杨明, 刘巨保^,2), 岳欠杯, 丁宇奇, 王明东北石油大学机械科学与工程学院, 黑龙江大庆 163318

NUMERICAL SIMULATION ON THE VORTEX-INDUCED COLLISION OF TWO SIDE-BY-SIDE CYLINDERS ¹⁾

Yang Ming, Liu Jubao^,2), Yue Qianbei, Ding Yuqi, Wang MingCollege of Mechanical Science and Engineering, Northeast Petroleum University, Daqing 163318, Heilongjiang, China

通讯作者: 2) 刘巨保, 教授, 主要研究方向: 流固耦合, 流致振动, 石油石化装备力学分析. E-mail:jslx2000@163.com

收稿日期:2019-08-19接受日期:2019-11-15网络出版日期:2019-11-15

基金资助:

1) 国家自然科学基金.11972114 1) 国家自然科学基金.51904075 1) 国家自然科学基金.51604080 黑龙江省博士后资金.LBH-Z17037 中国博士后科学基金.2017M621240 中国博士后科学基金.2019M661248 黑龙江省国家自然科学青年基金培育基金.2017PYQZL-09 黑龙江省高校青年创新人才培养计划.UPYSCT-2018045 黑龙江省高校青年创新人才培养计划.2017036 东北石油大学研究生教育创新工程资助项目.JYCX_CX04_2018

Received:2019-08-19Accepted:2019-11-15Online:2019-11-15
作者简介 About authors


摘要
圆柱类结构物的涡激振动是工程中较为常见的一种现象,如果圆柱结构物之间的距离较小, 就会产生涡激诱导碰撞现象,而涡激碰撞会比涡激振动对结构物疲劳破坏产生更严重的威胁.采用浸入边界法模拟流体中的动边界问题,避免了传统贴体网格方法在求解流体中存在固体间碰撞问题时出现数值求解不稳定问题,采用有限元方法对圆柱的运动和碰撞进行求解,通过数据回归方法建立了流体流动条件下的润滑模型,对不同间隙比下涡激诱导并列双圆柱振动及碰撞过程进行了数值模拟, 数值结果表明,如果两圆柱产生了碰撞将会有连续的碰撞发生, 碰撞时出现了多阶频率,振动主频率要比无碰撞时大, 两圆柱碰撞时的相对速度比自由来流速度小;当两圆柱相互接近时, 随着涡环分离角度的逐渐倾斜, 横向流体力先逐渐减小,当两圆柱间涡环开始相互影响发生挤压时, 横向流体力开始逐渐增大;当两圆柱开始反弹时, 两圆柱间形成了低压区, 改变了横向流体阻力的方向,使两圆柱又产生了接近运动,如此反复从而产生了碰撞后横向流体力和圆柱速度的振荡现象.
关键词： 涡激振动;碰撞;浸入边界法;有限元法;润滑模型

Abstract
Vortex-induced vibration of cylindrical structures is a common phenomenon in engineering. If the distance between cylindrical structures is small, vortex-induced collision will occur. Vortex-induced collision is more serious than vortex-induced vibration on the fatigue damage of the structures. The immersed boundary method was used to simulate the dynamic boundary problem in the fluid which avoided the numerical instability problem when the traditional boundary-fitting method was used to solve the collision problem between solids. The finite element method was used to simulate the motion and collision of the cylinders. The lubrication model under fluid flow condition was established by data regression method. The vortex-induced vibration and collision of two side-by-side cylinders at different initial gap ratios were simulated numerically. The numerical results show that if the collision occurs, there will be a continuous collision. Multiple frequencies occur in collisions and the main frequency of vibration is higher than that without collision. When the two cylinders collide, the relative velocity is smaller than that of free flow. When two cylinders are close to each other, the transverse fluid force decreases with the gradual inclination of vortex ring separation angle. When the vortex rings between two cylinders start to influence each other and squeeze, the transverse fluid force starts to increase gradually. When the two cylinders start to rebound, a low pressure area is formed between the two cylinders, which changes the direction of the transverse fluid force and makes the two cylinders move close to each other again. This repetition results in the oscillation of transverse fluid force and cylinder velocity after collision.
Keywords：vortex-induced vibration;collision;immersed boundary method;finite element method;lubrication model


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本文引用格式
杨明, 刘巨保, 岳欠杯, 丁宇奇, 王明. 涡激诱导并列双圆柱碰撞数值模拟研究 ¹⁾. 力学学报[J], 2019, 51(6): 1785-1796 DOI:10.6052/0459-1879-19-224
Yang Ming, Liu Jubao, Yue Qianbei, Ding Yuqi, Wang Ming. NUMERICAL SIMULATION ON THE VORTEX-INDUCED COLLISION OF TWO SIDE-BY-SIDE CYLINDERS ¹⁾. Chinese Journal of Theoretical and Applied Mechanics[J], 2019, 51(6): 1785-1796 DOI:10.6052/0459-1879-19-224

引言

圆柱类结构物的涡激振动是工程中较为常见的一种现象, 被国内外****广泛关注^[1-4],如机械工程中的热交换管、海洋工程中的隔水管和海底管线、石油工程中的输油管道等的涡激诱导振动问题.当圆柱的涡脱频率与圆柱结构物的自然频率相接近时, 结构响应会发生锁定,即在一定的流速范围内, 圆柱体的振动频率接近其自然频率并产生大位移振动,如果圆柱结构物之间的距离较小, 就会产生涡激诱导碰撞现象. 在结构疲劳破坏中,涡激碰撞比涡激振动更为严重, 会使结构物寿命进一步缩短.由于碰撞的非线性特性以及流体域网格拓扑结构的变化,因此采用数值模拟方法研究涡激诱导碰撞具有很大的挑战.

并列双圆柱作为圆柱群中的基本组成单元, 是工程中常见的一种结构形式,目前对并列双圆柱已进行了一定的研究. 当间隙比$g^*(g^* = G/D$,$G$为圆柱之间的最小间隙)较小($G/D < 0.1$)时, 间隙内没有旋涡脱落,仅在两圆柱的外侧有交替旋涡脱落, 尾流模态表现为单阻流体模态^[5-6].当间隙比增大到0.2$\sim $1.0时, 两圆柱后的尾流出现不对称现象,间隙流周期性偏向其中一个圆柱, 并在该圆柱后形成较窄的尾流,而在另一个圆柱后则形成宽尾流^[7-8]. 当圆柱间距进一步增长到$g^* > 1.5$时,两圆柱后尾流形成两列平行涡街, 两圆柱的尾流且具有相同脱涡频率^[9].当圆柱自由振动时, 由于与流体的相互作用, 使得流场更加复杂.Zhao^[10]数值研究了在雷诺数为150时不同间隙比对仅可横向振动的并列双圆柱涡激振动的影响,间隙比$g^*$最小为0.5, 但两圆柱是刚性耦合连接具有相同的位移,所以两圆柱并没有发生碰撞. 陈威霖等^[11]对雷诺数为100, $g^* = 1.5$和4.0时仅可横向振动的并列双圆柱涡激振动进行了数值模拟研究,在锁频区间内发现了不对称振动和对称性迟滞现象,并结合尾涡模式对其产生机制进行了阐述. 刘爽^[12]对雷诺数为100, $g^* = 1.0$$\sim$4.0时仅可横向振动并列双圆柱涡激振动进行了数值模拟,系统分析了折合流速和间隙比的变化对升阻力系数和无量纲振幅的影响.Liu和Jaiman^[13]研究了雷诺数为100, $g^* = 0.5$$\sim$2.5时的并列双圆柱涡激振动, 其中一个圆柱是静止的而另一个仅可横向振动.陈文曲^[14]对雷诺数为200, $g^* = 0.8$$\sim$3.0时二维并列双圆柱涡激振动进行了数值模拟, 探究了不同间距比下, 质量比对于圆柱的振动特性产生的影响. Wang等^[15]对高雷诺数下并列双圆柱涡激振动进行了数值模拟,由于自由流湍流的影响从而使得涡激力增大.Huera-Huarte和Gharib^[16]实验研究了高雷诺数下柔性并列双圆柱流致振动,发现当间隙比大于2.5时, 两圆柱之间的干扰耦合影响较小.由以上可以看出对并列双圆柱的涡激诱导碰撞问题还鲜有研究.

