Lattice Boltzmann method for studying dynamics of single rising bubble in shear-thickening power-law fluids

Xu Xin-Meng; Lou Qin

doi:10.7498/aps.73.20240394

Abstract

Authors and contacts

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1. 引　言
2. 数值方法
3. 物理问题及数值方法
3.1 物理问题描述
3.2 模型验证
4. 数值结果与分析
4.1 气泡变形
4.2 气泡终端速度
4.3 流场剪切速率
4.4 Ga, Eo及n对因变量的影响程度对比
5. 结　论

References

Abstract

Bubble motion in non-Newtonian fluids is widely present in various industrial processes such as crude oil extraction, enhancement of boiling heat transfer, CO₂ sequestration and wastewater treatment. System containing non-Newtonian liquid, as opposed to Newtonian liquid, has shear-dependent viscosity, which can change the hydrodynamic characteristics of the bubbles, such as their size, deformation, instability, terminal velocity, and shear rate, and ultimately affect the bubble rising behaviors. In this work, the dynamic behavior of bubble rising in a shear-thickened fluid is studied by using an incompressible lattice Boltzmann non-Newtonian gas-liquid two-phase flow model. The effects of the rheological exponent n, the Eötvös number (Eo), and the Galilei number (Ga) on the bubble deformation, terminal velocity, and the shear rate are investigated. The numerical results show that the degree of bubble deformation increases as Eo grows, and the effect of n on bubble deformation degree relates to Ga. On the other hand, the terminal velocity of the bubbles increases monotonically and nonlinearly with Ga for given Eo and n, and the effect of n on the terminal velocity of the bubbles turns stronger as Ga increases. When Ga is fixed and small, the terminal velocity of the bubble increases and then decreases with the increase of n at small Eo, and increases with the increase of n when Eo is large; but when Ga is fixed and large, the terminal velocity of the bubbles increases with the increase of n in a more uniform manner. In addition, regions with high shear rates can be found near the left end and right end of the bubble. The size of these regions grows with Eo and Ga, exhibiting an initial increase followed by a decrease as n increases. Finally, the orthogonal experimental method is used to obtain the influences of the aforementioned three factors on the shear rate and terminal velocity. The order of influence on shear rate is n, Ga and Eo which are arranged in descending order. For the terminal velocity, Ga has the greatest influence, followed by n, and Eo has the least influence.

Keywords:

PACS:

47.11.-j	(Computational methods in fluid dynamics)
47.20.Bp	(Buoyancy-driven instabilities (e.g., Rayleigh-Benard))
47.45.Ab	(Kinetic theory of gases)

Authors and contacts

School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

Corresponding author: Lou Qin, qlou@usst.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 52376068, 51976128) and the Pujiang Program of Shanghai, China (Grant No. 22PJD047).

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1. 引　言

非牛顿流体中的气泡运动问题广泛存在于原油开采^[1]、曝气工艺^[2]、泡沫浮选工艺^[3]、沸腾传热强化^[4,5]、微气泡减阻等工业过程. 在石油化工领域, 重质石油作为典型的非牛顿流体, 其与裂解气体乙烯、丙烯等之间的接触面积和传质强度对化工产物的纯度和原油裂解反应产率^[6]有较大影响; 另一方面, 油气混合物在管道输运过程^[7]中表现出非牛顿特性, 在该问题中, 气泡在非牛顿流体中的输运特性是设计增压系统需要考虑的重要因素. 综上所述, 大量工业过程都涉及气泡在非牛顿流体中的输运过程, 其中气泡形状、上升速度、剪切速率等变化对过程非常复杂, 并对工业过程设备性能有重要影响. 目前, 大量学者开展了对非牛顿流体中气泡运动行为的研究.

近年来, 众多学者采用数值模拟和实验方法对非牛顿流体中的气泡变形规律进行了研究. Amirnia等^[8]采用实验方法研究了黄原胶和羧甲基纤维素溶液(CMC, 剪切变稀幂律流体)中的单气泡上升过程, 发现随着气泡直径的增大, 气泡形状经历从球形变为泪滴状再恢复为球形的过程. 此外, 小气泡呈垂直的上升路线, 而大气泡呈螺旋或之字形上升. Li等^[9]通过实验方法进一步补充研究了不同黏弹性CMC溶液中上升气泡的变形, 得到了气泡变形与惯性力、黏性力、表面张力的关系, 并建立了球形和非球形气泡的阻力相关性模型, 模拟结果与Zhang等^[10]在0.05< Reynolds数(Re) <300范围内的实验结果表现出良好的一致性. Premlata等^[11]数值模拟研究了单个气泡在静止剪切变稀流体中的形变机制, 发现增大剪切减薄效应会增大上升速度并减小形变程度, 他们还发现了表面张力的增大对气泡的形变和移动存在抑制作用. Pang等^[12]采用流体体积(VOF)方法分析了不同作用力下的气泡形变程度, 他们指出气泡的变形程度随Eo和Ga的增大以及n的减小而增大. Pan和Chen^[13]采用前沿跟踪方法研究了Bond数(Bo)对气泡形变的影响, 发现气泡的变形能力随Bo的增大而提高, 并观察到上升过程中的局部摆动, 他们还发现气泡上升过程中的摆动是由气泡侧向接触吸收平动能并将其转化为表面能的结果.

