STUDY ON THE WAKE INTERFERENCE AND DYNAMIC EVOLUTION CHARACTERISTICS OF THE FLUID-STRUCTURE INTERACTION OF THREE TANDEM CIRCULAR CYLINDERS
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摘要: 基于有限体积法应用开源OpenFOAM软件对串列三圆柱的涡激振动响应进行了数值模拟研究, 分析了雷诺数Re = 150、间距比介于2 ~ 6的串列三圆柱在约化速度2 ~ 16范围内的流固耦合动态响应特性. 研究观察到: 当间距比为2时, 上游圆柱振动锁定的区域明显变宽, 并且在锁定区域内振幅明显高于大间距工况, 表明小间距布置时, 下游圆柱的存在对上游圆柱振动起增强作用. 由于尾流诱导的作用, 中间圆柱和下游圆柱振幅的最大值高于上游圆柱. 串列三圆柱的尾流干涉模式存在拓展体模式、持续再附着模式、交替再附着模式、拟同脱落模式和同脱落模式5种, 并且由于圆柱振动过程中振幅的动态变化, 尾流干涉模式出现不稳定切换. 在间距比为2, 约化速度为7时观察到振动存在“拍”现象, 在间距比为6, 约化速度为4时观察到多频参与振动的大周期现象, 两种现象发生时都会引起柱体升力和振幅的波动. 当中间圆柱和下游圆柱的尾流干涉处于拟同脱落模式时, 尾迹中的双排涡合并形成二次涡街, 而双排涡的融合发生在同侧相邻的两个旋涡或3个旋涡之间, 与圆柱的间距比相关.Abstract: Based on the finite-volume-method (FVM), the vortex-induced vibration (VIV) of three tandem circular cylinders is numerically simulated using the open source OpenFOAM. The fluid-structure interaction and dynamic response characteristics are analyzed with Reynolds number Re = 150 in the spacing ratio range of 2 ~ 6 and reduced velocity range of 2 ~ 16. The results show that, when the spacing ratio is 2, the lock-in region of the upstream cylinder vibration becomes significantly wider than those at other spacing ratios, and the amplitude of the upstream cylinder in the lock-in region is significantly higher than the larger spacing ratio cases. It indicates that the vibration of the upstream cylinder is enhanced at small spacing-ratio arrangement. Due to the effect of wake-induced vibration, the maximum amplitude of the middle cylinder and the downstream cylinder are higher than that of the upstream cylinder. Five wake interference modes are identified, including the overshoot mode (OS), continuous reattachment mode (CR), alternate reattachment mode (AR), quasi-co-shedding mode (QCS) and co-shedding mode (CS). Nevertheless, due to the dynamic alteration of amplitude during the vortex-induced vibration process, the wake interference mode undergoes unstable switching. When the spacing ratio is 2 with the reduced velocity of 7, a "beating" phenomenon is observed in the response. When the spacing ratio is 6 with the reduced velocity of 4, a large periodic phenomenon is observed due to multiple frequencies participating in the vibration. Both the two phenomena cause the time-varying lift coefficient and response amplitudes of the cylinders. When the wake interference mode between the middle cylinder and the downstream cylinder presents the quasi-co-shedding pattern, the two-layered vortices in the wake merge to form the secondary vortex street. Such a merging occurs among two or three adjacent vortices on the same side, which is related to the spacing ratio of the cylinders.
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引 言
圆柱结构的流固耦合问题广泛存在于实际工程中, 如电缆线、风机桩腿、海洋立管和换热管等[1]. 气体或液体流经圆柱时, 可能存在尾涡脱落激发的振动响应[2]和噪音问题[3], 对结构本身构成了疲劳失效的风险. 前人对单一圆柱的涡激振动开展了大量的实验测试和数值模拟研究, 明晰了尾涡脱落模式、升力和阻力系数的脉动、振动响应幅度和频率等变化规律[4-8]. 然而, 实际工程中的圆柱体并非单一存在, 多柱并存时的尾流干涉和振动响应方面的报道还较少, 为此, 本文针对串列布置的三圆柱尾流流场及其引起的涡激振动问题开展数值模拟研究.