传统的流固耦合数值方法通常是基于贴体网格, 最典型的例子是任意拉格朗日-欧拉(ALE)界面跟踪方法^[17],并已得到了广泛的应用, 该方法的主要缺点是在处理大变形和动态边界问题时需要不断更新网格, 需要在新旧网格上交换各种数据, 不仅增加了计算量, 而且降低了求解的精度和稳定性, 特别是当流体中存在固体间碰撞问题时, 会出现数值求解不稳定问题. 而基于固定网格的浸入边界流动求解方法^[18], 为此可提供了一个有效的解决途径, 适用于复杂和动边界问题, 但存在固体边界与流体网格线不一致问题, 需对浸入边界附近的控制方程进行局部修正, 这些修正在一定程度上施加了流体域中所需的边界条件^[19].关于流体中的碰撞问题已经做了大量的研究, 但大部分将流体的影响或简化为附加质量系数^[20-21],或简化为相间参数化的阻力系数^[22], 采用界面解析直接数值模拟方法进行求解还少有报道.本文采用基于浸入边界$\!-\!$有限元法的流固耦合碰撞数值模拟方法^[23]对弹性支承低雷诺数下并列双圆柱涡激诱导碰撞过程进行了数值模拟,虽然大多数工程应用中的圆柱体为大长径比结构, 但弹性支承结构允许在可负担得起的计算时间内, 在不同条件下进行广泛的数值模拟, 并且层流状态下可以在不受三维影响的情况下, 识别涡激诱导碰撞特性和机制, 研究结果可为未来实际工程仿真提供指导.

1 数值方法

1.1 控制方程

在采用浸入边界法求解流固耦合问题时, 流体域处理为连续体, 其中不可压缩黏性流体的控制方程可表示为

(1)$\frac{\partial u_i }{\partial x_i } = 0$
(2)$\frac{\partial u_i }{\partial t} + \frac{\partial (u_i u_j )}{\partial x_j }= - \frac{1}{\rho }\frac{\partial p}{\partial x_i } + \mu \frac{\partial}{\partial x_j }\lt(\frac{\partial u_i }{\partial x_j })$
式中, $u_i $代表速度分量, $\rho $是流体密度, $p$为流体压力, $\mu $是动力黏度. 控制方程的离散采用同位网格方法将所有原始变量存储于单元中心, 采用分裂步方法对控制方程进行时间推进, 分裂步方法由三个子步骤组成,在第一个子步骤中, 求解动量方程, 得到中间速度$u_i^ * $, 在这一步中只考虑对流项与扩散项, 对流项采用了二阶Adams-Bashforth格式离散, 扩散项采用隐式Crank-Nicolson格式离散, 离散方程为

(3)$$\begin{eqnarray}&&\frac{u_i^ * - u_i^n }{\Delta t} + \frac{1}{2}\left[ {3\frac{\partial (U_j u_i )^n}{\partial x_j } - \frac{\partial (U_j u_i )^{n - 1}}{\partial x_j }} \right] =\\&&\qquad \frac{\mu }{2}\left[ {\frac{\partial }{\partial x_j }\lt(\frac{\partial u_i^ * }{\partial x_j }) + \frac{\partial }{\partial x_j }\lt(\frac{\partial u_i^n }{\partial x_j })} \right] \end{eqnarray}$$
式中, $u_i^n $为当前时间步的节点速度, $U_j $为沿$j$方向的节点速度线性平均值计算得到的界面速度. 方程(3)采用迭代法进行求解.在第二个子步骤中, 由泊松方程得到压力的近似解

(4)$\begin{eqnarray} \label{eq1} \frac{\partial }{\partial x_j }\left( {\frac{\partial p^{n + 1}}{\partial x_j }} \right) = \frac{\rho }{\Delta t}\frac{\partial U_j^ * }{\partial x_j} \end{eqnarray}$
(5)$\begin{eqnarray} \label{eq2} u_i^{n + 1} = u_i^ * - \frac{\Delta t}{\rho }\frac{\partial p^{n + 1}}{\partial x_i } \end{eqnarray}$
采用清晰界面浸入边界法^[24]模拟流固界面解析时浸入的固体边界将切割流体网格,如图1所示, 由于浸入边界与网格不协调, 当一个网格节点拥有处于解域外的相邻节点时, 该节点上的流动变量无法通过求解控制方程获得, 如图1中所示的NB(near boundary)节点. 在NB节点,用插值重构方法由过该点的浸入边界法向矢量与浸入边界交点IB(intersecting boundary)速度与虚拟流体点IF(intersecting fluid cell)速度之间的线性插值得到其速度

(6)$\begin{eqnarray}\label{eq3}u_{{\rm NB}} = v_{{\rm IB}} + \frac{y_{{\rm NB}}}{y_{{\rm IF}} }(u_{{\rm IF}} - v_{{\rm IB}} )\end{eqnarray}$
式中, $v_{{\rm IB}} $和$u_{{\rm IF}} $分别为IB点和IF点的速度矢量, $y_{{\rm IF}} $是IF点和IB点之间的距离, $y_{{\rm NB}} $是NB节点和IB点之间的距离, IB点的速度由$v_{{\rm IB}} $固体域求解得到, IF点的速度由其所在流体单元中的节点通过线性插值求得. 在每个时间步重复该过程, 并将其值用作求解流体域的边界条件.

图1

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图1清晰界面浸入边界法示意图结构动力学方程控制方程可表示为

Fig.1Schematic diagram of sharp interface immersed boundary method



(7)$\begin{eqnarray} \label{eq4} M\ddot{d}^{\rm s}(t) + C\dot{d}^{\rm s}(t) + \left(K + K_{\rm c} (t) \right)d^{\rm s}(t) = F_{\rm f} \left( t \right) + F_{\rm c} (t) \end{eqnarray}$
式中, $d^{\rm s}$是位移矢量, $M$是质量矩阵, $C$是阻尼矩阵, $K$是刚度矩阵, $K_{\rm c} (t)$是接触刚度矩阵, $F_{\rm f} \left( t \right)$为流固耦合界面上的流体力, $F_{\rm c} \left( t \right)$为碰撞时的接触力, 采用有限元法中的罚函数方法进行求解^[23].采用Newmark法和Newton-Raphson迭代法相结合完成动力学方程(7)的求解^[25].