以上研究主要围绕剪切变稀流体中上升气泡的形变展开. 同时, 还有一部分学者研究了气泡在剪切变稀流体中上升时速度变化特性. Tripathi等^[14]数值研究了Bingham流体中气泡上升阶段的速度变化, 发现在高屈服应力或低表面张力条件下, 气泡速度存在非定常波动. 表面张力较弱时, 气泡上升呈匀速与加速交替的变化, 其加速阶段时长与气泡纵横比的振荡周期一致, 随表面张力的减小而增大. Pillapakkam等^[15]研究了Oldroyd-B流体中的气泡上升过程, 发现小于临界气泡体积时, 终端速度会随气泡体积的增大急剧增大. Amirnia等^[8]实验研究了不同尺寸气泡在剪切变稀聚合物溶液中的上升速度, 发现即使对于很小的气泡, 其上升速度都与体积呈幂律关系, 且气泡体积的增大会使终端速度逐渐趋于一个恒定值. Xu等^[16]实验研究了不同浓度CMC溶液中单个气泡上升的速度与变形, 发现在浓度较低的CMC溶液中, 气泡最初以椭球状直线上升, 垂直上升速度加快, 随之发生不规则变形伴随垂直速度不规则振荡, 最终过渡至明显的振荡阶段; 而在浓度较高的CMC溶液中, 气泡形状稳定, 上升速度恒定.

为了进一步研究非牛顿流体中气泡形状和速度的变化规律, 部分学者研究了气泡周围剪切速率的分布. Pang等^[12]采用VOF方法详细分析了n, Eo和Ga对气泡尾迹特性的影响, 发现在剪切减薄流体中, 高剪切速率的区域尺寸随Ga的增大而增大, 随n和Eo的增大而减小. Battistella等^[17]采用前沿跟踪Front-Tracking模型研究了非牛顿流体中的气泡形变和黏度分布, 发现气泡在强剪切变稀流体中蜿蜒上升, 并发现气泡前部和尾部存在高剪切速率区域.

以上工作调查研究了剪切减薄非牛顿流体中关键参数对气泡动力学行为、气泡速度变化以及气泡剪切速率的影响规律, 为剪切减薄非牛顿流体中气泡输运特性提供了理论指导. 然而与剪切减薄非牛顿流体对应的另一类非牛顿流体: 剪切增稠非牛顿流体在工业工程中也非常常见, 如高体积分数的胶体悬浮液^[18,19]. Ohta等^[20]结合VOF与水平集方法研究了上升气泡在剪切变稀和剪切增稠流体中黏度的变化, 发现在剪切变薄流体中气泡上升的黏度分布比在剪切增稠流体中的尺度更为广泛, 并发现剪切增稠效应使气泡上方区域黏度增大, 但抑制了尾迹部分的黏度增大区域. 非牛顿剪切增稠流体中上升气泡的变形、速度以及剪切速率受流体黏度、表面张力以及体积力的影响, 输运规律复杂, 目前对于剪切增稠流体中气泡的运动机理研究尚不充分. 基于此, 本文采用格子Boltzmann方法^[21–23]研究气泡在剪切增稠非牛顿流体中的运动过程, 重点分析流变指数n, Eo和Ga对气泡动力学运动特性的影响, 旨在进一步完善气泡在非牛顿流体中的运动机理.

2. 数值方法

格子Boltzmann方法是一种介观数值方法, 与传统数值方法相比, 具有算法简单、易于并行化、数值稳定性好、边界条件易于处理以及不需要追踪界面等^[24–29]优势, 已被广泛应用于非牛顿多相流领域相关研究. 本文采用Lou等^[30]提出的不可压非牛顿流体气液两相流格子Boltzmann模型来研究剪切增稠幂律流体中单气泡上升问题.

该模型中有两个分布函数f_α和g_α, 其中f_α描述指标参数演化过程, g_α描述速度/压力演化过程, 分别表示为

$\begin{split} &{f_\alpha }\left( {{\boldsymbol{x}} + {{\boldsymbol{e}}_\alpha }{\delta _{\text{t}}},t + {\delta _{\text{t}}}} \right) - {f_\alpha }({\boldsymbol{x}},{{t}}) \\ = \;&- \frac{{{f_\alpha }({\boldsymbol{x}},t) - f_\alpha ^{{\text{eq}}}({\boldsymbol{x}},t)}}{\tau } \\ &- \frac{{(2\tau - 1)}}{{2\tau }}\frac{{\left( {{{\boldsymbol{e}}_\alpha } - {\boldsymbol{u}}} \right) \cdot \nabla \psi (\phi )}}{{RT}}{\varGamma _\alpha }({\boldsymbol{u}}){\delta _{\text{t}}}, \end{split}$

(1)

$\begin{split} &{g_\alpha }\left( {{\boldsymbol{x}} + {{\boldsymbol{e}}_\alpha }{\delta _t},t + {\delta _{\text{t}}}} \right) - {g_\alpha }({\boldsymbol{x}},t) \\ = \;&- \frac{{{g_\alpha }({\boldsymbol{x}},t) - g_\alpha ^{{\text{eq}}}({\boldsymbol{x}},t)}}{\tau } + \frac{{2\tau - 1}}{{2\tau }}\left( {{{\boldsymbol{e}}_\alpha } - {\boldsymbol{u}}} \right) \\ &\times \left[ {{\varGamma _\alpha }({\boldsymbol{u}})({F_{\text{s}}} + G) - \left( {{\varGamma _\alpha }({\boldsymbol{u}}) - {\varGamma _\alpha }(0)} \right)\nabla \psi (\rho )} \right]{\delta _{\text{t}}}{,} \end{split} _{ }$

(2)

其中α = 0, 1, 2, ⋅⋅⋅, b–1, b为离散速度方向个数; x和t分别表示粒子运动的位置和时间; u为流体速度; е_α为离散速度; F_s用来描述气液表面张力, ${F_{\text{s}}} = \kappa \rho \nabla {\nabla ^2}\rho$ , κ代表表面张力强度系数; 浮力 $G = ({\rho _{\text{g}}} - {\rho _{\text{l}}})g$ , 其中ρ_g和ρ_l分别代表气相流体和液相流体密度; $\psi (\rho ) = p - \rho c_{\text{s}}^{2}$ , 其中p为流体压力; τ为松弛时间, 与运动黏度ν的关系表示为 $\nu = (\tau - 0.5)c_{\text{s}}^{2}{\delta _{\text{t}}}$ , ${\delta _{\text{t}}}$ 代表时间步长, $c_{\text{s}}^{2} = {{{c^2}}/3}$ 是与格子速度 $c = {{{\delta _x}} \mathord{\left/ {\vphantom {{{\delta _x}} {{\delta _{\text{t}}}}}} \right. } {{\delta _{\text{t}}}}}$ 相关的模型常数, ${\delta _x}$ 代表空间步长. 本文采用Carnahan-Starling^[31]状态方程, 对应演化方程中的ψ( $\phi$ )为如下形式:

$\psi (\phi ) = {\phi ^2}RT\frac{{4 - 2\phi }}{{{{(1 - \phi )}^3}}} - a{\phi ^2},$

(3)

其中a决定分子间相互吸引力强度, $\phi$ 代表宏观量指标参数, R为气体体积常数, T为流体温度. 演化方程(1)中的函数Γ_α(u)为

${\varGamma _\alpha }({\boldsymbol{u}}) = {\omega _\alpha }\left[ {1 + \frac{{3{{\boldsymbol{e}}_\alpha } \cdot {\boldsymbol{u}}}}{{{c^2}}} + \frac{{9{{\left( {{{\boldsymbol{e}}_\alpha } \cdot {\boldsymbol{u}}} \right)}^2}}}{{2{c^4}}} - \frac{{3{{\boldsymbol{u}}^2}}}{{2{c^2}}}} \right],$

(4)

其中ω_α为权重系数. 演化方程()和方程()中 $f_\alpha ^{{\text{eq}}}$ 和 $g_\alpha ^{{\text{eq}}}$ 为 $f_\alpha ^{}({\boldsymbol{x}}, t)$ 和 $g_\alpha ^{}({\boldsymbol{x}}, t)$ 的平衡态分布函数, 其形式为

$f_\alpha ^{{\text{eq}}} = {\omega _\alpha }\phi \left[ {1 + \frac{{3{{\boldsymbol{e}}_\alpha } \cdot {\boldsymbol{u}}}}{{{c^2}}} + \frac{{9{{\left( {{{\boldsymbol{e}}_\alpha } \cdot {\boldsymbol{u}}} \right)}^2}}}{{2{c^4}}} - \frac{{3{{\boldsymbol{u}}^2}}}{{2{c^2}}}} \right],$

(5)

$g_\alpha ^{{\text{eq}}} = {\omega _\alpha }\left[ {p + \rho \left( {\frac{{3{{\boldsymbol{e}}_\alpha } \cdot {\boldsymbol{u}}}}{{{c^2}}} + \frac{{9{{\left( {{{\boldsymbol{e}}_\alpha } \cdot {\boldsymbol{u}}} \right)}^2}}}{{2{c^2}}} - \frac{{3{{\boldsymbol{u}}^2}}}{{2{c^2}}}} \right)} \right].$

(6)

宏观量指标参数 $\phi$ , 压力p和流体速度u可根据分布函数得到,

$\begin{split} &\phi = \sum {{f_\alpha }} , \\ & p = \sum {{g_\alpha }} - \frac{1}{2}{\boldsymbol{u}} \cdot \nabla \psi (\rho ){\delta _t}, \\ &\rho RT{\boldsymbol{u}} = \sum {{{\boldsymbol{e}}_\alpha }{g_\alpha } + \frac{{RT}}{2}} (\kappa \rho \nabla {\nabla ^2}\rho + G){\delta _t}. \end{split}$

(7)

流体密度ρ( $\phi$ )和运动黏度ν( $\phi$ )可由指标参数 $\phi$ 计算:

$\begin{split} &\rho (\phi ) = {\rho _{\text{g}}} + \frac{{\phi - {\phi _{\text{l}}}}}{{{\phi _{\text{h}}} - {\phi _{\text{l}}}}}({\rho _{\text{l}}} - {\rho _{\text{g}}}), \\ &\nu (\phi ) = {\nu _{\text{g}}} + \frac{{\phi - {\phi _{\text{l}}}}}{{{\phi _{\text{h}}} - {\phi _{\text{l}}}}}({\nu _{\text{l}}} - {\nu _{\text{g}}}), \end{split}$

(8)

其中ϕ_h和ϕ_l为指标参数的最大值和最小值. 通过Chapman-Enskog分析, 可得到方程(1)和方程(2)对应的宏观方程:

$\begin{split} &\frac{{\partial \phi }}{{\partial t}} + \nabla \cdot (\phi {\boldsymbol{u}}) = - \lambda \nabla \cdot \left[ {\frac{\phi }{\rho }\nabla p(\rho ) - \nabla p(\phi )} \right], \\ &\frac{1}{{\rho RT}}\frac{{\partial p}}{{\partial t}} + \nabla \cdot {\boldsymbol{u}} = 0, \\ &\rho \left[ {\frac{{\partial {\boldsymbol{u}}}}{{\partial t}} + ({\boldsymbol{u}} \cdot \nabla ){\boldsymbol{u}}} \right] = - \nabla p + \nabla \cdot \boldsymbol\varPi + \kappa \rho \nabla {\nabla ^2}\rho + G, \end{split}$

(9)

式中λ为迁移率, ${\boldsymbol{\varPi}}$ 为黏性应力张量, 表达式为 ${\boldsymbol{\varPi}} = \eta (\nabla u + u\nabla )$ . 这里, η为动力黏度:

$\eta = {\eta _0}{\gamma ^{n - 1}} = {\eta _0}{({{\boldsymbol{S}}_{\alpha \beta }}{{\boldsymbol{S}}_{\alpha \beta }})^{(n - 1)/2}},$

(10)

其中η₀为稠度系数; n是流体的流变指数, 当n > 1时, 流体为剪切变稠非牛顿流体, 其动力黏度随剪切速率的增大而增大; 当n = 1时, 为牛顿流体; 当n < 1时, 流体为剪切变稀非牛顿流体, 其动力黏度随剪切速率的增大而减小. (10)式中剪切速率γ可根据流体速度计算^[12]:

$\gamma = \sqrt {2\left[ {{{\left( {\frac{{\partial {{\boldsymbol{u}}_x}}}{{\partial {\boldsymbol{x}}}}} \right)}^2} + {{\left( {\frac{{\partial {{\boldsymbol{u}}_y}}}{{\partial {\boldsymbol{y}}}}} \right)}^2} + \frac{1}{2}{{\left( {\frac{{\partial {{\boldsymbol{u}}_x}}}{{\partial {\boldsymbol{y}}}} + \frac{{\partial {{\boldsymbol{u}}_y}}}{{\partial {\boldsymbol{x}}}}} \right)}^2}} \right]} .$

(11)

应变张量 ${{\boldsymbol{S}}_{\alpha \beta }} = \dfrac{1}{2}\left( {\dfrac{{\partial {{\boldsymbol{u}}_\alpha }}}{{\partial {{\boldsymbol{x}}_\beta }}} + \dfrac{{\partial {{\boldsymbol{u}}_\beta }}}{{\partial {{\boldsymbol{x}}_\alpha }}}} \right)$ 可由分布函数g_α得到^[30,32]:

${{\boldsymbol{S}}_{\alpha \beta }} = - \frac{{\displaystyle\sum\limits_{\alpha = 0}^8 {{{\boldsymbol{e}}_\alpha }{{\boldsymbol{e}}_\alpha }{g_\alpha } - p{\delta _{\alpha \beta }}} + G}}{{\tau {\delta _{\mathrm{t}}}RT\rho }}.$

(12)

本文使用D2 Q9模型来进行数值模拟研究, 权重系数ω_α分别为: 当α = 0时, ω_α = 4/9; 当α = 1—4时, ω_α = 1/9; 当α = 5—8时, ω_α = 1/36; 离散速度е_α表达式如下:

${{\boldsymbol{e}}}_{\alpha }=\left\{\begin{aligned} &(0,0),&&\alpha =0,\\ &\Bigr\{{\mathrm{cos}}\left[(\alpha -1)\text{π}/2\right],{\mathrm{sin}}\Bigr\}c,&&\alpha =1—4,\\ &\sqrt{2}\Bigr\{{\mathrm{cos}}\left[(\alpha -5)\text{π}/2+\text{π}/4\right],{\mathrm{sin}}\left[(\alpha -5)\text{π}/2+\text{π}/4\right]\Bigr\}c,&&\alpha =5—8.\end{aligned}\right.$

(13)

3. 物理问题及数值方法

3.1 物理问题描述

本工作研究的物理问题示意图如所示, 在长为 ${L_x}$ , 宽为 ${L_y}$ 的计算域内放置直径为d的气泡(如中黑色轮廓所示), 气泡中心距离底部壁面的距离为h, 气泡中心的坐标为( ${{{L_x}} \mathord{\left/ {\vphantom {{{L_x}} 2}} \right. } 2}$ , ${{{L_y}} \mathord{\left/ {\vphantom {{{L_y}} 5}} \right. } 5}$ ). 通道上下为周期边界, 左右为无滑移边界. 数值模拟中, 计算域长 ${L_x} = 200$ , 宽 ${L_y} = 500$ , 气泡初始半径 $d = 40$ , 气泡密度 ${\rho _{\text{g}}} = 0.1$ , 气泡周围剪切增稠非牛顿流体密度 ${\rho _{\text{l}}} = 0.5$ , 气液黏度比 ${\mu _{\text{r}}} = {{{\mu _{\text{g}}}} \mathord{\left/ {\vphantom {{{\mu _{\text{g}}}} {{\mu _{\text{l}}}}}} \right. } {{\mu _{\text{l}}}}} = 2.5$ , ${\delta _{\text{t}}}$ 与 ${\delta _x}$ 都为1, 模型参数κ取0.0056, a取4.0, RT赋值为 ${1 \mathord{\left/ {\vphantom {1 3}} \right. } 3}$ , ϕ_h和ϕ_l分别为0.250291和0.022838, 松弛时间 $\tau = 5.6$ .

$图 1 模拟问题示意图\r\nFig. 1. Schematic illustration of simulation problem.$

图 1 模拟问题示意图

Fig. 1. Schematic illustration of simulation problem.

下载: 全尺寸图片幻灯片

本文研究中, 主要考虑惯性力、表面张力以及黏度对气泡形状的影响, 其相对大小通过无量纲数Ga, Eo以及流变指数n表征, 其中 $Ga = {{{\rho _{\text{g}}}\sqrt {gR} \cdot R} \mathord{\left/ {\vphantom {{{\rho _{\text{g}}}\sqrt {gR} \cdot R} {{\mu _{\text{g}}}}}} \right. } {{\mu _{\text{g}}}}}$ 表征浮力与黏性力的相对大小, Ga越大表示气泡所受浮力相对于黏性力越大; $Eo = {{g({\rho _{\text{l}}} - {\rho _{\text{g}}}){d^2}} \mathord{\left/ {\vphantom {{g({\rho _{\text{l}}} - {\rho _{\text{g}}}){d^2}} \sigma }} \right. } \sigma }$ 描述为浮力和表面张力的比, Eo越小即表征表面张力的影响较浮力越突出. 本文主要考虑剪切增稠非牛顿流体, n分别取1.0, 1.2, 1.4, 1.6和1.8五种情况. 为便于与前人结果进行对比, 下文中对气泡速度和剪切速率进行归一化处理. 气泡速度(U)和剪切速率(γ)归一化因子分别选取为最大速度U_max和最大剪切速率γ_max.

