SIMULATION OF THE COLLAPSE EVOLUTION OF ATTACHED CAVITATION BUBBLE AND STUDY ON THE MECHANISM OF WALL WETTABILITY BASED ON IMPROVED MULTI RELAXATION PSEUDO POTENTIAL MODEL
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摘要: 结合改进后的多松弛伪势模型, 使用格子Boltzmann方法对附壁空化泡溃灭演化过程进行了数值模拟及分析, 着重研究了壁面润湿性对附壁空化泡溃灭的影响规律. 首先, 对改进后的多松弛伪势模型开展了热力学一致性验证, 通过对比气液共存密度的数值解与解析解, 选取了满足热力学一致性需求的参数, 使得模型能够实现大密度比多相流动的数值计算; 然后, 对不同壁面润湿性条件下的附壁空化泡平衡接触角进行了数值模拟, 分析了壁面润湿性与附壁空化泡平衡接触角的关系, 分析表明: 随着壁面与流体相互作用强度的增大, 平衡接触角逐渐增大, 壁面润湿能力逐渐减弱. 进一步, 对附壁空化泡的溃灭问题进行建模, 处理了相应的边界格式, 完成了附壁空化泡溃灭过程的数值计算, 并分析了不同壁面润湿能力对附壁空化泡溃灭的影响规律; 当初始空化泡内外压差及形态相同时, 增强壁面疏水性缩短了空化泡溃灭的时间. 特别是, 壁面润湿性由亲水逐渐变为疏水时, 最大溃灭压力先降低后又逐渐升高, 而最大溃灭速度随着壁面疏水性的增强而持续升高. 研究结果揭示了壁面润湿性对附壁空化泡溃灭的影响规律, 为进一步抑制附壁空化泡溃灭的破坏效应及利用空化泡溃灭效应实现工程应用提供了理论指导.Abstract: Combined with the improved multi-relaxation pseudo-potential model, the lattice Boltzmann method was used to numerically simulate and analyze the evolution process of the collapse of attached cavitation bubble, with a focus on studying the influences of wall wettability on the collapse of attached cavitation bubble. First, a thermodynamic consistency verification was conducted on the improved multi-relaxation pseudo-potential model. By comparing the numerical and analytical solutions of gas-liquid coexistence density, the parameters that satisfy the thermodynamic consistency requirements were chosen to enable the model to achieve numerical computation of multiphase flows with a large density ratio. Then, the equilibrium contact angle of attached cavitation bubble was numerically simulated for different wall wettability, and the relationship between wall wettability and the equilibrium contact angle of attached cavitation bubbles was analyzed. The analysis show that the equilibrium contact angle gradually increases and the wall wetting ability gradually becomes weaken, as the interaction strength between the wall and the fluid increase. Further, the collapse process of attached cavitation bubble was modeled, and the corresponding boundary formats were processed to complete the numerical calculation of the collapse of attached cavitation bubble. The influence of different wall wetting conditions on the collapse of attached cavitation bubble was analyzed in detail. The analysis show that the enhancement of wall hydrophobicity shortens the collapse time when the initial pressure difference and morphology of the cavitation bubble are the same. In particular, as the wall wettability gradually changed from hydrophilic to hydrophobic, the maximum collapse pressure decreases first and then gradually increases, while the maximum collapse velocity continued to increase with the enhancement of the wall hydrophobicity. The results reveal the influence law of wall wettability on the collapse of attached cavitation bubble, providing theoretical guidance for further suppressing the destructive effects of the collapse of attached cavitation bubble and the engineering application of utilizing the cavitation bubble collapse effects.
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引 言
空化是发生在液体介质中的一种特有物理现象: 当液体的局部压力降低至饱和蒸汽压以下, 液体内部的气核急剧膨胀成为空化泡的现象称为空化[1]. 空化现象是海洋船舶工程、石油工程、水利水电工程和流体机械工程等众多领域的研究难题, 其形成于低压区的空化泡随着液体介质的流动进入高压区域后, 在压差的作用下发生溃灭, 空化泡溃灭瞬间产生的高速射流及高压现象将直接损伤周围的设备[2-6]. 相反地, 若合理利用空化泡溃灭时产生的能量亦可使空化现象的某些特征发挥出有益的效应, 如空化射流清洁技术、空化射流钻井技术等[7-8].