假设弹性支承的圆柱结构物仅可横向振动, 通过引入无量纲参数, 可得到动力学方程(7)的量纲归一化形式为

(8)$\begin{eqnarray} \label{eq5} \ddot{Y} + \frac{4\pi \varsigma }{U_{{\rm red}} }\dot{Y} + \frac{4\pi ^2}{U_{{\rm red}}^2 }Y = \frac{2C_Y }{\pi M_{{\rm red}} }+\frac{2C_{\rm C} }{\pi M_{{\rm red}} } \end{eqnarray}$
式中, $Y$为圆柱结构物的横流向位移; $U_{{\rm red}} = U /(f_{\rm n} D)$为折流速度, 其中$U$为流场自由流速度, $f_{\rm n} $为圆柱的自然频率, $D$为圆柱的直径; $M_{\rm r} = 4m / (\pi \rho LD^{2})$为质量比, 其中$m$为圆柱体质量, $\rho $为流体密度, $L$为圆柱体展向长度; $\zeta = c / (2\sqrt {km} )$为阻尼比, 其中$c$为阻尼系数, $k$为弹簧刚度; $C_{\rm L} = F_y/(0.5\rho U^{2} DL)$为升力系数; $C_{\rm C} = F_C /(0.5\rho U^2 DL)$为碰撞力系数.

1.2 润滑模型

在黏性流体中当一个物体以有限的相对速度接近另一个物体或壁面时, 物体将会受到润滑力的影响^[26], 当两物体间隙不大于$\delta _{\Delta x} $时, 由于存在多个浸入边界, 使得浸入边界间缺乏流体网格分辨率, 传递边界参数的插值方程无法建立, 不能够准确计算此时的润滑力, 如图2所示.而在黏性流体中固体与固体碰撞的模拟中, 考虑润滑力的影响对计算真实的碰撞速度和反弹速度具有重要意义^[27].对于物体与固定壁面碰撞问题, 由于可以提前预测碰撞位置为固定壁面处, 可以借助于过度的网格细化来弥补此问题^[23], 而对于两物体或多个物体在黏性流体中碰撞位置不确定的情况,如采用过度的网格细化将会给计算量带来巨大的额外增加, 文献^[27--29]采用的一种方法是保持网格固定,并使用基于Stokes流动中润滑力解析解的渐近展开的润滑模型来补偿这种缺乏空间网格分辨率情况, 对静止流体中的颗粒碰撞问题进行了数值模拟, 但在实际问题中流体通常是流动的, 所以需建立一种新的适用于有流体流动条件下的润滑模型.

图2

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图2多个浸入边界示意图

Fig.2Schematic diagram of multiple immersed boundary



润滑理论表明, 由于固体间碰撞时间隙$\delta $趋于零, 润滑力将趋于无穷大, 致使理想的光滑物体不会达到实际的固固接触, 但实际中在润滑效应变得非常重要之前, 两物体可通过壁面的粗糙度$\delta_\varsigma $而产生接触碰撞. 在两物体接触过程中润滑力的最重要分量是沿两固体中心连线的挤压力,因为它的主导项是$1/\delta $, 而平移剪切力和旋转剪切力的主导项是发散较慢的$\ln \delta $^[30], 因此本文仅对润滑力的法向挤压力进行修正. 润滑模型示意图如图3所示, 参照Stokes流动中润滑模型的表达式^[31], 本文建立了两圆柱相互接近时法向润滑力系数$C_{\rm Ln}$的表达式为

(9)$\begin{eqnarray} \label{eq9} C_{\rm Ln} = \left\{ {{\begin{array}{ll} {0, } & {\delta > \delta _{\Delta x} } \\[2mm] {\dfrac{\mu V_{ij, n} }{0.5\rho V_i^2 L_i }f\left( {Re_{V_i } , \delta /D} \right), } & {\delta _\varsigma < \delta \leqslant \delta _{\Delta x} } \\[3mm] {\dfrac{\mu V_{ij, n} }{0.5\rho V_i^2 L_i }f\left( {Re_{V_i } , \delta _\varsigma /D} \right), } & {0 \leqslant \delta \leqslant \delta _\varsigma } \\ \end{array} }} \right. \end{eqnarray}$
式中, $V_{ij, n} $为两圆柱相互接近时法向的相对速度; $L_i $为圆柱体的长度; 绕流雷诺数为$Re_{V_i } = \rho D_i V_i / \mu $, 其中$V_i $为圆柱相对流体的运动速度; $\delta _\varsigma $一般由固体表面的绝对粗糙度和碰撞临界状态来确定, $\delta _\varsigma $越小越不容易产生碰撞, 为了能够产生碰撞取$\delta _\varsigma = 0.001D$, 并在$\delta _\varsigma $以下取$C_{\rm Ln} $的值为$\delta = \delta _\varsigma $时的值.

图3

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图3润滑模型示意图

Fig.3Schematic diagram of lubrication model



$\delta _{\Delta x}$的值可以通过模拟两圆柱相互接近来确定, 为减小计算量, 本文采用圆柱附近$\Delta x = \Delta y = 0.0\mbox{2}D$网格大小模拟两圆柱的相互接近过程, 并进一步在两圆柱中间采用平滑过渡方法进行局部网格加密, 第一层网格高度为$h$, 增长率为1.1, 图4为$h$ = 0.001$D$时的计算网格, 图5为两圆柱在无背景流速时以相同恒定速度0.5 m/s相互接近接触,不同$h$工况下圆柱的法向润滑力系数$C_{\rm Ln} $随间隙$\delta $的变化, 由图中可以看出在$\delta > 0.04D$时, 即两圆柱间有两个以上的完整流体单元, 不同$h$工况下$C_{\rm Ln} $基本一致; 当$\delta \leqslant 0.04D$时,不同$h$下$C_{\rm Ln} $差别较明显, 表明在$\Delta x = \Delta y = 0.0\mbox{2}D$网格大小下$\delta _{\Delta x} $为0.04$D$, 并且随着$h$的减小, $C_{\rm Ln} $在接触时呈现逐渐增大的趋势, 也与润滑理论相吻合.