串列双圆柱的尾流干涉模式随间距比L/D 圆心间距L与圆柱直径D之比)、不同直径和雷诺数(Re = uD/υ, u为来流速度, υ为运动黏度)的变化而变化[9-12]. Zdravkovich[13]将流动结构归为3种模式: 当L/D < 1.2 ~ 1.8时, 为拓展体模式, 即上游圆柱分离的剪切层跨过下游圆柱, 在下游圆柱后方卷曲形成旋涡, 两个圆柱犹如一个拓展的整体, 两个圆柱之间形成了停滞的间隙流动; 当1.2 ~ 1.8 < L/D < 3.4 ~ 3.8时, 为再附着模式, 即上游圆柱分离的剪切层交替或连续地附着于下游圆柱壁面, 形成准稳态的间隙流动; 当L/D > 3.4 ~ 3.8时, 为共同脱落模式, 即两个圆柱分离的剪切层均能卷曲形成旋涡并脱落. Alam等[11]进一步将再附着模式分为交替和稳定两种, 其临界间距约为L/D = 3.0, Zhu等[14]将后者称为持续再附着模式.
在串列双柱下游再放置一个圆柱时, 尾流干扰更为复杂. Zhu等[15]基于等距串列三柱的绕流模拟结果, 提出了一个新的尾流干涉模式: 准(拟)共同脱落模式, 该模式发生在3.5 < L/D < 6.5时, 下游圆柱被中间圆柱两侧分离的剪切层夹持, 且每侧剪切层相邻的两个同向旋涡在远场合并成一个更大的旋涡, 形成二次涡街. Jiang等[16-17]系统分析了单一圆柱在Re = 100 ~ 200时的二次涡街形成机理, 认为水动力不稳定性和双排涡的合并是产生二次涡街的主要原因, 即二次涡街的形成是由剪切层的对流不稳定性引起, 随着尾流从初始卡门涡街过渡至双排涡, 剪切层的剪切速率明显增加, 进一步诱发了双排涡向二次涡街的演变. 目前, 尚罕见关于串列多柱尾部二次涡街形成机理的研究报道.
Yu等[18]数值研究了L/D = 4和质量比m* = 4/π的串列三柱在Re = 100, 150时的双自由度涡激振动, 与双柱相比, 最大横向振幅增加约25%, 且同时存在与横向振幅相当的流向振幅. 串列三柱的振动轨迹被描述为“有界随机运动”, 通过相图和庞加莱图证明其比双柱更为复杂, 即使在低雷诺数时, 也出现了类似混沌的自由振动响应. 陈威霖等[19]对L/D = 1.2和m* = 2的串列三圆柱在Re = 100时的单自由度涡激振动进行了数值研究, 在这样的小间距比下, 当约化速度增大到一定值后, 3个圆柱的振动响应均呈现随约化速度增大而增大的弛振现象, 平衡位置偏移、低频振动及旋涡脱落与圆柱运动间的相位差共同引起了弛振现象的出现. 其进一步研究发现, 串列三柱的弛振现象仅出现在m* ≤ 2.0和Re ≤ 100时. 当质量比m* = 5, 10时, 串列三柱的平衡位置固定不变, 且圆柱振动不规律, 使得旋涡脱落与圆柱运动的相位差不断变化.
串列三柱的振动响应很大程度上取决于间距比, Chen等[20]发现, 在Re = 100和L/D = 1.2时, 由于尾流与下游圆柱间的强烈干涉, 激发了下游圆柱的驰振; 在L/D = 1.5 ~ 5时, 驰振现象消失, 上游圆柱体的振动响应类似于单圆柱的涡激振动. 然而, 在Zhu等[21]的研究结果中发现, 在Re = 150和L/D = 4时, 由于圆柱间的流场干涉结构发生动态变化, 上游圆柱的振动规律也会发生变化, 并出现多个频率. 张志猛等[22]将上游圆柱固定, 中间圆柱和下游圆柱作横向单自由度振动, Re = 100和L/D = 1.2, 2.0, 5.0时, 中游和下游圆柱的振幅明显大于单一圆柱的振幅. 当振幅较小时, 上游圆柱的剪切层将3个圆柱包裹, 尾流与绕流时相似, 表现为经典的卡门涡街; 当振幅较大时, 上游圆柱的旋涡或剪切层撞击或重附着于下游圆柱, 圆柱之间存在强烈的干涉作用, 尾流呈现双排涡结构. Behara等[23]和Gao等[24]在Re = 150的串列三柱双自由度涡激振动研究中发现, 中间圆柱和下游圆柱存在多周期振动, Behara等[23]认为是由于不规则的涡脱导致的, 而Gao等[24]和涂佳黄等[25]基于频谱分析认为是由尾流结构动态演变引起的.