3.2 模型验证

为验证模型的正确性, 将不同无量纲参数(Eo, Ga)下气泡稳定发展后的形变与Pang和Lu^[12]在研究剪切变薄非牛顿流体中气泡上升时的形变结果进行对比. 在其物理问题中, 气泡的初始直径为d, 计算域的长度和宽度分别为25d和40d, 气泡中心到底壁的距离为20d/3. 气液密度比 ${\rho _{\text{r}}} = {{{\rho _{\text{g}}}} \mathord{\left/ {\vphantom {{{\rho _{\text{g}}}} {{\rho _{\text{l}}}}}} \right. } {{\rho _{\text{l}}}}} = 0.001$ , 气液黏度比μ_r = 0.01, 计算域中上下壁面采用周期性边界条件, 左右壁面均为无滑移边界. 表1列出了所得到的结果, 可以看出本文得到的结果与前人结果吻合较好.

表 1 不同参数下得到的气泡终端形状　(a) Eo = 200; (b) Ga = 3.0

Table 1. Bubble terminal deformation at different values of Eo and Ga: (a) Eo = 200; (b) Ga = 3.0.

Case A	Eo = 200		Case B	Ga = 3.0
Case A	Ref. [12]	本文	Case B	Ref. [12]	本文
Ga = 2.1			Eo = 5
Ga = 2.6			Eo = 20
Ga = 3.0			Eo = 80

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4. 数值结果与分析

4.1 气泡变形

本节研究Ga, Eo以及n对气泡形变的影响. 表2为不同参数下气泡达到充分发展时的形状图, 从表中结果可知, 固定n, Ga不变, Eo增大过程中气泡偏离初始形状向球帽形过渡, 变形程度加剧. 这是因为表面张力越小界面越容易变形, Eo增大使得表面张力相对浮力减小, 降低了气泡维持初始圆形的稳定性. 从表2还可以看出, n对气泡形状的影响与Ga有关, 为体现两者关联作用, 固定Eo为30, 分别分析不同Ga下n对气泡形变程度的影响. 当Ga = 22, 32时, 随n的增大, 气泡的长径比增大, 变形程度为增大趋势; 随着Ga增大(Ga = 39, 45), 气泡形状反而随着n值的增大先呈左右两端裙边下垂的球帽状(n = 1, 1.2, 1.4时), 后呈变形截面曲线较平缓的球帽状(n = 1.6, 1.8时), 此时变形程度稍稍收敛. 这是由于较大Ga对应相对高的浮力水平, 气泡上升速度增大从而伴随较大的剪切速率, 致使流体表观黏度明显增大, 抑制气泡的形态变化.

表 2 气泡形状图

Table 2. Bubble shape map.

	Ga = 22					Ga = 32
	n = 1.0	n = 1.2	n = 1.4	n = 1.6	n = 1.8	n = 1.0	n = 1.2	n = 1.4	n = 1.6	n = 1.8
Eo = 5
Eo = 10
Eo = 20
Eo = 30
	Ga = 39					Ga = 45
	n = 1.0	n = 1.2	n = 1.4	n = 1.6	n = 1.8	n = 1.0	n = 1.2	n = 1.4	n = 1.6	n = 1.8
Eo = 5
Eo = 10
Eo = 20
Eo = 30

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4.2 气泡终端速度

本节分析Eo, Ga和n三个参数影响下气泡终端速度的变化规律. 图2展示了不同Eo下Ga和n对气泡终端速度的影响. 可以观察到, 对于4种不同Eo情况, 气泡终端速度随Ga的增大呈非线性单调增大的趋势. 另一方面, 还可以看出, n对终端速度的影响随着Ga增大也在逐渐增大. 以图2(b)中Ga = 22和Ga = 45为例, Ga = 22时, n从1.0增至1.8对应的气泡终端速度分别为0.51×10^–2, 0.57×10^–2, 0.58×10^–2, 0.59×10^–2, 0.60×10^–2, n取1.8对应的速度值相对n取1.0增值0.09×10^–2, 增幅为17.64%; 而当Ga = 45时, 随着n从1.0增至1.8, 气泡终端速度分别为2.02×10^–2, 2.34×10^–2, 2.64×10^–2, 2.79×10^–2, 2.93×10^–2, n = 1.8对应的速度值相对n = 1.0增值0.91×10^–2, 增幅为45.05%. 因此, Ga越大, 气泡终端速度随n的增大趋势越明显.

$图 2 不同Eo下Ga和流变指数n对气泡终端速度的影响　(a) Eo = 5; (b) Eo = 10; (c) Eo = 20; (d) Eo = 30\r\nFig. 2. Effects of Ga and power-law index n on bubble terminal velocity at different Eo numbers: (a) Eo = 5; (b) Eo = 10; (c) Eo = 20; (d) Eo = 30.$

图 2 不同Eo下Ga和流变指数n对气泡终端速度的影响　(a) Eo = 5; (b) Eo = 10; (c) Eo = 20; (d) Eo = 30

Fig. 2. Effects of Ga and power-law index n on bubble terminal velocity at different Eo numbers: (a) Eo = 5; (b) Eo = 10; (c) Eo = 20; (d) Eo = 30.