图4

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图4第一层网格高度$h $= 0.001$D$时的计算网格

Fig.4Computational grid with the first layer of grid height $h$ = 0.001$D$

图5

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图5不同$h$工况下润滑力系数随间隙的变化

Fig.5Variation of lubrication force coefficient with clearance under different $h$ cases



采用第一层网格高度$h = 0.001D$网格, 对背景流速垂直于两圆柱运动方向时两圆柱相互接近接触进行了数值模拟, 得到了背景流速$U = 0.1\sim1.0$ m/s时(间隔0.1 m/s)和两圆柱相同恒定速度$V_{\rm s} = 0.1\sim1.0$ m/s (间隔0.1 m/s)共100种工况下在$0.001D \leqslant \delta \leqslant 0.04D$时润滑力系数$C_{\rm Ln} $随间隙$\delta $的变化, 并采用1stOpt软件中准牛顿法(BFGS) +通用全局优化法(UGO), 回归拟合得到了背景流速垂直于两圆柱运动方向时$C_{\rm Ln} $计算公式为

(10)$\begin{eqnarray} \label{eq7} && C_{\rm Ln} = \frac{\mu V_{ij, n} }{0.5\rho V_i^2 L}\Big( p_1 + p_2 Re_{V_i} + p_3 Re_{V_i}^2+p_4 \delta /D+\\&&\qquad p_5\delta^{2}/D^2 \Big)\Big/\left( 1 + p_6Re_{V_i }+p_7 \delta /D +p_8\delta ^{2}/D^2 \right) \end{eqnarray}$
式中, $p_1$$\sim$$p_8$的值如表1中所示. 该计算公式的均方差(RSME)为3.412,相关系数($R$)为0.978, 决定系数(DC)为0.956. 图6为背景流速$U =0.5$ m/s、两圆柱相同恒定速度$V_{\rm s} = 0.5$ m/s工况下, 无润滑修正、有润滑修正和两圆柱中间采用网格加密的数值模拟结果比较, 由图中可以看出所建立的润滑模型与网格加密结果较为接近, 表明所建模型是可靠的.

Table 1
表1
表1润滑力系数公式参数值
Table 1The parameters of lubrication force coefficient formula

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图6

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图6无润滑修正、有润滑修正以及加密网格的数值模拟结果比较

Fig.6Comparison of numerical simulation results of without lubrication correction, with lubrication correction and refinement mesh

2 数值方法验证与问题描述

文献^[8]通过与静止流体中球形颗粒与壁面正碰撞和斜碰撞的实验数据对比, 验证了建立的数值模拟方法对碰撞问题模拟的正确性, 为进一步验证数值方法的正确性, 对弹性支承仅可横向振动的单圆柱单圆柱在雷诺数为150, $M_{\rm r} = 8 / \pi $和$\zeta = 0$时不同折流速度下的涡激振动进行了数值模拟, 示意图如图7所示.圆柱最大横向位移$Y_{\max } / D$随$U_{{\rm red}} $的变化趋势如图8所示, 从图中可以看出在$U_{{\rm red}} \in [4, 7]$为 "锁频"区间, 并可观察到较大的响应振幅(超过直径的20%), 而在该区域之外,最大横向位移则大幅降低, 在图8中也给出了相同参数下Ahn和Kallinderis^[32]基于贴体网格ALE流固耦合的数值模拟结果以及Borazjani等^[33]同样采浸入边界方法得到的数值模拟结果,从图中可以看出, 除$U_{{\rm red}} = \mbox{7}$工况外, 三者之间的变化趋势吻合较好.

图7

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图7单自由度涡激振动示意图

Fig.7Schematic diagram of vortex-induced vibration with single degree of freedom

图8

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图8不同折流速度下的最大位移

Fig.8The maximum displacement under different reduced velocities



图9为$U_{{\rm red}} \in [3, 8]$圆柱处于最小位置时的瞬时涡量云图, 在$U_{{\rm red}} = 3$时, 对应最小振幅的振荡, 圆柱尾迹呈现出类似静止圆柱由交替脱落涡组成的单排涡街; 当$U_{{\rm red}}$增大到4.0时, 漩涡脱落频率与圆柱固有频率最为接近, 圆柱在流体力作用下发生了共振, 振动幅值有一个巨大的阶跃, 并在圆柱附近尾迹形成了双排涡街, 但这种双排涡街是不稳定的; 当$U_{{\rm red}}$进一步增大到5.0时, 圆柱尾流显示了一个相对稳定的双排涡街, 它可以持续到尾流区下游很远的地方; 当$U_{{\rm red}} \geqslant \mbox{6}$时, 漩涡脱落频率逐渐远离圆柱固有频率, 振动幅值也逐渐减小,圆柱尾流又呈现出单排涡街结构.

图9

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图9不同折合流速度下的瞬时涡量云图

Fig.9The instantaneous vorticity contours under different reduced velocities



由图8可知, 弹性支承的单圆柱在雷诺数为150, $M_{\rm r} = 8 / \pi $, $\zeta = 0$和$U_{\rm red} = 4$时, 其横向最大响应振幅已超过0.5$D$, 本文对此条件下并列双圆柱在间隙比$g^* = G/D =1.0$, 0.75, 0.5和0.25可能发生的涡激诱导碰撞现象进行了数值模拟, 图10为计算域示意图, 两圆柱体相同且均由弹簧单元支撑, 仅可横向振动, 在进口边界上指定了狄利克雷(Dirichlet)速度边界条件, 在出口边界处设定了诺依曼(Neumann)压力边界条件, 上下边界为自由滑移边界条件, 流体域与固体域展向长度$L$为$\pi D$, 均匀布置了10层网格, 在垂直展向的平面$[-1.5D, 1.5D ]\times [-3.0D, 3.0D]$范围内采用相同网格大小$\Delta x = \Delta y = 0.02D$进行了局部加 密, 浸入的圆柱体由有限元实体单元进行划分网格, 圆周上的节点数为120, 润滑模型采用通过回归方法得到的式(10). 不同间隙比下的计算初始流场如图11所示, 由图中可以看出, 不同间隙比下均产生了间隙流, 两圆柱的尾流流场并不是对称的, 间隙流出现了不同程度的偏斜流态^[5], 并随着间隙比减小两圆柱的间隙流逐渐减弱.

图10

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图10并列双圆柱计算域示意图

Fig.10Schematic diagram of the computational field of two side-by-side cylinders

图11

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图11不同间隙比下的初始流场

Fig.11Initial flow field at different clearance ratios

3 结果分析

并列双圆柱在雷诺数为150, $M_{\rm r} = 8 / \pi $, $\zeta= 0$, $U_{\rm r} = 4$时, 不同初始间隙比下两圆柱横向位移随时间变化如图12所示, 由图可知当初始间隙比$g^*\geqslant 0.75$时, 两圆柱不会发生碰撞, 间隙比越大, 其振动越接近单圆柱的涡激振动; 当初始间隙比为$g^* \leqslant 0.5$时, 两圆柱会发生碰撞, 并且碰撞发生时两圆柱会产生连续的碰撞, 也可以从图中看出在初始间隙比$g^* = 1.0$和0.75时, 两圆柱在稳定后为同相位振动,而在初始间隙比$g^* = 0.5$和0.25时两圆柱为反相位振动.

图12

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图12横向位移随时间变化

Fig.12Time history of transverse displacement



从数值模拟结果来看, 圆柱1与圆柱2的振动频率基本相同, 图13为不同初始间隙比下通过圆柱1的位移曲线进行快速傅里叶变换得到的振幅频谱图,由图中可以看出, 在初始间隙比$g^* = 1.0$和0.75时振动较为规则, 主频率也较为接近,而在初始间隙比$g^* = 0.5$和0.25时由于碰撞的发生, 出现了多阶频率, 并且振动主频率要比无碰撞时大, 而且$g^* = 0.25$振动主频率比$g^* = 0.5$振动主频率小.

