Numerical modelling of sloshing responses in a cylindrical tank under seismic excitations
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摘要:
为了研究地震载荷下圆柱型储罐内液体的晃荡特性，选用15种典型的地震信号，采用计算流体力学软件Fluent进行数值仿真，以探究地震的频率、频率成分、峰值速度以及峰值加速度对晃荡波高和水动压的作用规律。结果表明：① 地震主频是影响自由液面响应的主要因素之一，当其接近储液的一阶固有频率时，会激发强烈的非线性晃荡现象，工程中应添加减晃装置；② 波高与地震峰值速度呈较强的正相关，并且低频成分的地震信号激发的波浪较其它频率成分的地震信号更为剧烈；③ 水动压在储罐上部呈对流模式分布，主要受地震主频和频率成分的影响，并与地震峰值速度和频率成分呈正相关；④ 水动压在储罐中、下部为脉冲模式分布，与地震峰值加速度呈线性正相关且下部的水动压增长速率明显大于中部。因此在抗震设计中，应加强罐壁下部的强度，尤其是峰值加速度较大的储罐放置场地。
Abstract:China is a country with frequent and high-intensity earthquakes and has arranged a large amount of storage tanks to conserve liquid materials, which can cause violent liquid sloshing in tanks, resulting in wall buckling, roof breaking and overflow. To comprehensively investigate the sloshing characteristics in cylindrical tanks under seismic excitations, seismic parameters covering a broad range of seismic frequency, frequency content, peak ground velocity （PGV） and peak ground acceleration （PGA）, which were four main concerns of seismic excitations, need to be traversed. Then, totally 12 seismic events involving 15 kinds of seismic records at home and abroad were selected to explore as much as possible about the key factors affecting the sloshing responses. The seismic frequency ranged from 0.11 Hz to 2.545 Hz, which covered the several natural frequencies of the fluid. The frequency content of all seismic records included the low frequency, intermediate frequency and high frequency. PGA increased from 0.081g to 1.79g, and PGV ranged from 0.22 m/s to 1.76 m/s. Numerical simulations, which could avoid the constraints of the theoretical assumption and experimental facility in prototype tanks, were carried out by using the computational fluid dynamics software Fluent to investigate the effects of seismic frequency, frequency content, PGV and PGA on the sloshing height and hydrodynamic pressure. After full validations against available numerical and experimental results in literatures, systematic simulations were carried out. The results suggest that: ① the dominant frequency of the seismic excitation is one main factor affecting the free-surface response, when it is close to the first-order natural frequency of the liquid and meantime lasts a longer duration, a strong non-linear phenomenon occurs, thereby the inhibition devices should be introduced in application; ② the wave height and the PGV exhibit a strong positive correlation in most cases; and the low-frequency content can excite more intense sloshing wave than those with other frequency contents, since the frequency content is a representative index to identify the overall situation of the seismic frequency; ③ the hydrodynamic pressure in the upper part of the tank shows a convective mode, namely the dominant response frequency of the pressure is the natural frequency of the fluid and the secondary response frequencies are the seismic frequencies, which is mainly affected by the dominant frequency and frequency content, and is also positively correlated with the PGV and frequency content; ④ the hydrodynamic pressures in the middle and lower parts are linearly and positively correlated with the PGA and remain pulse-like which is extremely obvious in the lower parts since the impulsive mode is dominated in the lower parts, and the growth rate in the lower part is also significantly higher than that in the middle part. Thereby, to ensure the tank safety, in the seismic design, the lower part of the tank sidewall should be strengthened especially in site conditions with potential strong PGA.