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为了进一步分析Eo, Ga和n三个参数对气泡终端速度的影响, 图3展示了4种Ga数值下气泡终端速度随Eo和n的变化趋势. 如图3(a)所示, 在Ga = 22条件下, Eo = 5和Eo = 10两种情况下当n从1.0增大到1.2时气泡终端速度有所增大, 而在1.2向1.8增大的流变指数范围内, 气泡终端速度随着n值的增大呈下降趋势. 这是因为此时Ga, Eo较小, 浮力对气泡运动特性的影响较弱, 而随着n的增大, 阻力耗散作用增强, 导致终端速度降低; Eo增大至20, 30, 速度曲线则随n的增大上升, 这是因为随着Eo的增大气泡所受浮力增大, 终端速度最终增大. 当Ga = 32时(图3(b)), 不同流变指数n和Eo下气泡终端速度的变化趋势与Ga = 22时相似, 当Eo = 5和10时气泡终端速度随着n的增大先增大再减小, 而当Eo = 20和30时, 气泡终端速度随着n的增大而增大. 相对于Ga = 22的情况, Ga = 32时得到的结果对应的增大和减小的趋势都逐渐变缓. 当Ga增大到39和45时, 分别如图3(c), (d)所示, 不同Eo对应的速度曲线随n值增大而增大的趋势较为统一, 即对于所有的Eo取值情况, 气泡终端速度随着n的增大而增大. 以图3(c)中Eo = 10和Eo = 30为例, 当Eo = 10时, 随着n从1.0增至1.8, 气泡终端速度分别为1.48×10^–2, 1.78×10^–2, 1.87×10^–2, 1.91×10^–2, 1.92×10^–2, 分别增大了20.27%, 5.06%, 2.14%, 0.52%; 而Eo = 30时, 气泡终端速度分别为1.49×10^–2, 1.73×10^–2, 1.85×10^–2, 1.91×10^–2, 1.94×10^–2, 增幅依次为16.11%, 6.94%, 3.24%, 1.57%.

$图 3 Eo和流变指数n对气泡终端速度的影响　(a) Ga = 22; (b) Ga = 32; (c) Ga = 39; (d) Ga = 45\r\nFig. 3. Effects of Eo and power-law index n on bubble terminal velocity: (a) Ga = 22; (b) Ga = 32; (c) Ga = 39; (d) Ga = 45.$

图 3 Eo和流变指数n对气泡终端速度的影响　(a) Ga = 22; (b) Ga = 32; (c) Ga = 39; (d) Ga = 45

Fig. 3. Effects of Eo and power-law index n on bubble terminal velocity: (a) Ga = 22; (b) Ga = 32; (c) Ga = 39; (d) Ga = 45.

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综上所述, 对于不同表观黏度的剪切增稠流体, 气泡终端速度在Ga和Eo都较小时随流变指数n的增大先增大后减小; 反之, 气泡终端速度随流变指数n的增大而增大. 由此可得, 气泡终端速度的增大或减小并不单一取决于浮力水平、液体黏度或表面张力, 而是以上3种因素的交互作用.

4.3 流场剪切速率

本节分析Eo, Ga和n对气泡周围剪切速率的影响. 图4展示了不同Eo (固定Ga = 22)以及不同Ga (固定Eo = 5)情况下, 气泡充分发展时剪切速率随流变指数n的变化趋势. 如图4所示, 对于不同Eo和Ga的情况, 剪切速率随n的增大而增大, 较大n值处(1.6, 1.8)剪切速率的增幅相对明显. 如图4(a)中, 当Eo = 30时, 随着n从1.0增至1.8, 剪切速率分别为6.85×10^–4, 7.90×10^–4, 8.44×10^–4, 1.29×10^–3, 2.25×10^–3, 其增幅分别为15.35%, 6.87%, 39.35%, 75.10%; 而在图4(b)中, 当Ga = 39时, n从1.0增至1.8的过程中, 剪切速率分别为1.43×10^–3, 1.71×10^–3, 1.99×10^–3, 3.46×10^–3, 6.21×10^–3, 增幅分别为19.58%, 16.60%, 73.53%, 79.51%. 从图4(b)还可以发现, Ga = 45的曲线在n较大(1.6, 1.8)时几乎与表示Ga = 39的曲线重合, 这是由于随着黏性耗散的增大, 浮力对剪切速率的影响变得不显著.

$图 4 剪切速率随Ga, Eo和n的变化趋势　(a) Ga = 22; (b) Eo = 5\r\nFig. 4. Effects of Ga, Eo and power-law index n on shear rate distribution: (a) Ga = 22; (b) Eo = 5.$

图 4 剪切速率随Ga, Eo和n的变化趋势　(a) Ga = 22; (b) Eo = 5

Fig. 4. Effects of Ga, Eo and power-law index n on shear rate distribution: (a) Ga = 22; (b) Eo = 5.

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表3展示了最大和最小Ga/Eo在不同流变指数n时得到的剪切速率分布图, 其中白色为气泡区域, 其余部分表示剪切增稠流体所在区域. 如表中结果所示, 气泡在向上运动时, 周围流体产生的速度梯度在气泡周围形成涡旋, 导致剪切速率增大且分布不均匀, 且气泡左右两侧剪切速率较高的区域尺寸随Eo, Ga的增大而增大, 但随n的增大先增大后缩小. 结合图4中气泡剪切速率曲线的单调递增趋势, 可知气泡周围剪切速率较大的区域尺寸并不一定与剪切速率最大值成正比. 如表3所示, 当n < 1.4时, 高剪切速率分布的范围随n的增大更为广泛; 而1.4 < n < 1.8时, 高剪切速率范围随n的增大略有缩减. 这是因为对于剪切增稠非牛顿流体, 流体黏度随剪切速率的增大而增大, 在较大的n值情况下, 剪切增稠流体的黏度耗散更大, 导致剪切速率分布范围的缩减.

表 3 气泡周围剪切速率分布

Table 3. Shear rate distribution around the bubble.

	Eo = 5		Eo = 30
	Ga = 22	Ga = 45	Ga = 22	Ga = 45
n = 1
n = 1.2
n = 1.4
n = 1.6
n = 1.8

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4.4 Ga, Eo及n对因变量的影响程度对比

为全面分析上文中Ga, Eo和n对剪切速率和终端速度的交互影响, 选取以上3个参数作为考察因素, 采用正交实验法量化并比较它们对气泡剪切速率和终端速度的影响程度.