图13

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图13不同初始间隙比下圆柱1的振幅频谱图

Fig.13Amplitude spectrum diagram of cylinder 1 at different initial clearance ratios



图14为初始间隙比$g^* = 0.5$和0.25碰撞发生时两圆柱无量纲相对速度$V_{ij, n}/U$与碰撞频次$n$的关系曲线, 由图中可以看出两圆柱碰撞时的相对速度$V_{ij, n}$要比自由来流速度$U$小, 而$g^* = 0.25$碰撞时的相对速度普遍比$g^* = 0.5$相对速度小; 在相同总计算时间内, 由于$g^* = 0.25$时两圆柱间的距离较小, 开始发生碰撞的时间较早,从而导致其碰撞发生的次数比$g^* = 0.5$碰撞次数多, 小间隙比时的碰撞更为频繁.

图14

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图14碰撞时两圆柱的相对速度与碰撞频次关系从

Fig.14The relation between relative velocity and collision frequency



数值模拟结果来看不同间隙比碰撞时横向流体力和圆柱速度变化规律基本相同, 图15给出了间隙比$g^{\ast}=0.5$时一个周期内包含第九次碰撞的两圆柱横向流体力系数$C_{\rm L}$和无量纲化横向速度$V_{y}/U$随时间变化曲线, 由图中可以看出两圆柱的变化趋势相同, 当两圆柱从最大横向位移开始逐渐互相接近时, 横向流体力方向始终与速度相反,$C_{\rm L}$先是逐渐减小, 当两圆柱间隙约为0.2$D$时又逐渐增大, 在碰撞前两圆柱的横向流体力系数$C_{\rm L}$约为10;当两个圆柱发生碰撞时会产生碰撞力, 由于碰撞力较大,如在第九次碰撞时最大碰撞力系数$C_{\rm c}$为145,导致两圆柱反弹时的瞬时加速度较两圆柱接近时大,并且阻碍圆柱反弹的瞬时横向流体力也随之增大,如两圆柱在第九次碰撞反弹时的瞬时$C_{\rm L}$约为19.5, 相当于两圆柱接近时的两倍;碰撞后, 横向流体力出现振荡但随两圆柱间隙的增大振荡逐渐减小,然后横向流体力方向与速度方向同向, 流体对圆柱做正功向圆柱输入能量.随着两圆柱从最大位移逐渐接近, 圆柱的速度先逐渐增大后又逐渐减小,但由于阻碍两圆柱接近的排斥力作用时间较短, 冲量不能减小为零,所以两圆柱仍会产生碰撞, 由于流体的黏性耗散,碰撞后的回弹速度约为碰撞前速度的三分之一,碰撞后圆柱的横向速度如横向流体力一样振荡, 并随着间隙的增加振荡也逐渐减小,当两圆柱相互远离时圆柱的速度也是先逐渐增大后又逐渐减小.

图15

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图15横向流体力系数和无量纲化横向速度随时间变化

Fig.15Time history of the transverse force coefficient and dimensionless velocity



图16为$g^* = 0.5$时包含第九次碰撞过程中不同时刻的涡量、顺流向流速和压力云图,当两圆柱相互接近时, 随着涡环分离角度的逐渐倾斜, 横向流体力先逐渐减小,当两圆柱间涡环开始相互影响发生挤压时, 横向流体力又逐渐增大,并且两圆柱间涡环由于相互挤压和碰撞改变了其脱落形态; 当两圆柱逐渐分离时,二次涡环逐渐增长而主涡环逐渐脱落, 随着涡环倾斜角度的逐渐减小,横向流体力逐渐增大. 随着两圆柱逐渐接近, 流体受到挤压,两圆柱间的流速逐渐增大, 压力也随之增大,从而使两圆柱逐渐接近时的速度逐渐减小, 并在碰撞时流场压力达到最大值,圆柱的速度达到最小值; 当两圆柱碰撞后开始反弹时,在两圆柱间形成了回流区和低压区, 改变了横向流体阻力的方向,由于碰撞后圆柱的回弹速度较小, 在横向流体阻力作用下两圆柱又产生了接近运动,如此反复从而产生了碰撞后横向流体力和圆柱速度的振荡现象,直至两圆柱间的距离约为0.5$D$时振荡现象消失.

图16

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图16碰撞过程中不同时刻的涡量、顺流向流速和压力云图

Fig.16The contours of the vorticity, downstream velocity and pressure at different times during the collision



图17为$g^* = 1.0$时一个周期内两圆柱的涡量云图, 从图中可以看出两圆柱为同相位振动,意味着圆柱之间始终保持近似相同的距离, $g^* = 0.75$时两圆柱也为同相位振动,所以两圆柱不会产生碰撞.

图17

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图17$g^* = 1.0$时一个周期的涡量云图

Fig.17Vorticity contours of a period when $g^* = 1.0$

4 结 论

本文采用基于浸入边界-有限元的流固耦合分析方法, 将流体区域离散在简单且固定网格上,避免了传统贴体网格方法求解流体中存在固体间碰撞问题时出现负体积网格问题, 并通过回归方法建立了有流体流动条件下的润滑模型, 对不同小间隙比下的涡激诱导并列双圆柱振动及其碰撞进行了数值模拟,得到以下结论:

(1) 小间隙比下如果两圆柱产生了碰撞将会有连续的碰撞发生, 两圆柱为反相位振动;

(2) 碰撞时出现了多阶频率, 并且振动主频率要比无碰撞时大;

(3) 两圆柱碰撞时的相对速度比自由来流速度小,较小间隙比碰撞时的相对速度普遍比较大间隙比碰撞时小, 但碰撞更为频繁;

(4) 当两圆柱互相接近时, 横向流体力先逐渐减小后又逐渐增大,碰撞反弹时的瞬时流体力约为两圆柱接近时最大横向流体力的两倍,碰撞后横向流体力出现振荡但随两圆柱间隙的增大振荡逐渐减小;

(5) 当两圆柱互相接近时, 圆柱的速度先逐渐增大后又逐渐减小,碰撞后的回弹速度约为碰撞前速度的三分之一,碰撞后圆柱的横向速度如横向流体力一样振荡, 并随着间隙的增加振荡也逐渐减小;

(6) 当两圆柱相互接近时, 随着涡环分离角度的逐渐倾斜, 横向流体力逐渐减小;当两圆柱间涡环开始相互影响发生挤压时, 横向流体力开始逐渐增大,两圆柱间涡环由于相互挤压和碰撞改变了其脱落形态;

(7) 当两圆柱开始反弹时, 两圆柱间形成了低压区, 改变了横向流体阻力的方向,在横向流体阻力作用下两圆柱又产生了接近运动,如此反复从而产生了碰撞后横向流体力和圆柱速度的振荡现象,直至两圆柱间的距离约为0.5$D$时振荡现象消失.

