LNG tank model test[18]. There was no clear tendency for the case where the tank was 30% full but a 12%-19% reduction of impact pressure was observed for the case where the tank was 95% full. The dynamical strain of the copper panel on the rigid LNG tank was analyzed recently[19].

Fig.1 The sketch of the rectangular tank and the setup of the instrumentation (m)

Fig.2 The setup of the CCD camera and its use in the test

Fig.3 A typical experiment acquisition signal of the sweep test (h =0.250 m,A=0.010 m)

Despite many studies in these directions, the fundamental physics of sloshing in elastic tanks is stillnot well understood. In this paper, a systematic experiment is conducted to explore this subject, and a statistical approach is adopted to analyze the hydroelastic effect of the elastic tank. For this purpose, a physical model test is conducted to study the sloshing in an elastic rectangular tank under translational harmonic excitations. Different liquid depths and excitation frequencies are considered, and then the characteristics of the wave elevation and the sloshing pressure in the tank are discussed. The results are compared with the rigid experimental results and the theoretical calculations.

Table 1 The lowest-order natural frequency for the liquid depths of h=0.080 m and 0.250 m

1. Experimental setup

1.1The model tank and the instrumentation

The sloshing experiments under translational harmonic excitations are conducted at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The physical model tanks are mounted on a shaking table, which is capable of performing harmonic and random motions with three degrees of freedom. The working surface of the slip table is 0.500 m×0.500 m, and the range of the excitation amplitude and frequency are 0 m-0.0254 m and 0 Hz-1 000 Hz, respectively.

Two types of rectangular tanks, rigid and elastic, are modeled in our experiment. The rigid model is made of 0.012 m plexiglass sheets, while for the elastic model, the top, left and right bulkheads are made of 0.002 m plexiglass sheets, and the other parts are made of 0.012 m plexiglass sheets. The internal tank dimensions are L×H×B=0.5 m×0.5 m×0.1m . A sketch of the tank and the coordinate system xoy are shown in Fig.1.

Clean tap water at room temperature is used as the liquid in the tank. Its density and viscosity are ρ=1.0× 103kg/m3and ν=1.005× 103N· s/m2, respectively. In order to catch the free surface, the red fluorescent dye rhodamine is added to the water; this dye does not affect the density and viscosity of the water.

The sloshing pressures are measured by using the DS30-MD14 multi-point pressure-measuring system, made by the Research Institute of Water Transportation of Tianjin. All pressure signals are sampled at 1 000 Hz with a precision of ±0.01kPa . Nine pressure transducers are fixed on the bulkhead at different positions in the tank, as shown in Fig.1.

An image acquisition and data analysis system is designed to catch the free surface, as shown in Fig.2. The CM-140MCL industrial CCD camera from the Japanese company JAI and the 09C lens from Computar in Japan are used in this system, with the Coreco image acquisition card from DALSA Corporation in Canada. Shade cloth is used to block external sources of light and make the room dark, and a projector is set in the upper part of the tank, projecting a sheet of light on the free surface of the liquid from above.

With the red fluorescent dye Rhodamine in the water, a band of light will be formed on the free surface, while other parts of the liquid below the surface will still be dark. The camera is placed in front of the tank, capturing the band of light on the free surface at 30 frames per second. The images are collected through the acquisition software Sapera CamExpert. Later, a specially written MATLAB procedure for correcting distortions and detecting wave elevations is used to process and analyze the acquired images, and then the wave elevations at the desired points on the right part of the free surface are obtained. Two measuring points, E01 and E02, are shown in Fig.1 with the x coordinates being 0.125 m and 0.240 m, respectively.

1.2Test cases

Several cases are tested for two different liquid depths: the shallow liquid depth h=0.080 m (h/ l= 0.16) and the intermediate liquid depth h=0.250 m (h/ l=0.50). The amplitude A is set to several different values: 0.004 m, 0.006 m, 0.008 m and 0.010 m. In order to obtain the periodic steady-state sloshing waves, most test runs last at least 100 s, and the real running time of the test is determined by the excitation period and the specific circumstances of the test.

Fig.4 Snapshots and comparisons of the experimental and theoretical wave elevations

Around the primary resonance frequency f0, a relatively wide range of excitation frequencies is selected, between 0.5 Hz and 1.7 Hz. Each test is repeated at least three times, and the mean value of the wave elevation and the sloshing pressure is taken over the three test runs.