表4是因素水平表, 包含n, Ga和Eo 3个因素, 因素n存在5个水平, 因素Ga, Eo水平均为4, 制作25次试验正交表并对试验结果进行处理, 如表5—表8所示, 其中表5、表7分别列出了不同试验因素对应的剪切速率和终端速度, 表6和表8分别为对表5和表7数据分析的结果, 其中k1—k5行数据代表各影响因素分别为①—⑤水平时剪切速率/终端速度的平均值, R行数据对应各因素不同水平下剪切速率/终端速度均值的极差. 数据极差越小, 则表示数据受对应因素变化越不显著, 即该因素影响程度越低; 反之, 影响程度越高.

表 4 因素水平表

Table 4. Table of factor levels.

水平	因素
水平	n	Ga	Eo
①	1.0	22	5
②	1.2	32	10
③	1.4	39	20
④	1.6	45	30
⑤	1.8	—	—

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表 5 针对剪切速率的试验正交表

Table 5. Orthogonal table of tests for shear rate.

试验次数	因素			剪切速率(×10^–6)
试验次数	n	Ga	Eo	剪切速率(×10^–6)
1	1.0	22	5	718
2	1.0	32	20	1152
3	1.0	39	30	1443
4	1.0	45	10	1743
5	1.0	45	30	1673
6	1.2	22	30	798
7	1.2	32	10	1325
8	1.2	39	30	1659
9	1.2	45	5	2078
10	1.2	45	20	2035
11	1.4	22	30	837
12	1.4	32	5	1625
13	1.4	39	20	1904
14	1.4	45	30	2033
15	1.4	45	10	1872
16	1.6	22	20	1464
17	1.6	32	30	1747
18	1.6	39	10	2532
19	1.6	45	30	2732
20	1.6	45	5	3707
21	1.8	22	10	3329
22	1.8	32	30	2903
23	1.8	39	5	6200
24	1.8	45	20	4016
25	1.8	45	30	3519

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表 8 针对终端速度的数据分析

Table 8. Data analysis for terminal speed.

试验均值/极差	因素
试验均值/极差	n	Ga	Eo
k1	1390.0	540.0	1707.0
k2	1608.8	1193.6	1623.6
k3	1783.8	1792.8	1735.2
k4	1851.4	2507.6	1737.9
k5	1907.6	—	—
R	517.6	1967.6	114.3

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表 7 针对终端速度的试验正交表

Table 7. Orthogonal table of tests for terminal speed.

试验次数	因素			终端速度(×10^–6)
试验次数	n	Ga	Eo	终端速度(×10^–6)
1	1.0	22	5	479
2	1.0	32	20	976
3	1.0	39	30	1487
4	1.0	45	10	2001
5	1.0	45	30	2007
6	1.2	22	30	575
7	1.2	32	10	1163
8	1.2	39	30	1734
9	1.2	45	5	2250
10	1.2	45	20	2322
11	1.4	22	30	614
12	1.4	32	5	1181
13	1.4	39	20	1889
14	1.4	45	30	2654
15	1.4	45	10	2581
16	1.6	22	20	565
17	1.6	32	30	1322
18	1.6	39	10	1906
19	1.6	45	30	2787
20	1.6	45	5	2677
21	1.8	22	10	467
22	1.8	32	30	1326
23	1.8	39	5	1948
24	1.8	45	20	2924
25	1.8	45	30	2873

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表 6 针对剪切速率的数据分析

Table 6. Data analysis for shear rate.

试验均值/极差	因素
试验均值/极差	n	Ga	Eo
k1	1345.8	1429.2	2865.6
k2	1579.0	1750.4	2160.2
k3	1654.2	2747.6	2114.2
k4	2436.4	2540.8	1934.4
k5	3993.4	—	—
R	2647.6	1318.4	931.2

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如表6所示, 不同水平的n, Ga和Eo对应剪切速率平均值的极差分别为2647.6, 1318.4, 931.2, 可得n对剪切速率的影响最大, Ga次之, Eo影响程度最小; 对于终端速度, 表8中n, Ga和Eo对应的水平极差值分别为517.6, 1967.6, 114.3, 由1967.6 > 517.6 > 114.3的排序, 可知Ga对剪切速率的影响最大, n次之, Eo影响程度最小.

5. 结　论

采用不可压非牛顿流体气液两相流格子Boltzmann模型研究了剪切增稠流体中单个气泡的上升行为, 分析了流变指数n, Eöυös数(Eo)和Galilei数(Ga)对气泡运动特性的影响规律, 得出以下结论:

1) 当n一定时, 气泡变形程度随Eo, Ga的增大而增大, 其中n对气泡变形的影响与Ga有关. 当Ga较小时, 随n的增大, 气泡变形程度为增大趋势; 另一方面随着Ga继续增大, 气泡形变程度随着n的增大先增大后减小.

2) 对于所研究的4种Eo的情况, 气泡终端速度随着Ga增大而增大, 且n对气泡终端速度的影响依赖于Ga, 表现为Ga较大时终端速度随n增大的趋势更明显. 另一方面, 对于给定Ga的情况, 当Ga取值较小时, 随着剪切增稠非牛顿流体黏性耗散作用的增强, Eo取值较小时的气泡终端速度先增大后减小, 而Eo取值较大时的气泡由于受到的浮力较大, 因而终端速度随n增大非线性增大; 对于Ga取值较大的情况, 不同Eo对应的速度曲线都随n值的增大较为统一地呈上升趋势.

3) 气泡在剪切增稠流体中上升时左右两端存在剪切速率较高的区域, 该区域尺寸随Eo, Ga的增大而增大; 另一方面, 由于较大n值下剪切增稠流体的黏度耗散增大, 高剪切速率分布的范围随n的增大先扩大后缩减.

4) 利用正交试验法分析了Ga, Eo, n三个参数对剪切速率和终端速度的耦合影响, 可知流变指数n对剪切速率的影响最大, Ga数次之, Eo数影响程度最小; 对于终端速度, 影响程度由大到小的因素依次为Ga, n, Eo.