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Abstract

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[10] Zhao M. Flow induced vibration of two rigidly coupled circular cylinders in tandem and side-by-side arrangements at a low Reynolds number of 150
Physics of Fluids, 2013,25(12):123601
DOIURLPMID [本文引用: 1]
Hydraulic permeabilities of fiber networks are of interest for many applications and have been studied extensively. There is little work, however, on permeability calculations in three-dimensional random networks. Computational power is now sufficient to calculate permeabilities directly by constructing artificial fiber networks and simulating flow through them. Even with today's high-performance computers, however, such an approach would be infeasible for large simulations. It is therefore necessary to develop a correlation based on fiber volume fraction, radius, and orientation, preferably by incorporating previous studies on isotropic or structured networks. In this work, the direct calculations were performed, using the finite element method, on networks with varying degrees of orientation, and combinations of results for flows parallel and perpendicular to a single fiber or an array thereof, using a volume-averaging theory, were compared to the detailed analysis. The detailed model agreed well with existing analytical solutions for square arrays of fibers up to fiber volume fractions of 46% for parallel flow and 33% for transverse flow. Permeability calculations were then performed for isotropic and oriented fiber networks within the fiber volume fraction range of 0.3%-15%. When drag coefficients for spatially periodic arrays were used, the results of the volume-averaging method agreed well with the direct finite element calculations. On the contrary, the use of drag coefficients for isolated fibers overpredicted the permeability for the volume fraction range that was employed. We concluded that a weighted combination of drag coefficients for spatially periodic arrays of fibers could be used as a good approximation for fiber networks, which further implies that the effect of the fiber volume fraction and orientation on the permeability of fiber networks are more important than the effect of local network structure.

[11] 陈威霖, 及春宁, 徐万海 . 并列双圆柱流致振动的不对称振动和对称性迟滞研究
力学学报, 2015,47(5):731-739
DOIURL [本文引用: 1]
对雷诺数Re = 100 间距比s/D = 2.5 和5.0 的并列双圆柱流致振动进行了数值模拟研究, 其中圆柱质量比m = 2.0, 折合流速U_r 在2.0~10.0 之间, 两圆柱仅能做横流向振动. 研究发现, 当间距比s/D = 2.5 时, 在折合流速4.4 < U_r< 4.8区间内, 两圆柱流致振动响应出现不对称振动现象, 在折合流速4.4 < U_r< 4.8 区间内, 两圆柱流致振动响应出现对称性迟滞现象; 而当间距比s/D = 2.5时, 圆柱流致振动响应与单圆柱涡激振动响应相似, 没有出现不对称振动和对称性迟滞现象. 在不对称振动区间内, 两圆柱的升、阻力参数也出现了不相等的情况. 此外, 当两圆柱不对称振动时, 圆柱间隙流稳定地偏斜向其中的一个圆柱; 相应地, 尾涡也出现了宽窄不等的模式. 窄尾流圆柱的振幅和升、阻力均较宽尾流圆柱的大. 通过对比不对称振动现象发生前后的尾涡模式, 对新现象的产生机制进行了阐述.
( Chen Weilin, Ji Chunning, Xu Wanhai, Numerical investigation on the asymmetric vibration and symmetry hysteresis of flow-induced vibration of two side-by-side cylinders
Chinese Journal of Theoretical and Applied Mechanics, 2015,47(5):731-739 (in Chinese))
DOIURL [本文引用: 1]
对雷诺数Re = 100 间距比s/D = 2.5 和5.0 的并列双圆柱流致振动进行了数值模拟研究, 其中圆柱质量比m = 2.0, 折合流速U_r 在2.0~10.0 之间, 两圆柱仅能做横流向振动. 研究发现, 当间距比s/D = 2.5 时, 在折合流速4.4 < U_r< 4.8区间内, 两圆柱流致振动响应出现不对称振动现象, 在折合流速4.4 < U_r< 4.8 区间内, 两圆柱流致振动响应出现对称性迟滞现象; 而当间距比s/D = 2.5时, 圆柱流致振动响应与单圆柱涡激振动响应相似, 没有出现不对称振动和对称性迟滞现象. 在不对称振动区间内, 两圆柱的升、阻力参数也出现了不相等的情况. 此外, 当两圆柱不对称振动时, 圆柱间隙流稳定地偏斜向其中的一个圆柱; 相应地, 尾涡也出现了宽窄不等的模式. 窄尾流圆柱的振幅和升、阻力均较宽尾流圆柱的大. 通过对比不对称振动现象发生前后的尾涡模式, 对新现象的产生机制进行了阐述.

[12] 刘爽 . 低雷诺数下并列圆柱涡激振动的数值模拟及其机理研究
[硕士论文]. 天津: 天津大学, 2014
[本文引用: 1]

( Liu Shuang . Numerical investigation on vortex-induced vibrations of two side-by-side arranged circular cylinders at a low Reynolds number
[Master Thesis]. Tianjin: Tianjin University, 2014 (in Chinese))
[本文引用: 1]

[13] Liu B, Jaiman RK. Interaction dynamics of gap flow with vortex-induced vibration in side-by-side cylinder arrangement
Physics of Fluids, 2016,28(12):127103
DOIURL [本文引用: 1]

[14] 陈文曲 . 二维串并列圆柱绕流与涡致振动研究
[硕士论文]. 杭州: 浙江大学, 2005
[本文引用: 1]

( Chen Wenqu . Study on flow and vortex induced vibration of two 2-D cylinders
[Master Thesis]. Hangzhou: Zhejiang University, 2005 (in Chinese))
[本文引用: 1]

[15] Wang XQ, So RMC, Xie WC, et al. Free-stream turbulence effects on vortex-induced vibration of two side-by-side elastic cylinders
Journal of Fluids and Structures, 2008,24:664-679
DOIURL [本文引用: 1]

Abstract

The effect of free-stream turbulence on vortex-induced vibration of two side-by-side elastic cylinders in a cross-flow was investigated experimentally. A turbulence generation grid was used to generate turbulent incoming flow with turbulence intensity around 10%. Cylinder displacements in the transverse direction at cylinder mid-span were measured in the reduced velocity range 1.45<U_r0<12.08, corresponding to a range of Reynolds number (Re), based on the mean free-stream velocity and the diameter of the cylinder, between Re=5000–41 000. The focus of the study is on the regime of biased gap flow, where two cylinders with pitch ratio (T/D) varying from 1.17 to 1.90 are considered. Results show that the free-stream turbulence effect is to enhance the vortex-induced force, thus to restore the large-amplitude vibration associated with the lock-in resonance. However, the enhancement is significant at a different Strouhal number (St) for different pitch ratios. When the spacing between two cylinders is relatively small (1.17<T/D<1.50), the enhancement is significant at St≈0.1. When the spacing is increased, the Strouhal number at which the enhancement is significant shifts to St≈0.16. This enlarges the range of reduced velocity to be concerned. An energy analysis showed that free-stream turbulence feeds energy to the cylinder at multiple frequencies of vortex shedding. Therefore, the lock-in region is still of main concern when the approach flow is turbulent.

[16] Huera-Huarte FJ, Gharib M. Flow-induced vibrations of a side-by-side arrangement of two flexible circular cylinders
Journal of Fluids and Structures, 2011,27(3):354-366
DOIURL [本文引用: 1]
Laboratory experiments with a side-by-side arrangement of two vertical, high aspect ratio (length over diameter) and low mass ratio (mass over mass of displaced fluid) cylinders, pin-jointed at the ends and vibrating at low mode number, were carried out in a free-surface water channel. The dynamic response of the models under two different wake interference situations is presented here. Initially, one of the cylinders was fixed and the other was completely free to move. In a second battery of experiments both cylinders were free to vibrate. A very large parameter space was covered by varying the free-stream flow speeds, the natural frequencies of the system and the separation between the models, allowing the identification of vortex-induced vibrations (VIV) and wake-coupled VIV (WCVIV). Amplitudes, frequencies and phase synchronisation between the models are presented. (C) 2011 Elsevier Ltd.