综上所述, 目前关于串列三柱流固耦合响应的尾流干涉与动态演变机制方面的研究还较少, 本文通过有限体积法对均匀来流作用下的串列三圆柱单自由度涡激振动响应进行数值模拟研究, 重点分析振动响应随间距比和约化速度的变化规律, 探究振动过程中圆柱的尾流干涉模式与动态演变过程.
1. 数值方法
1.1 控制方程
通过求解非定常二维不可压缩Navier-Stokes (N-S)方程, 捕捉串列三圆柱周围的流动, 包括
$$ \frac{{\partial {u_i}}}{{\partial t}} + {u_j}\frac{{\partial {u_i}}}{{\partial {x_j}}} = - \frac{1}{\rho }\frac{{\partial p}}{{\partial {x_i}}} + \upsilon \frac{{{\partial ^2}{u_i}}}{{\partial {x_i}{x_j}}} $$ (1) $$ \frac{{\partial {u_i}}}{{\partial {x_i}}} = 0 $$ (2) 其中, xi为坐标轴, ui为流体在xi方向上的速度, t为时间, ρ为流体密度, p为压力, υ为流体的运动黏度.
将3个圆柱均视为弹簧-质量系统, 由弹簧弹性支撑, 通过求解结构的运动方程, 计算每个圆柱在横向上的运动响应
$$ m\ddot Y + c\dot Y + kY = {F_L}(t) $$ (3) 其中, m, c和k分别为每个圆柱的质量、阻尼和刚度, $ \ddot Y $, $ \dot Y $和Y分别表示横向振动的加速度、速度和位移, FL表示圆柱体受到的升力(横向流体作用力, 由流场计算得到圆柱表面的压应力和切应力积分后得到).
利用开源计算流体力学求解器OpenFOAM进行求解计算, 该工具由C++库组成, 压力-速度耦合是基于压力隐式算子分裂(PISO)方法和SIMPLE算法. 采用pimpleDyMFoam求解器求解结构周围的网格运动[26]. 利用高斯线性插值对对流项进行离散, 以及高斯线性格式对拉普拉斯项和压力项进行离散, 采用欧拉隐式格式进行时间离散. 由Newmark-β法求解3个弹簧支撑圆柱的运动响应.
1.2 问题描述
本文数值模拟研究的雷诺数为Re = 150, 圆柱的质量比为2.0 (m* = m/mf, 其中mf为等体积流体质量). 相邻两圆柱之间的间距相等, L为圆心间距, D为圆柱直径, 间距比L/D = 2.0, 3.0, 4.0, 5.0, 6.0共5组. 为激发较大的圆柱振动响应, 特将阻尼比设为0. 此外, 为避免计算过程中圆柱发生碰撞, 三圆柱均仅作横向单自由度振动. 表1对比了不同计算域尺寸及阻塞率的计算结果, L/D = 2.0和Ur = 4时各柱体的AyRMS在阻塞率降至1/40时达到收敛, 故本文采用40D作为计算域的横向宽度尺寸.
表 1 计算域对结果的影响Table 1. Computational domain effect on resultsDomain size Block ratio Ay1RMS Ay2RMS Ay3RMS 20D × 70D 1/20 0.363 0.134 0.179 40D × 70D 1/40 0.366 0.147 0.201 60D × 70D 1/60 0.367 0.150 0.201 如图1所示, 模拟计算域设为40D × 70D的矩形, 从上游至下游3个圆柱依次记为C1, C2和C3, 其中C1的圆心设为坐标原点.
左侧入口边界设为Dirichlet型边界(u = uin, v = 0), 右侧出口边界设为Neumann型边界(∂u/∂x = 0, ∂u/∂y = 0), 上、下边界设为对称边界(∂u/∂y = 0, v = 0). 为保证数值收敛, 选定时间步长∆t = 0.0005, 以满足CFL条件, 即Umax·∆t/∆x ≤ 0.5.
2. 模型验证
首先, 通过单圆柱的涡激振动模拟验证本文采用的数值方法在预测结构涡激振动方面的准确性. 在雷诺数Re = 150、质量比m* = 2.0和阻尼比0 (与本文工况条件相同)时, 将本文模拟的单圆柱振动结果与文献[21, 27-29]对比, 如图2所示, 横向振幅随约化速度的变化曲线与已报道的结果吻合较好.
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图 2 单圆柱涡激振动模拟验证Figure 2. Validation of the implemented numerical algorithm for the vibration of an isolated cylinder此外, 将Re = 100, L/D = 4, m* = 2.0和Ur = 5时的串列三柱振动结果与Chen等[20]的研究报道对比, 见表2, 横向最大振幅和均方根振幅均相差很小, 表明本文采用的数值方法准确可行.