人们对空化现象的研究已超百年, 空泡动力学的研究在许多方面也取得了重大的进步. Rayleigh结合理想气体状态方程, 建立了描述自由单个球形空泡在压力场中半径演化的空泡动力学方程, 即经典的Rayleigh方程[9], 该方程为后续空泡动力学的深入研究奠定了基础. 此后, Plesset等[10-11]和Keller等[12]考虑了对空泡演化的不同影响因素, 提出了各自修正的空泡动力学模型. 对于近壁面空化泡溃灭的研究方法主要有实验研究和数值模拟研究. Philipp等[13]和Brujan等[14]通过实验方法对空化泡溃灭的动态过程进行了捕捉, 并对溃灭产生的射流速度、冲击压力进行了分析. 与此同时, 随着计算机技术的快速发展, 数值计算方法已成为研究空化泡溃灭问题的一种有效手段. 目前, 多相流领域内主流的数值计算方法有边界积分法(BIM)、流体体积法(VOF)、水平集法(LS)和格子Boltzmann方法(LBM)等[15]. 在众多数值方法中, LBM由于其物理过程清晰、并行性好、边界条件易处理等诸多优点[16-18], 在复杂多相流动的数值计算中显现出了明显优势. LBM体系中的伪势模型[19-20](Shan-Chen (SC)模型)引入了粒子间相互作用力, 能够自动追踪相界面运动, 被广泛应用于多相流动的研究. 对于空化泡溃灭的数值模拟而言, 关键难点在于能否保持溃灭过程中的数值稳定性. Li等[21-22]对伪势模型的单松弛外力格式和多松弛外力格式均做出了改进, 提出了一种热力学一致性可调的外力格式来提升两相流动模拟过程中的数值稳定性. 胡青晨等[23]使用改进的单松弛外力格式对近壁面空化泡溃灭流动进行了建模, 建模计算结果与实验结果吻合; 邹建锋等[24]使用改进的多松弛外力格式对近壁面空化泡溃灭流动进行了建模, 并探究了表面张力变化对溃灭产生的微射流及溃灭压力带来的影响; Liu等[25]和Yang等[26-27]使用改进的多松弛外力格式并耦合温度分布函数模拟了空化泡在不同的壁面边界下的溃灭演化, 对空化泡溃灭流动及传热过程进行了分析. Peng等[28]使用改进的多松弛外力格式对刚性壁面附近的单空化泡和双空化泡溃灭进行了研究.
壁面润湿性表征了液体在壁面上的铺展能力, 与壁面结构、壁面材料等有关, 是影响空化与空蚀过程的一个重要因素. 然而, 在空化泡溃灭流动研究中, 大多数的研究都基于壁面附近的空化泡, 仅将壁面当成简单的反弹边界处理, 没有考虑空化泡附于壁面的情况及壁面属性对空化泡溃灭的影响. 在本研究中, 采用能够模拟大密度比的改进多松弛格子Boltzmann方法对附壁空化泡的平衡接触角进行了分析, 并对附壁空化泡溃灭过程进行了模拟计算. 在此基础上, 进一步探究了在不同压力条件下壁面润湿性对附壁空化泡溃灭产生的影响规律. 为进一步抑制附壁空化泡溃灭的破坏效应及利用空化泡溃灭效应实现工程应用提供了理论指导.