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图 1 1995年神户M7地震的位移时程（a）以及神户地震载荷下罐壁右侧波高的时程曲线对比（b）
Figure 1. Time history of ground displacement of Kobe seismic excitation （a） and comparison of time histories of wave heights at the right sidewall of the cylindrical tank under 1997 M7 Kobe seismic excitation （b）
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图 2 本文选取地震的加速度时程曲线
Figure 2. Time histories of considered seismic accelerations used in this study
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图 3 圆柱型储罐的整体视图（a）和顶视图（b）
Figure 3. The overall-view （a） and top-view （b） geometries of the cylindrical tank
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图 4 芦山53Elk-SN载荷下储罐东侧We （a）和北侧Wn （b）处不同网格数下波高的对比
Figure 4. Comparison of time histories of the wave heights at We （a） and Wn （b） from various grids under Lushan 53Elk SN excitation
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图 5 不同地震载荷下储罐东侧We和北侧Wn处的波高时程曲线
（a） 芦山53Elk-EW；（b） 芦山53Elk-SN；（c） 芦山53Yjb-EW；（d） 埃尔森特罗-EW；（e） 神户地震；（f） 埃尔津；（g） 北岭-1；（h） 唐山；（i） 康奎特；（j） 埃尔森特罗-SN；（k） 塔夫脱；（l） 曼杜西诺角；（m） 汶川卧龙-EW；（n） 柯依娜-SN；（o） 芦山53Yjb-SN
Figure 5. Time histories of wave heights under various seismic excitations
（a） Lushan53Elk-EW；（b） Lushan53Elk-SN；（c） Lushan53Yjb-EW；（d） El Centro-EW；（e） Kobe；（f） Erzican；（g） Northridge-1；（h） Tangshan；（i） Concrete；（j） El Centro-SN；（k） Taft；（l） Cape Mendocino；（m） WenchuanWolong-EW；（n） Koyna-SN；（o） Lushan53Yjb-SN
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图 6 储罐东侧We处峰值波高与地震主频的关系
Figure 6. The relationship between the maximum wave height at east side We of the tank and dominant frequency of earthquake
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图 7 储罐东侧We处峰值波高与PGV的关系
Figure 7. The relationship between the maximum wave height at the east side We of the tank and PGV
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图 8 塔夫脱地震载荷下的自由液面
Figure 8. Free surface snapshots under excitation of Taft earthquake
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图 9 塔夫脱地震载荷下自由液面的波高相位图
Figure 9. Phase diagrams of wave heights under excitation of Taft earthquake
（a） 0—15 s；（b） 15—30 s；（c） 20—45 s
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图 10 芦山53Elk-SN地震载荷下的自由液面
Figure 10. Free surface snapshots under Lushan 53Elk-SN excitation
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图 11 峰值水动压沿罐壁的分布
Figure 11. Distributions of peak hydrodynamic pressures along the tank sidewall
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图 12 芦山53Elk-SN地震载荷下罐内水动压分布
Figure 12. Distributions of hydrodynamic pressures in the tank under Lushan 53Elk-SN excitation
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13 储罐内水动压的快速傅里叶分析结果
13. Fast Fourier transform results of hydrodynamic pressures in the tank
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13 储罐内水动压的快速傅里叶分析结果
13. Fast Fourier transform results of hydrodynamic pressures in the tank
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图 14 各工况下PGV和PGA与峰值水动压的关系
Figure 14. The relationships between PGV （or PGA） and peak hydrodynamic pressure for all cases
表 1 本研究涉及地震的相关参数
Table 1 The characteristics of considered seismic excitations in this study
地震信号 频率类别 主要频率/Hz PGA/g PGV/（m·s−1） PGA/PGV 芦山53Elk-EW 低频 0.425，0.625 0.081 0.377 0.215 芦山53Elk-SN 低频 0.325，0.4 0.224 0.866 0.259 芦山53Yjb-EW 低频 0.658，0.703 0.075 0.232 0.323 埃尔森特罗（El Centro）-EW 低频 0.449，0.842 0.214 0.48 0.446 神户（Kobe） 低频 0.269，0.537，0.879 0.611 1.27 0.48 埃尔津（Erzican） 低频 0.42，0.563，0.798 0.52 0.84 0.619 北岭-1（Northridge-1） 低频 0.353，0.873，1.08 0.607 0.8 0.754 唐山 中频 0.65，0.8，1.2 1.268 1.584 0.8 康奎特（Concrete） 中频 0.49，0.6，0.98，1.1 0.63 0.758 0.83 埃尔森特罗（El Centro）-SN 中频 0.561，0.839，1.16 0.349 0.38 0.91 塔夫脱（Taft） 中频 0.11，0.37，0.61，1.2 1.79 1.76 1.02 曼杜西诺角（Cape Mendocino） 中频 0.868，1.43，2.14 1.45 1.26 1.15 汶川卧龙-EW 高频 0.622，1.38，2.35 0.97 0.585 1.66 柯依那（Koyna）-SN 高频 0.87，1.55 0.585 0.31 1.89 芦山53Yjb-SN 高频 0.841，2.545 0.473 0.22 2.15 注：第一列中的53Elk，53Yjb和卧龙为站点名称，EW和SN为地震信号的东西分量和南北分量。 
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表 2 地震主频、PGV和东侧We处相应的峰值波高
Table 2 The dominant frequency and PGV of seismic excitations and corresponding maximum wave heights at east side We of the tank
地震信号 主频/Hz PGV/（m·s−1） 峰值波高/m 地震信号 主频/Hz PGV/（m·s−1） 峰值波高/m 塔夫脱 0.11 1.76 4.15 埃尔森特罗-SN 0.561 0.38 0.484 神户 0.269 1.27 2.02 汶川卧龙-EW 0.622 0.585 0.26 芦山53Elk-SN 0.325 0.866 3.574 唐山 0.65 1.584 2.3 北岭-1 0.353 0.8 1.48 芦山53Yjb-EW 0.658 0.232 0.17 埃尔津 0.42 0.84 1.06 芦山53Elk-SN 0.841 0.22 0.39 芦山53Elk-EW 0.425 0.377 0.713 曼杜西诺角 0.868 1.26 1.35 埃尔森特罗-EW 0.449 0.48 0.664 柯依那-SN 0.87 0.31 0.334 康奎特 0.49 0.758 0.675
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