表 2 串列三圆柱涡激振动验证Table 2. Comparison of results of vortex-induced vibration of three tandem circular cylinders3. 研究结果与讨论
3.1 结构振动幅值
如图3所示, 与单圆柱涡激振动不同, 串列布置的3个圆柱各自振动但又相互影响. L/D = 2时, 上游圆柱C1的振动锁定区明显变宽, 在Ur > 4时, 振幅高于其他大间距时的振幅, 表明小间距布置时, 下游圆柱的存在对上游圆柱振动起增强作用. 在L/D = 4和Ur = 4时, 由于C1的剪切层持续地再附着于C2, 剪切层不能得到充分发展, 致使C1受到的横向脉动升力减小, 因此在相同的Ur条件下, C1的振幅远小于其他间距比时的振幅.
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图 3 串列三圆柱横向均方根振幅(AyRMS)Figure 3. The root-mean-squared transverse amplitude (AyRMS) of three tandem circular cylinders对于中间圆柱C2而言, 其均方根振幅值(AyRMS)先随着Ur的增大而增大, 在Ur = 7或8时达到最大值, 而后随着Ur的增大而减小. L/D = 2时, 振幅在Ur > 8时减小, 其余间距比组次均在Ur > 7时减小. 下游圆柱C3的AyRMS在L/D = 2, 3, 4时呈现和C2相同的变化规律, 随着Ur的增大先增大后减小, 最大值分别出现在Ur = 9, 8, 8时. 而L/D = 5, 6时, AyRMS随着Ur的增大而增大, 但在Ur > 8时增幅明显减小. 相较C1, C2和C3的均方根振幅最大值均更大, 表明尾流诱导的振动响应更剧烈.
3.2 结构振动频率
图4给出了串列三圆柱横向振动的无量纲主导频率(f/fn), 其中虚线为Jiang等[30]计算的单圆柱绕流的涡脱频率(斯特劳哈尔数St = 0.185). Ur < 5时, 各圆柱振动的频率均随Ur的增大而增大, 且处于St = 0.185附近. 其中, L/D = 2和Ur = 3时, 由于C2振动影响了上游圆柱C1分离的剪切层, 使得C1后方剪切层的变化周期相比于远场旋涡脱落的周期缩小了一半(如图4所示), 对应的频率则增大了一倍. Ur ≥ 5时, 出现f/fn ≈ 1的频率锁定区. L/D = 2布置的C1和C2在 Ur > 8时振动频率脱离固有频率(脱离锁定区), 而L/D > 2时, 在Ur > 7即脱离锁定, 相较而言, C3达到锁定的约化速度更高.
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图 4 串列三圆柱横向无量纲振动主频(f/fn)Figure 4. The transverse normalized dominant frequency (f/fn) of three tandem circular cylinders3.3 尾流干涉模式
在本文单自由度各自振动的串列三圆柱中, 也发现了拓展体模式(overshoot mode, OS)、持续再附着模式(continuous reattachment mode, CR)、交替再附着模式(alternate reattachment mode, AR)、拟同脱落模式(quasi-co-shedding mode, QCS)和同脱落模式(co-shedding mode, CS). 图5罗列了C1和C2之间及C2和C3之间的尾流干涉模式, 其中, 部分工况可能出现2个甚至3个模式(如L/D = 6和Ur = 4), 表明尾流模式不稳定, 在2个或3个模式间切换.
下文将C1和C2的尾流干涉模式与C2和C3的尾流干涉模式连写, 如AR-QCS表示C1和C2的尾流干涉模式与C2和C3的尾流干涉模式分别为交替再附着和拟同脱落模式. 如图6所示, L/D = 2和Ur = 2时, 尾流干涉模式呈OS-OS模式, 从上游圆柱C1分离的剪切层未卷曲形成旋涡, 直接略过中间圆柱C2和下游圆柱C3, 在C3的后方卷曲形成旋涡. 在相同的间距比下, Ur增加到3时, 出现CR-CR模式, 此时上游圆柱C1分离的剪切层在向后运移并卷曲的过程中, 持续性地附着于中间圆柱C2表面, 并在C2后产生交替分离的剪切层, 该剪切层又持续性地再附着于下游圆柱C3的表面, 随后在C3的远场尾流中形成旋涡脱落. 其与OS-OS模式的区别在于C2和C3的横向振动影响了上游圆柱C1分离的剪切层, 造成了后者的弯曲, 因而旋涡在更下游的位置形成. 当Ur进一步增加到5时, 出现AR-QCS模式, 上游圆柱C1的剪切层交替地再附着于中间圆柱C2表面, 而后迁移掠过C3, 尽管C3后方可以看到旋涡的形成, 但并非C3自身剪切层卷曲形成. Ur = 7时, 尾流干涉模式在AR-QCS模式和AR-AR模式间切换, 这与圆柱振动引起的间距动态变化有关. AR-AR模式的上游圆柱C1的剪切层交替再附着于C2, 促使C2的边界层分离, 而后再次交替附着于C3. Ur ≥ 8时, 尾流干涉模式稳定于AR-AR模式.