1. 改进格式多松弛伪势模型的建立
LBM体系中流体运动被描述为离散的密度分布函数演化过程. 在密度比大、相界面变形严重的空化泡溃灭流动中, 具有多松弛时间的伪势模型具有较好的数值稳定性, 其分布函数演化方程为[29-30]
$$ \begin{split} & f_\alpha ^{}({\boldsymbol{x}} + {\boldsymbol{e}}_\alpha ^{}\Delta t,t + \Delta t) = f_\alpha ^{}({\boldsymbol{x}},t) -\\ &\qquad ({\boldsymbol{M}}_{}^{{{ - 1}}}{\boldsymbol{\varLambda M)}}_{\alpha \beta }^{}(f_\beta ^{} - f_\beta ^{eq}) + \Delta tF'_\alpha \end{split} $$ (1) 其中, $ {f_\alpha } $为沿$\alpha $方向的密度分布函数; $t$为对应时刻; $ \Delta t $为时间步长; $ {\boldsymbol{e}}_\alpha ^{} $为沿$\alpha $方向的离散速度; $ {\boldsymbol{M}} $为正交变换矩阵, $ {\boldsymbol{M}}_{}^{ - 1} $为其逆矩阵; $ {\boldsymbol{\varLambda }} $为对角矩阵; $ F'_\alpha $为速度空间中的外力项; $ f_{}^{eq} $为平衡密度分布函数. 对于本研究采用的D2Q9模型, 平衡密度分布函数$ f_{}^{eq} $为[31]
$$ f_{}^{eq} = \rho w_\alpha ^{}\left[ {1 + 3\frac{{{\boldsymbol{e}}_\alpha ^{} \cdot {\boldsymbol{\nu }}}}{{c_{}^2}} + \frac{9}{2}\frac{{({\boldsymbol{e}}_\alpha ^{} \cdot {{{\boldsymbol{\nu}} )}}_{}^2}}{{c_{}^4}} - \frac{3}{2}\frac{{{\boldsymbol{\nu }}_{}^2}}{{c_{}^2}}} \right] $$ (2) 离散速度$ {\boldsymbol{e}}_\alpha ^{} $的取值为
$$ {\boldsymbol{e}}_\alpha ^{} = \left\{ \begin{gathered} (0,0){,^{}}\begin{array}{*{20}{c}} {} \end{array}\alpha = 0 \\ ( \pm 1,0)c,(0, \pm 1)c,\begin{array}{*{20}{c}} {} \end{array}\alpha = 1,2,3,4 \\ ( \pm 1, \pm 1)c,\begin{array}{*{20}{c}} {} \end{array}\alpha = 5,6,7,8 \\ \end{gathered} \right. $$ (3) 权系数$ w_\alpha ^{} $的取值为
$$ w_\alpha ^{} = \left\{ \begin{gathered} {4 \mathord{\left/ {\vphantom {4 9}} \right. } 9},\begin{array}{*{20}{c}} {} \end{array}{\left| {{\boldsymbol{e}}_\alpha ^{}} \right|^2} = 0 \\ {1 \mathord{\left/ {\vphantom {1 9}} \right. } 9},\begin{array}{*{20}{c}} {} \end{array}{\left| {{{\boldsymbol{e}}_\alpha }} \right|^2} = {c^2} \\ {1 \mathord{\left/ {\vphantom {1 {36,}}} \right. } {36,}}\begin{array}{*{20}{c}} {} \end{array}{\left| {{{\boldsymbol{e}}_\alpha }} \right|^2} = 2{c^2} \\ \end{gathered} \right. $$ (4) 其中, $c = \Delta x/\Delta t$为粒子迁移速率; $ \Delta x和\Delta t $分别为网格步长和时间步长, 取$\Delta x = \Delta t = 1$.