2. Results and discussions

A linear analytical solution for a rigid tank derived by Faltinsen[20]is employed and compared with the experiment results. This solution is widely used, such as Wu et al.[21], Liu and Lin[22], Ming and Duan[23]and Xue and Lin[24], to verify experimental data and numerical simulations. Here, we compare this theoretical solution with our experimental data, in order to highlight the nonlinearities in the experiment, such as the nonlinear phenomenon of the hydraulic jump, the double-peak phenomenon and the asymmetry of the wave crest and the trough. An artificial damping term is introduced in this approach, which is chosen based on trial-and-error from a variety of parameters in the tank. Different damping coefficients are used in the calculations, and in the case of the damping coefficient equal to 5%, the theoretical solution and the experimental data are most close to each other. Thus, the artificial damping coefficient of 5% is adopted in our calculations. Due to the limitations of the theoretical solution, there are no theoretical values at the measuring positions above the free surface. The experimental wave elevations and sloshing pressures in the elastic tank are analyzed in the time domain, and are compared with the experimental results and the analytical solutions in the rigid tank. The excitation amplitude A maintains the same value (0.006 m) in the following analysis.

Table 2 The average values of the experimental and theoretical wave elevations (h=0.080 m)

2.1Free surface elevations

As discussed in Section 2.1, only the right half of the tank is captured in the CCD images, which are presented as the snapshots of the wave elevation here. Figure 4 shows a snapshot of the wave elevation in the rigid experiment and a comparison of the experimental and theoretical elevations in the tank for the depth h=0.080 m . Under the resonant condition (f= 0.92 Hz, a double-peak phenomenon is observed at the crest of the E01 (x=0.125 m), which results from the joint action of the traveling wave and the water jump. First, a traveling wave propagates forward in the tank, generating the first elevation peak at E01 (see Fig.4(a), t2=83.52s), and then a hydraulic jump is formed when the traveling wave arrives at the bulkhead (t3=83.71s). The second peak is from another traveling wave, which is formed after the collision with the bulkhead and the collapse of the hydraulic jump on the bulkhead, propagating in the opposite direction (t4=83.87 s). By adopting the PIV technique, Lugni[6]described a similar process of the wave impact in a rigid tank. The physical phenomenon of the experimental wave elevation in the elastic tank is almost the same as that in the rigid tank, but its magnitude is slightly less than the rigid result by 7.19%, as shown in Table 2. The analytical solution is a regular cosine curve, and its magnitude is less than the magnitude in the test. At the position E02 (x=0.240 m), the crest is larger and sharper, while the trough is smaller and flatter, which is due to the nonlinear effect of the hydraulic jump near the bulkhead. In this case, the experimental elevation in the elastic tank is also less than that in the rigid tank by 21.31%, as shown in Table 2. The analytical solution is still a regular cosine curve, the magnitude of which is smaller than the experimental result.

Fig.5 Snapshots and comparisons of the experimental and theoretical wave elevations

Figure 5 shows snapshots of the wave elevation in the rigid experiment and comparisons of the experimental and theoretical elevations in the tank of the depth h=0.250 m . Under the resonant condition (f=1.11Hz), a large-amplitude asymmetric standing wave is observed on the free surface, accompanying the traveling wave, and overturns the free surface, as well as causing waves to impact on the tank roof. (Thedash-dotted line in Fig.5(b) represents the position of the tank roof, y=0.250 m ) This hit on the tank ceiling was also observed by Faltinsen[5]for a rigid tank. The physical process of the experimental wave elevation in the elastic tank is almost the same as that in the rigid tank. The wave elevations in the elastic tank are less than those in the rigid tank by 6.86% at the position E01 and by 3.10% at the position E02, and the experimental value is much greater than the theoretical value, as shown in Table 3. This suggests that the wave elevation of the liquid in the model tank is changed due to the influence of the fluid–tank interaction. The maximum wave crest (0.25 m) of the experimental result is 4.7 times as much as that (0.0533 m) of the analytical solution.