如图7所示, L/D = 4和Ur = 4时, 尾流干涉模式呈CR-AR模式, 上游圆柱C1的剪切层持续地再附着于C2, 而后C2分离的剪切层交替地再附着于C3. Ur增至5时, 出现CS-QCS模式, 上游圆柱C1和中间圆柱C2的剪切层都卷曲形成了旋涡并在间隙中脱落, 而C3的旋涡脱落受到了抑制. Ur进一步增至6时, 呈现CS-CS模式, 3个圆柱自身的剪切层均能卷曲并脱落产生旋涡. 值得注意的是, 当L/D = 6和Ur = 4时, 尾流干涉模式在QCS-QCS, QCS-CS和AR-AR间切换. QCS-QCS模式时, 从C1脱落的旋涡抑制了C2和C3的旋涡脱落. 而QCS-CS模式时, 从C1脱落的旋涡抑制了C2的旋涡脱落, 但在C2和C3间存在旋涡的融合, 未影响C3的旋涡脱落.
3.4 振动响应的动态切换
由于圆柱尾流干涉模式不稳定, 对应的振动响应也不稳定. 本文主要发现了两种代表性的不稳定振动现象. 一种是“拍”现象(beating phenomenon), 如图8所示; 二是不稳定的多频振动, 如图9所示.
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图 8 L/D = 2和Ur = 7时, 串列三圆柱的升力系数(CL)和振动位移(Ay)时程曲线Figure 8. Time histories of lift coefficient (CL) and amplitude (Ay) of three tandem circular cylinders during the switching between different wake interference modes at L/D = 2 and Ur = 7![]()
图 9 L/D = 6和Ur = 4时, 串列三圆柱的升力系数(CL)和振动位移(Ay)时程曲线Figure 9. Time histories of lift coefficient (CL) and amplitude (Ay) of three tandem circular cylinders during the switching between different wake interference modes at L/D = 6 and Ur = 4当同方向的两个频率相差不大的简谐波叠加时, 叠加后的波动幅值将随时间作强弱的周期性变化, 这种现象称之为“拍”[31-32]. 如图8所示, 当L/D = 2和Ur = 7时, 圆柱的升力系数(CL)和横向振幅(Ay)均存在“拍”现象, 此时, 上游圆柱C1和中间圆柱C2的尾流干涉模式始终保持为AR, 所以C1振幅受到的干扰较小; 而中间圆柱C2和下游圆柱C3的尾流干涉模式在AR和QCS之间来回切换, AR模式较QCS模式出现的时间占比短, 但AR模式时C3的振幅更大, 这是由于QCS模式时C3的旋涡脱落被抑制所致.
在L/D = 6和Ur = 4时, 中间圆柱C2和下游圆柱C3受到的升力与横向振幅存在多频参与现象, 如图9所示. 这是由于C2和C3的尾流干涉模式在QCS, AR和CS三者之间切换, 在一个变换的大周期内, QCS模式时间占比最长, 由其引起的C3振幅最小; 当尾流干涉模式切换至AR模式时, C3振幅明显增大; 进入CS模式时, C3振幅又开始减小. 此外, C1和C2的尾流干涉模式在QCS和AR之间动态切换, 且QCS模式占主导, 相比AR模式, C1和C2振幅在QCS模式出现时较大.
3.5 远场演变
Zhu等[15]的研究报道指出, 串列三柱绕流出现双排涡(two-layered vortex, TV)结构时, 远场存在旋涡融合形成的二次涡街(secondary vortex street, SV). 这一现象同样在本文研究的单自由度振动串列三柱中发现. 当C2和C3的尾流干涉模式中出现QCS模式时, 在下游圆柱的尾迹中即可发现二次涡街. 此外, 模拟还发现了以2S (一个周期内脱落两个方向相反的单涡)模式脱落的初始卡门旋涡(primary vortex street, PV)形成双排涡(TV)后向下游运移的过程. 图10统计了由双排涡演变为二次涡街或初始卡门旋涡演变为双排涡的组次.