在D2Q9模型中, 式(1)中对角矩阵表示为
$$ {{\boldsymbol{\varLambda}} } = {\rm{diag}}\left( {\tau _\rho ^{ - 1},\tau _e^{ - 1},\tau _\zeta ^{ - 1},\tau _j^{ - 1},\tau _q^{ - 1},\tau _j^{ - 1},\tau _q^{ - 1},\tau _\upsilon ^{ - 1},\tau _\upsilon ^{ - 1}} \right) $$ (5) 在正交变换矩阵$ {\boldsymbol{M}} $作用下, 通过$ {\boldsymbol{m}} = {\boldsymbol{M}}f $和$ {\boldsymbol{m}}_{}^{eq} = {\boldsymbol{M}}f_{}^{eq} $将分布函数$ {f_{}} $及其平衡分布函数$ f_{}^{eq} $投影到矩空间中, $ {\boldsymbol{m}}_{}^{eq} $计算式为
$$ {\boldsymbol{m}}_{}^{eq} = \rho \left\{ \begin{gathered} 1 \\ - 2 + 3\left| {\boldsymbol{\nu }} \right|_{}^2 \\ 1 - 3\left| {\boldsymbol{\nu }} \right|_{}^2 \\ v_x^{} \\ - v_x^{} \\ v_y^{} \\ - v_y^{} \\ v_x^2 - v_y^2 \\ v_x^{}v_y^{} \\ \end{gathered} \right\} $$ (6) 式中, $ \rho $为宏观密度; $ {\boldsymbol{\nu }} $为流体宏观速度, $ v_x^{} $和$ v_y^{} $分别为宏观速度在$ x $和$ y $方向上的速度分量. 引入正交变换矩阵后, 式(1)的碰撞过程表示为
$$ {\boldsymbol{m}}_{}^* = {\boldsymbol{m}} - {\boldsymbol{\varLambda (m}} - {\boldsymbol{m}}_{}^{eq}{\boldsymbol{)}} + \Delta t{\boldsymbol{(I}} - {\boldsymbol{\varLambda }}/2{\boldsymbol{)S}} $$ (7) 式中, $ {\boldsymbol{I}} $为单位矩阵; $ {\boldsymbol{S}} $为矩空间的外力项.
式(1)的迁移过程可表示为
$$\qquad\qquad f_\alpha ^{}({\boldsymbol{x}} + {{\boldsymbol{e}}_\alpha }\Delta t,t + \Delta t) = f_\alpha ^*({\boldsymbol{x}},t) $$ (8) $$\qquad\qquad f_{}^* = {\boldsymbol{M}}_{}^{ - 1}{\boldsymbol{m}}_{}^* $$ (9) 式(6)中的流体宏观密度和宏观速度计算式为
$$ \boldsymbol{ }\rho = {\displaystyle \sum _{\alpha }{f}_{\alpha }^{}}{, }\quad \boldsymbol{v} = \left({\displaystyle \sum _{\alpha }{\boldsymbol{e}}_{\alpha }^{}}{f}_{\alpha }^{} + \frac{\Delta t}{2}\boldsymbol{F}\right)\Bigg/\rho $$ (10) 不考虑重力作用, $ {\boldsymbol{F}} = {\boldsymbol{F}}_m^{} + {\boldsymbol{F}}_{ad}^{} $为粒子所受到的总作用力, $ {\boldsymbol{F}}_m^{} $表示流体间相互作用力, $ {\boldsymbol{F}}_{ad}^{} $表示流体与壁面间的相互作用力. 流体间相互作用力$ {\boldsymbol{F}}_m^{} $计算式为[32-33]
$$ {\boldsymbol{F}}_m^{} = - G\varphi (x)\sum\limits_\alpha {w\left(\left| {{\boldsymbol{e}}_\alpha ^{}} \right|_{}^2\right)} \varphi (x + {\boldsymbol{e}}_\alpha ^{}){\boldsymbol{e}}_\alpha ^{} $$ (11) 式中, $ G $为相互作用强度, 取$ G = - 1 $; $ \varphi (x) $为流体间相互作用势; $ w $为权重系数, $ w(1) $ = 1/3, $ w(2) $ = 1/12; 基于非理想状态方程(EOS)的伪势[34]由下式计算
$$ \varphi (x) = \sqrt {2(p_{{\mathrm{EOS}}}^{} - \rho c_s^2)/Gc_{}^2} $$ (12) 式中, $ p_{{\mathrm{EOS}}}^{} $由非理想气体状态方程获得, 采用Carnahan-Starling (C-S)状态方程, 其表达式为
$$ p_{{\mathrm{EOS}}}^{} = \frac{{\rho RT\left[ {1 + b\rho /4 + {{\left( {b\rho /4} \right)}^2} - {{\left( {b\rho /4} \right)}^3}} \right]}}{{{{\left( {1 - b\rho /4} \right)}^3}}} - a{\rho ^2} $$ (13) 式中, $ R $ = 1气体常数; $ a = 0.496\;3(RT_c^{})_{}^2/p_c^{} $ = 0.5; $ b = 0.187\;3 (RT_c^{})/ p_c^{} $ = 4; $ T_c^{} $和$ p_c^{} $分别为临界温度和临界压力, $ T_c^{} = 0.047\;2 $, $ p_c^{} = 2.2 \times 10_{}^{ - 3} $.