[17] Fabian D, Raul G, Srinivasan N. Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries
Computer Methods in Applied Mechanics and Engineering, 2004,193(45-47):4819-4836
DOIURLPMID [本文引用: 1]
A computational model of flagellar motility is presented using the finite element method. Two-dimensional traveling waves of finite amplitude are propagated down the flagellum and the swimmer is propelled through a viscous fluid according to Newto's second law of motion. Incompressible Navier-Stokes equations are solved on a triangular moving mesh and arbitrary Lagrangian-Eulerian formulation is employed to accommodate the deforming boundaries. The results from the present study are validated against the data available in the literature and close agreement with previous works is found. The effects of wave parameters as well as head morphology on the swimming characteristics are studied for different swimming conditions. We have found that the swimming velocities are linear functions of finite amplitudes and that the rate of work is independent of the channel height for large amplitudes. Furthermore, we have also demonstrated that for the range of wave parameters that are often encountered in human sperm motility studies, the propulsive velocity versus the wavelength exhibits dissimilar trends for different channel heights. Various head configurations were analyzed and it is also observed that wall proximity amplifies the effects induced by different head shapes. By taking non-Newtonian fluids into account, we present new efficiency analyzes through which we have found that the model microorganism swims much more efficiently in shear-thinning fluids.

[18] Peskin CS. Flow patterns around heart valves: A numerical method
Journal of Computational Physics, 1972,10(2):252-271
DOIURLPMID [本文引用: 1]
The blood flow patterns in the region around the aortic valve depend on the geometry of the aorta and on the complex flow-structure interaction between the pulsatile flow and the valve leaflets. Consequently, the flow depends strongly on the constitutive properties of the tissue, which can be expected to vary between healthy and diseased heart valves or native and prosthetic valves. The main goal of this work is to qualitatively demonstrate that the choice of the constitutive model of the aortic valve is critical in analysis of heart hemodynamics. To accomplish that two different constitutive models were used in curvilinear immersed boundary-finite element-fluid-structure interaction (CURVIB-FE-FSI) method developed by Gilmanov et al. (2015, &amp;quot;A Numerical Approach for Simulating Fluid Structure Interaction of Flexible Thin Shells Undergoing Arbitrarily Large Deformations in Complex Domains,&amp;quot; J. Comput. Phys., 300, pp. 814-843.) to simulate an aortic valve in an anatomic aorta at physiologic conditions. The two constitutive models are: (1) the Saint-Venant (StV) model and (2) the modified May-Newman&amp;amp;Yin (MNY) model. The MNY model is more general and includes nonlinear, anisotropic effects. It is appropriate to model the behavior of both prosthetic and biological tissue including native valves. Both models are employed to carry out FSI simulations of the same valve in the same aorta anatomy. The computed results reveal dramatic differences in both the vorticity dynamics in the aortic sinus and the wall shear-stress patterns on the aortic valve leaflets and underscore the importance of tissue constitutive models for clinically relevant simulations of aortic valves.

[19] Mittal R, Iaccarino G. Immersed boundary methods.
Annu. RevFluid Mech, 2005,37:239-261
DOIURLPMID [本文引用: 1]
Fluid-structure systems occur in a range of scientific and engineering applications. The immersed boundary (IB) method is a widely recognized and effective modeling paradigm for simulating fluid-structure interaction (FSI) in such systems, but a difficulty of the IB formulation of these problems is that the pressure and viscous stress are generally discontinuous at fluid-solid interfaces. The conventional IB method regularizes these discontinuities, which typically yields low-order accuracy at these interfaces. The immersed interface method (IIM) is an IB-like approach to FSI that sharply imposes stress jump conditions, enabling higher-order accuracy, but prior applications of the IIM have been largely restricted to numerical methods that rely on smooth representations of the interface geometry. This paper introduces an immersed interface formulation that uses only a C⁰ representation of the immersed interface, such as those provided by standard nodal Lagrangian finite element methods. Verification examples for models with prescribed interface motion demonstrate that the method sharply resolves stress discontinuities along immersed boundaries while avoiding the need for analytic information about the interface geometry. Our results also demonstrate that only the lowest-order jump conditions for the pressure and velocity gradient are required to realize global second-order accuracy. Specifically, we demonstrate second-order global convergence rates along with nearly second-order local convergence in the Eulerian velocity field, and between first- and second-order global convergence rates along with approximately first-order local convergence for the Eulerian pressure field. We also demonstrate approximately second-order local convergence in the interfacial displacement and velocity along with first-order local convergence in the fluid traction along the interface. As a demonstration of the method's ability to tackle more complex geometries, the present approach is also used to simulate flow in a patient-averaged anatomical model of the inferior vena cava, which is the large vein that carries deoxygenated blood from the lower extremities back to the heart. Comparisons of the general hemodynamics and wall shear stress obtained by the present IIM and a body-fitted discretization approach show that the present method yields results that are in good agreement with those obtained by the body-fitted approach.

[20] 朱帅帅 . 圆柱耐压结构碰撞响应研究
[硕士论文]. 镇江: 江苏科技大学, 2019
[本文引用: 1]

( Zhu Shuaishuai . Research on collision response of cylindrical pressure-resistant structure
[Master Thesis]. Zhenjiang: Jiangsu University of Science and Technology, 2019 (in Chinese))
[本文引用: 1]

[21] 张娅 . 船舶碰撞过程中的附加质量研究
[硕士论文]. 武汉: 武汉理工大学, 2016
[本文引用: 1]

( Zhang Ya . The added mass research in the process of ship collision
[Master Thesis]. Wuhan: Wuhan University of Technology, 2016 (in Chinese))
[本文引用: 1]

[22] 章和兵 . 叶轮流道内颗粒碰撞及磨损机理研究
[硕士论文]. 杭州: 浙江理工大学, 2018
[本文引用: 1]

( Zhang Hebing . Research on particle collision and erosion mechanism in the channel of impeller. [Master Thesis]. Hangzhou:
Zhejiang Sci-Tech University , 2018 (in Chinese))
[本文引用: 1]

[23] 杨明, 刘巨保, 岳欠杯 等. 基于浸入边界-有限元法的流固耦合碰撞数值模拟方法
应用数学和力学, 2019,40(8):880-892
[本文引用: 3]

( Yang Ming, Liu Jubao, Yue Qianbe, et al. Numerical simulation of fluid-solid coupling collision based on immersed boundary-finite element method
Applied Mathematics and Mechanics, 2019,40(8):880-892 (in Chinese))
[本文引用: 3]

[24] Gilmanov A, Sotiropoulos F, Balaras E. A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids
Journal of Computational Physics, 2003,191(2):660-669
DOIURL [本文引用: 1]

[25] 刘巨保, 罗敏 . 有限单元法及应用. 北京: 中国电力出版社, 2013
[本文引用: 1]

( Liu Jubao, Luo Min. Finite Element Method and Application. Beijing: China Electric Power Press, 2013 (in Chinese))
[本文引用: 1]