在Re = 100 ~ 200时, 单圆柱尾流二次涡街的形成位置发生在x/D > 50, 且随着Re数的减小, 形成位置越靠后[16]. 但对串列三柱绕流而言, 双排涡的涡融合形成二次涡街提前发生, 且随间距比L/D的增大而减小[15]. 振动的三圆柱尾流二次涡街的形成位置同样被提前, 如图11所示, 选取双排涡演变为二次涡街、无双排涡结构和初级涡街演变为双排涡的典型组次进行对比. 在双排涡演变为二次涡街的过程中, 出现了两种形式, 一种是双排涡同侧相邻的两个旋涡融合形成二次涡街的大旋涡(L/D = 3和Ur = 4); 另一种是由同侧相邻的3个旋涡融合形成二次涡街的大旋涡(L/D = 6和Ur = 4). 前者发生在下游圆柱的远场, 而后者紧邻下游圆柱发生, 这与圆柱间的间距比密切相关. L/D = 4和Ur = 4时, 尾流旋涡结构较杂乱, 而L/D = 2和Ur = 9时, 下游圆柱后方的初始涡街很快演变为整齐的双排涡结构. 因此, 圆柱间的尾流干涉模式不仅影响了结构的振动响应, 还影响了下游旋涡结构的演变.
4. 结 论
本文基于有限体积法, 应用开源软件OpenFOAM对串列三圆柱的单自由度涡激振动进行了模拟研究, 分析了间距比和约化速度对结构振动响应及尾流干涉模式的影响, 得到主要结论如下.
(1) 研究观察到OS, CR, AR, QCS和CS 5种尾流干涉模式, 且干涉模式存在不稳定切换的现象.
(2) 与单圆柱涡激振动不同, 串列三柱的振动存在相互影响, C1的均方根振幅随约化速度Ur的变化规律与单圆柱相似, 但受到C2和C3的干涉, 其数值较单圆柱小; 而C2和C3由于C1的遮蔽效应, 均方根振幅最大值出现在更高的约化速度, 但最大均方根振幅比单圆柱大. 3个圆柱锁定区的范围不同, 与间距比相关, C3进入锁定区的临界约化速度最大.
(3) 尾流干涉模式的动态切换存在两种形式: 一种是“拍”现象, 切换周期tuin/D < 100, 二是多频参与的大周期现象, 切换周期约为tuin/D = 150 ~ 250. 因尾流干涉模式的切换, 各圆柱的CL和Ay均有明显波动.
(4) 当C2和C3的尾流干涉模式存在QCS模式时, 在下游圆柱的尾迹中出现双排涡合并形成二次涡街的现象. 而双排涡融合可以发生在同侧相邻的两个旋涡或3个旋涡之间, 与圆柱的间距比密切相关.
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图 2 单圆柱涡激振动模拟验证
Figure 2. Validation of the implemented numerical algorithm for the vibration of an isolated cylinder
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图 3 串列三圆柱横向均方根振幅(AyRMS)
Figure 3. The root-mean-squared transverse amplitude (AyRMS) of three tandem circular cylinders
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图 4 串列三圆柱横向无量纲振动主频(f/fn)
Figure 4. The transverse normalized dominant frequency (f/fn) of three tandem circular cylinders
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图 8 L/D = 2和Ur = 7时, 串列三圆柱的升力系数(CL)和振动位移(Ay)时程曲线
Figure 8. Time histories of lift coefficient (CL) and amplitude (Ay) of three tandem circular cylinders during the switching between different wake interference modes at L/D = 2 and Ur = 7
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图 9 L/D = 6和Ur = 4时, 串列三圆柱的升力系数(CL)和振动位移(Ay)时程曲线
Figure 9. Time histories of lift coefficient (CL) and amplitude (Ay) of three tandem circular cylinders during the switching between different wake interference modes at L/D = 6 and Ur = 4
表 1 计算域对结果的影响
Table 1 Computational domain effect on results
Domain size Block ratio Ay1RMS Ay2RMS Ay3RMS 20D × 70D 1/20 0.363 0.134 0.179 40D × 70D 1/40 0.366 0.147 0.201 60D × 70D 1/60 0.367 0.150 0.201 
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