流体与壁面间的相互作用力$ {\boldsymbol{F}}_{ad}^{} $计算式为[35]
$$ {\boldsymbol{F}}_{ad}^{} = - G_w^{}\varphi (x)\sum\limits_\alpha {\omega _\alpha ^{}} S(x + {\boldsymbol{e}}_\alpha ^{}){\boldsymbol{e}}_\alpha ^{} $$ (14) 式中, $ G_w^{} $为流体与壁面间的相互作用强度; $ S(x + {\boldsymbol{e}}_\alpha ^{}) = \varphi (x)s(x + {\boldsymbol{e}}_\alpha ^{}) $, 开关函数$ s(x + {\boldsymbol{e}}_\alpha ^{}) $对固相和液相分别为1和0.
采用改进格式的多松弛伪势模型, 则式(7)中矩空间的外力项$ {\boldsymbol{S}} $表示为
$$ {\boldsymbol{S}}_{}^{} = \left\{ \begin{gathered} 0 \\ 6(v_x^{}F_x^{} + v_y^{}F_y^{}) + \frac{{12\sigma \left| {{\boldsymbol{F}}_m^{}} \right|_{}^2}}{{\varphi _{}^2\Delta t(\tau _e^{} - 0.5)}} \\ - 6(v_x^{}F_x^{} + v_y^{}F_y^{}) - \frac{{12\sigma \left| {{\boldsymbol{F}}_m^{}} \right|_{}^2}}{{\varphi _{}^2\Delta t(\tau _\zeta ^{} - 0.5)}} \\ F_x^{} \\ - F_x^{} \\ F_y^{} \\ - F_y^{} \\ 2(v_x^{}F_x^{} - v_y^{}F_y^{}) \\ v_x^{}F_y^{} + v_y^{}F_x^{} \\ \end{gathered} \right\} $$ (15) 其中, $ F_x^{} $和$ F_y^{} $分别为总作用力$ {\boldsymbol{F}} $在$ x $和$ y $方向上的分量; $ \sigma $通过调整力学稳定性条件来调节伪势模型的热力学一致性.
2. 伪势模型的热力学一致性验证
在伪势模型中, 当力学稳定性条件与Maxwell等面积法则保持一致, 并且状态方程与热力学中的状态方程也取得一致时, 称为伪势模型的热力学一致性. 保证伪势模型的热力学一致性, 有利于提升气液两相流模拟的密度比, 能够提高计算过程的稳定性.
定义无量纲温度$ k = T/T_{{c}}^{} $, 通过改变$ k $的值, 使用改进的多松弛伪势模型对系统稳定时的气、液相共存密度分别进行求解, 将得到的共存密度数值解与解析解进行对比来完成热力学一致性验证. 若无特殊说明, 本文单位均采用格子单位, 包括长度单位(lu)、质量单位(mu)、时间单位(ts)、速度单位(lu·ts−1)、密度单位(mu·lu−3)、压力单位(mu·lu−1·ts−2). 松弛系数选择为: $ \tau _\rho ^{ - 1} = 1.0 $, $ \tau _e^{ - 1} = 1.1 $, $ \tau _\zeta ^{ - 1} = 1.1 $, $ \tau _j^{ - 1} = 1.0 $, $ \tau _q^{ - 1} = 1.1 $, $ \tau _\upsilon ^{ - 1} = 1.25 $.