Table 3 The average values of the experimental and theoretical wave elevations (h=0.250 m)

The average value of the crest ηc,avgis chosen as the characteristic value to use in the statistical analysis of the wave elevation in the steady or quasi-steady state. Figure 6 shows the variation of the elevation versus different excitation frequencies at the liquid depths of h=0.080 m and 0.250 m. This indicates that the trends of the experimental results and that of the theoretical result are almost the same. When the excitation frequency f changes from the non-resonant frequency to the lowest-order natural frequency of the liquid, the wave crest increases gradually, achieving the maximum around the lowest-order resonance frequency f0. The increase of the wave crest is slow and flat in the non-resonant region, while it is substantial in the resonant region.

When the excitation frequency is far away from the lowest-order natural frequency of the liquid in the tank, the elastic experimental result, the rigid experimental result and the theoretical value are all relatively close to each other. As discussed in Section 2.2, the theoretical resonance frequency, the rigid and the elastic experimental resonance frequencies are 0.85 Hz, 0.92 Hz and 0.89 Hz, respectively, at the shallow liquid depth (h=0.080 m). For the intermediate liquid depth (h=0.250 m), the corresponding frequencies are 1.20 Hz, 1.11 Hz and 1.11 Hz, respectively. When the excitation frequency approaches the lowest-order natural frequency, the experimental elevations in the elastic tank are slightly less than those in the rigid tank. In addition, the theoretical values are smaller than the experimental results under the resonant frequencies at the two depths.

Fig.6 Wave elevations in the experiment and the analytical solution versus excitation frequencies

Fig.7 Comparisons of the experimental and theoretical pressures (h =0.080 m,f=0.89 Hz)

2.2 Sloshing pressure

At the static liquid level, the recorded value of the pressure transducer is zero when the water falls below this level. Therefore, for the measuring points at this level, only the positive pressures are analyzed when the analytical solution is compared with the experimental result. In order to facilitate the check and the comparison between the rigid and the elastic pressures, a phase is set to a reasonable value and modulated in the time-history curves, and the resulting phase difference does not appear in the actual data.

Table 4 The 1/3 maximum peak pressure of the theoretical and test results (h=0.080 m)

Fig.8 Comparisons of the experimental and theoretical pressures (h =0.250 m,f=1.11Hz)

Table 5 The 1/3 maximum peak pressure of the theoretical and test results (h=0.250 m)

When the excitation frequency is far away from the lowest-order natural frequency of the liquid, the elastic experimental result, the rigid experimental result and the theoretical values are all relatively close to each other. The theoretical resonance frequency and the rigid and elastic experimental resonance frequencies are given in Section 3.1 and are not repeated here. While the excitation frequency approaches the lowestorder natural frequency, the sloshing pressures on the elastic bulkhead are slightly smaller than those on the rigid bulkhead. In addition, the theoretical values are smaller than the experimental results under the resonant frequencies at the two depths.

3. Conclusions

A physical model test is conducted to study the hydroelastic effect of the sloshing in the elastic tank under translational harmonic excitation. The results are compared with the experimental results for a rigid tank and with theoretical calculations. The following conclusions can be drawn.

Fig.9 Pressures in the experiments and the analytical solution versus excitation frequencies

(1) There is no significant difference between the experimental lowest-order natural frequencies of the liquid in the elastic tank and in the rigid tank, but these two frequencies slightly deviate from the theoretical value in the rigid tank. At the shallow liquid depth (h/ l=0.16), the experimental frequencies are slightly greater than the theoretical one by about 5%, however, at the intermediate depth (h/ l=0.50), they are less than the theoretical one by 5%-10%.

(2) In the non-resonant cases, the elastic and the rigid experimental wave elevations and the theoretical values are all relatively close to each other. Under the resonant condition, the experimental elevations in the elastic tank are slightly less than those in the rigid tank. Further, the theoretical values are smaller than the experimental results at the two depths.

(3) In the non-resonant cases, the elastic and the rigid experimental sloshing pressures and the theoretical values are all relatively close to each other. Under the resonant condition, the experimental sloshing pressures on the elastic bulkhead are slightly smaller than those on the rigid bulkhead, and the theoretical values are smaller than the experimental results.

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10.1016/S1001-6058(14)60084-6

* Project supported by the National Natural Science Foundation of China (Grant No. 51179030, 51309038).

Biography: JIANG Mei-rong (1983-), Male, Ph. D. Candidate

REN Bing, E-mail: bren@dlut.edu.cn