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图 1 模拟问题示意图

Figure 1. Schematic illustration of simulation problem.

DownLoad: Full-Size Img PowerPoint

图 2 不同Eo下Ga和流变指数n对气泡终端速度的影响　(a) Eo = 5; (b) Eo = 10; (c) Eo = 20; (d) Eo = 30

Figure 2. Effects of Ga and power-law index n on bubble terminal velocity at different Eo numbers: (a) Eo = 5; (b) Eo = 10; (c) Eo = 20; (d) Eo = 30.

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图 3 Eo和流变指数n对气泡终端速度的影响　(a) Ga = 22; (b) Ga = 32; (c) Ga = 39; (d) Ga = 45

Figure 3. Effects of Eo and power-law index n on bubble terminal velocity: (a) Ga = 22; (b) Ga = 32; (c) Ga = 39; (d) Ga = 45.

DownLoad: Full-Size Img PowerPoint

图 4 剪切速率随Ga, Eo和n的变化趋势　(a) Ga = 22; (b) Eo = 5

Figure 4. Effects of Ga, Eo and power-law index n on shear rate distribution: (a) Ga = 22; (b) Eo = 5.

DownLoad: Full-Size Img PowerPoint

表 1 不同参数下得到的气泡终端形状　(a) Eo = 200; (b) Ga = 3.0

Table 1. Bubble terminal deformation at different values of Eo and Ga: (a) Eo = 200; (b) Ga = 3.0.

Case A	Eo = 200		Case B	Ga = 3.0
Case A	Ref. [12]	本文	Case B	Ref. [12]	本文
Ga = 2.1			Eo = 5
Ga = 2.6			Eo = 20
Ga = 3.0			Eo = 80

DownLoad: CSV

表 2 气泡形状图

Table 2. Bubble shape map.

	Ga = 22					Ga = 32
	n = 1.0	n = 1.2	n = 1.4	n = 1.6	n = 1.8	n = 1.0	n = 1.2	n = 1.4	n = 1.6	n = 1.8
Eo = 5
Eo = 10
Eo = 20
Eo = 30
	Ga = 39					Ga = 45
	n = 1.0	n = 1.2	n = 1.4	n = 1.6	n = 1.8	n = 1.0	n = 1.2	n = 1.4	n = 1.6	n = 1.8
Eo = 5
Eo = 10
Eo = 20
Eo = 30

DownLoad: CSV

表 3 气泡周围剪切速率分布

Table 3. Shear rate distribution around the bubble.

	Eo = 5		Eo = 30
	Ga = 22	Ga = 45	Ga = 22	Ga = 45
n = 1
n = 1.2
n = 1.4
n = 1.6
n = 1.8

DownLoad: CSV

表 4 因素水平表

Table 4. Table of factor levels.

水平	因素
水平	n	Ga	Eo
①	1.0	22	5
②	1.2	32	10
③	1.4	39	20
④	1.6	45	30
⑤	1.8	—	—

DownLoad: CSV

表 5 针对剪切速率的试验正交表

Table 5. Orthogonal table of tests for shear rate.

试验次数	因素			剪切速率(×10^–6)
试验次数	n	Ga	Eo	剪切速率(×10^–6)
1	1.0	22	5	718
2	1.0	32	20	1152
3	1.0	39	30	1443
4	1.0	45	10	1743
5	1.0	45	30	1673
6	1.2	22	30	798
7	1.2	32	10	1325
8	1.2	39	30	1659
9	1.2	45	5	2078
10	1.2	45	20	2035
11	1.4	22	30	837
12	1.4	32	5	1625
13	1.4	39	20	1904
14	1.4	45	30	2033
15	1.4	45	10	1872
16	1.6	22	20	1464
17	1.6	32	30	1747
18	1.6	39	10	2532
19	1.6	45	30	2732
20	1.6	45	5	3707
21	1.8	22	10	3329
22	1.8	32	30	2903
23	1.8	39	5	6200
24	1.8	45	20	4016
25	1.8	45	30	3519

DownLoad: CSV

表 8 针对终端速度的数据分析

Table 8. Data analysis for terminal speed.

试验均值/极差	因素
试验均值/极差	n	Ga	Eo
k1	1390.0	540.0	1707.0
k2	1608.8	1193.6	1623.6
k3	1783.8	1792.8	1735.2
k4	1851.4	2507.6	1737.9
k5	1907.6	—	—
R	517.6	1967.6	114.3

DownLoad: CSV

表 7 针对终端速度的试验正交表

Table 7. Orthogonal table of tests for terminal speed.

试验次数	因素			终端速度(×10^–6)
试验次数	n	Ga	Eo	终端速度(×10^–6)
1	1.0	22	5	479
2	1.0	32	20	976
3	1.0	39	30	1487
4	1.0	45	10	2001
5	1.0	45	30	2007
6	1.2	22	30	575
7	1.2	32	10	1163
8	1.2	39	30	1734
9	1.2	45	5	2250
10	1.2	45	20	2322
11	1.4	22	30	614
12	1.4	32	5	1181
13	1.4	39	20	1889
14	1.4	45	30	2654
15	1.4	45	10	2581
16	1.6	22	20	565
17	1.6	32	30	1322
18	1.6	39	10	1906
19	1.6	45	30	2787
20	1.6	45	5	2677
21	1.8	22	10	467
22	1.8	32	30	1326
23	1.8	39	5	1948
24	1.8	45	20	2924
25	1.8	45	30	2873

DownLoad: CSV

表 6 针对剪切速率的数据分析

Table 6. Data analysis for shear rate.

试验均值/极差	因素
试验均值/极差	n	Ga	Eo
k1	1345.8	1429.2	2865.6
k2	1579.0	1750.4	2160.2
k3	1654.2	2747.6	2114.2
k4	2436.4	2540.8	1934.4
k5	3993.4	—	—
R	2647.6	1318.4	931.2

DownLoad: CSV

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