在200 lu × 200 lu的计算域内对二维静态空化泡进行模拟, 流体域四周均采用周期性边界, 空化泡圆心位置$ (x_0^{},y_0^{}) = (100,100) $. 密度场根据式(16)进行初始化[36], 初始设置相界面宽度$ W = 5 $lu, 空化泡半径$ r_0^{} = 30 $lu, 如图1所示.
$$ \begin{split} & \rho (x,y) = \frac{{\rho _l^{} + \rho _g^{}}}{2} + \frac{{\rho _l^{} - \rho _g^{}}}{2} \cdot \\ &\qquad \tanh \left\{ {\frac{{2\left[ {\sqrt {(x - x_0^{})_{}^2 + (y - y_0^{})_{}^2} - r_0^{}} \right]}}{W}} \right\}\end{split}$$ (16) 式中, $ \rho _l^{} $和$ \rho _g^{} $分别表示液相及气相密度; $ \tanh (x) $函数为双曲正切函数, 其函数表达式为$ \tanh (x) = ({\mathrm{e}}_{}^{2 x} - 1)/({\mathrm{e}}_{}^{2 x} + 1) $.
图2给出的是在不同参量$ \sigma $下求得的共存密度数值解与Maxwell方程解析解的密度曲线图. 图中显示, 当$ \sigma = 0.11 $和$ 0.115 $时, 计算模型的液相求解值与解析值耦合度较高, 但气相求解值与解析值存在一定差异, 且差值随着两相密度比的增大而增大. 当$ \sigma = 0.12 $时, 气相及液相的求解值与解析值基本保持一致, 能够近似满足于热力学一致性需求. 因此, 为了使计算模型尽可能满足热力学一致性需求的同时实现较大气液密度比的计算, 本研究的参量$ \sigma $选择为0.12.
3. 壁面润湿性条件对平衡接触角的影响规律
不同的壁面润湿条件对附壁空化泡的形态、大小存在很大影响, 而不同形态下的附壁空化泡溃灭过程存在差异. 因此对表征润湿能力的平衡接触角进行探讨十分重要. 平衡接触角$ \theta _w^{} $的定义为在气、液、固三相交点处所作的气液界面的切线与壁面的夹角. 当$ \theta _w^{} < 90^\circ $时为润湿性壁面, 即亲水壁面; 当$ \theta _w^{} > 90^\circ $时为非润湿性壁面, 即疏水壁面, 如图3所示.
在300 lu × 100 lu的计算域内对二维静态空化泡进行模拟, 流体域上下边界采用标准反弹格式来实现无滑移边界条件, 左右边界采用周期边界格式, 空化泡圆心位置$ (x_0^{},y_0^{}) = (150,100) $, 按照式(16)进行密度初始化, 设置相界面宽度$ W = 5 $lu, 空化泡半径$ r_0^{} = 30 $lu. 原始伪势模型不能求解密度比大于10的两相流动, 但对于大多数实际两相流体而言, 气液系统的密度比通常超过100[37]. 设置初始两相密度比$ \rho _l^{}/\rho _g^{} = 135 $, 无量纲温度$ k = 0.6 $, 空化泡内外压力设置为对应温度下的平衡压力, 初始时刻接触角为90°, 如图4所示.
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图 4 初始时刻附壁空化泡密度分布Figure 4. Initialized density distribution of attached cavitation bubble通过设置不同的$ G_w^{} $来调整流体与壁面间的相互作用强度, 即调整壁面润湿性条件, 气液两相流动达到平衡后得到在各$ G_w^{} $下的平衡接触角如图5所示.