[26] Davis RH, Serayssol JM, Hinch EJ. The elastohydrodynamic collision of two spheres
Journal of Fluid Mechanics, 2006,163(163):479-497
DOIURLPMID [本文引用: 1]
Experimental evidence shows that the presence of an ambient liquid can greatly modify the collision process between two solid surfaces. Interactions between the solid surfaces and the surrounding liquid result in energy dissipation at the particle level, which leads to solid-liquid mixture rheology deviating from dry granular flow behaviour. The present work investigates how the surrounding liquid modifies the impact and rebound of solid spheres. Existing collision models use elastohydrodynamic lubrication (EHL) theory to address the surface deformation under the developing lubrication pressure, thereby coupling the motion of the liquid and solid. With EHL theory, idealized smooth particles are made to rebound from a lubrication film. Modified EHL models, however, allow particles to rebound from mutual contacts of surface asperities, assuming negligible liquid effects. In this work, a new contact mechanism, 'mixed contact', is formulated, which considers the interplay between the asperities and the interstitial liquid as part of a hybrid rebound scheme. A recovery factor is further proposed to characterize the additional energy loss due to asperity-liquid interactions. The resulting collision model is evaluated through comparisons with experimental data, exhibiting a better performance than the existing models. In addition to the three non-dimensional numbers that result from the EHL analysis--the wet coefficient of restitution, the particle Stokes number and the elasticity parameter--a fourth parameter is introduced to correlate particle impact momentum to the EHL deformation impulse. This generalized collision model covers a wide range of impact conditions and could be employed in numerical codes to simulate the bulk motion of solid particles with non-negligible liquid effects.

[27] Kempe T, Fr?Hlich J. Collision modelling for the interface-resolved simulation of spherical particles in viscous fluids
Journal of Fluid Mechanics, 2012,709:445-489
DOIURL [本文引用: 2]
The paper presents a model for particle-particle and particle-wall collisions during interface-resolving numerical simulations of particle-laden flows. The accurate modelling of collisions in this framework is challenging due to methodological problems generated by interface approach and contact as well as due to the greatly different time scales involved. To cope with this situation, multiscale modelling approaches are introduced avoiding excessive local grid refinement during surface approach and time step reduction during the surface contact. A new adaptive model for the normal forces in the phase of 'dry contact' is proposed, stretching the collision process in time to match the time step of the fluid solver. This yields a physically sound and robust collision model with modified stiffness and damping determined by an optimization scheme. Furthermore, the model is supplemented with a new approach for modelling the tangential force during oblique collisions which is based on two material parameters: a critical impact angle separating rolling from sliding and the friction coefficient for the sliding motion. The resulting new model is termed the adaptive collision model (ACM). All proposed sub-models only contain physical parameters, and virtually no numerical parameters requiring adjustment or tuning. The new model is implemented in the framework of an immersed boundary method but is applicable with any spatial and temporal discretization. Detailed validation against experimental data was performed so that a general and versatile model for arbitrary collisions of spherical particles in viscous fluids is now available.

[28] Ten Cate A, Nieuwstad CH, Derksen JJ, et al. Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity
Physics of Fluids, 2002,14(11):4012
DOIURL

[29] Br?Ndle d MJC, Breugem WP, Gazanion B, et al. Numerical modelling of finite-size particle collisions in a viscous fluid
Physics of Fluids, 2013,25(8):083302
DOIURL [本文引用: 1]

[30] Costa P, Boersma BJ, Westerweel J, et al. Collision model for fully resolved simulations of flows laden with finite-size particles
Physical Review E, 2015,92(5):053012
DOIURLPMID [本文引用: 1]
We present a collision model for particle-particle and particle-wall interactions in interface-resolved simulations of particle-laden flows. Three types of interparticle interactions are taken into account: (1) long- and (2) short-range hydrodynamic interactions, and (3) solid-solid contact. Long-range interactions are incorporated through an efficient and second-order-accurate immersed boundary method (IBM). Short-range interactions are also partly reproduced by the IBM. However, since the IBM uses a fixed grid, a lubrication model is needed for an interparticle gap width smaller than the grid spacing. The lubrication model is based on asymptotic expansions of analytical solutions for canonical lubrication interactions between spheres in the Stokes regime. Roughness effects are incorporated by making the lubrication correction independent of the gap width for gap widths smaller than ～1% of the particle radius. This correction is applied until the particles reach solid-solid contact. To model solid-solid contact we use a variant of a linear soft-sphere collision model capable of stretching the collision time. This choice is computationally attractive because it allows us to reduce the number of time steps required for integrating the collision force accurately and is physically realistic, provided that the prescribed collision time is much smaller than the characteristic time scale of particle motion. We verified the numerical implementation of our collision model and validated it against several benchmark cases for immersed head-on particle-wall and particle-particle collisions, and oblique particle-wall collisions. The results show good agreement with experimental data.

[31] Cox RG, Brenner H. The slow motion of a sphere through a viscous fluid towards a plane surface---II small gap widths, including inertial effects
Chemical Engineering Science, 1967,16(3):242-251
DOIURL [本文引用: 1]

[32] Ahn HT, Kallinderis Y. Strongly Coupled Flow/Structure Interactions with A Geometrically Conservative Ale Scheme on General Hybrid Meshes
Academic Press Professional Inc, 2006
[本文引用: 1]

[33] Borazjani I, Ge L, Sotiropoulos F. Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies
Journal of Computational Physics, 2008,227(16):7587-7620
DOIURLPMID [本文引用: 1]
The sharp-interface CURVIB approach of Ge and Sotiropoulos [L. Ge, F. Sotiropoulos, A Numerical Method for Solving the 3D Unsteady Incompressible Navier-Stokes Equations in Curvilinear Domains with Complex Immersed Boundaries, Journal of Computational Physics 225 (2007) 1782-1809] is extended to simulate fluid structure interaction (FSI) problems involving complex 3D rigid bodies undergoing large structural displacements. The FSI solver adopts the partitioned FSI solution approach and both loose and strong coupling strategies are implemented. The interfaces between immersed bodies and the fluid are discretized with a Lagrangian grid and tracked with an explicit front-tracking approach. An efficient ray-tracing algorithm is developed to quickly identify the relationship between the background grid and the moving bodies. Numerical experiments are carried out for two FSI problems: vortex induced vibration of elastically mounted cylinders and flow through a bileaflet mechanical heart valve at physiologic conditions. For both cases the computed results are in excellent agreement with benchmark simulations and experimental measurements. The numerical experiments suggest that both the properties of the structure (mass, geometry) and the local flow conditions can play an important role in determining the stability of the FSI algorithm. Under certain conditions unconditionally unstable iteration schemes result even when strong coupling FSI is employed. For such cases, however, combining the strong-coupling iteration with under-relaxation in conjunction with the Aitken's acceleration technique is shown to effectively resolve the stability problems. A theoretical analysis is presented to explain the findings of the numerical experiments. It is shown that the ratio of the added mass to the mass of the structure as well as the sign of the local time rate of change of the force or moment imparted on the structure by the fluid determine the stability and convergence of the FSI algorithm. The stabilizing role of under-relaxation is also clarified and an upper bound of the required for stability under-relaxation coefficient is derived.