当$ G_w^{} = - 3.0 $和$ - 3.2 $时, $ \theta _w^{} $ > $ 90^\circ $, 表征非润湿性壁面; 当$ - 3.8 < G_w^{} < - 3.4 $时, $ \theta _w^{} $ < $ 90^\circ $, 表征润湿性壁面; 当$ G_w^{} < - 3.8 $时, $ \theta _w^{} = 0 $, 空化泡脱离壁面. 图6展示了平衡接触角$ \theta _w^{} $随体与壁面间的相互作用强度$ G_w^{} $的变化图, 在$ - 3.8 < G_w^{} < - 3.0 $范围内, $ \theta _w^{} $与$ G_w^{} $呈线性相关, 随着$ G_w^{} $的增大, $ \theta _w^{} $逐渐增大, 壁面润湿能力逐渐变弱.
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图 6 $ \theta _w^{} $随$ G_w^{} $的变化Figure 6. Variation of $ \theta _w^{} $with$ G_w^{} $4. 附壁空化泡溃灭过程及影响规律研究
4.1 附壁空化泡溃灭过程的模拟
本节主要针对附壁空化泡的溃灭过程进行模拟计算, 设置计算域为500 lu × 500 lu, 上边界采用反弹边界格式, 左右边界采用周期边界格式, 下边界采用Guo等[38]提出的非平衡外推压力边界, 计算边界及计算域如图7所示. $ P_v^{} $与$ P_\infty ^{} $分别表示空化泡内压力和环境压力.
按照式(16)进行密度初始化, 设置相界面宽度$ W = 5 $lu, 空化泡半径$ r_0^{} = 70 $lu, 空化泡圆心位置$ (x_0^{},y_0^{}) = (250,450) $, 无量纲温度$ k = 0.6 $, $ G_w^{} = - 3.8 $, 空化泡内压力$ P_g^{} = 7.94 \times 10_{}^{ - 5} $mu·lu−1·ts−2, 空化泡外压力$ P_l^{} = 0.01 $mu·lu−1·ts−2, 初始两相密度比$ \rho _l^{}/\rho _g^{} = 150 $, 初始时刻状态如图8所示.
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图 8 初始时刻附壁空化泡密度分布Figure 8. Initialized density distribution of attached cavitation bubble模拟得到的附壁空化泡溃灭密度场演化过程如图9(a)所示. 为了保证空化泡密度演化图的分辨率, 此处只显示了空化泡附近的部分流动区域. 在Zwaan等[39]的实验中, 壁面设在底部, 与本次模拟计算的设置相反, 为了更清楚地对比实验结果和模拟结果, 将提取的模拟计算结果旋转180°.
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图 9 空化泡溃灭模拟结果与实验结果的对比Figure 9. Comparison between simulation results and experimental results of cavitation bubble collapse由于空化泡内外压差大于表面张力, 空化泡受压差作用开始向内收缩. 同时, 在流体与壁面的相互作用力下, 液相区域逐渐被壁面排开, 空化泡逐渐沿着壁面向两侧外扩. 由于空化泡各方向压差不对称, 造成收缩速率不同, 空化泡底部出现持续高压, 空化泡底部朝着壁面方向逐渐向内凹陷, 在t = 779 ts时, 底部的泡壁与顶部的壁面撞到一起, 空化泡从中间断裂溃灭, 溃灭后的空化泡在壁面中心处释放能量, 能量传播至溃灭点周围的液相区域, 形成了局部高密度区. 观察演化图像可得, 使用改进伪势模型模拟的空化泡形态演化过程与实验结果基本一致, 验证了模型的有效性.
图10给出了空化泡溃灭后不同时刻的壁面压力曲线, 根据壁面压力曲线可以看出, 溃灭过程中的最大溃灭压力达0.18 mu·lu−1·ts−2, 产生的高压集中在壁面中心处, 随着时间的增加, 溃灭点处的压力波向壁面两侧传播, 并且压力峰值逐渐衰减, 衰减率逐渐减小. t = 787 ts时, 在溃灭发生初期, 壁面中心处的压力衰减较快, 且压力波向两侧传播的速度较大, 在两种因素的作用下, 压力峰值曲线在溃灭点两侧存在较为明显的突降点.
4.2 壁面湿润性附壁空化泡溃灭过程的影响机制
由第3节可知, 不同壁面润湿性的条件下对应着不同的平衡接触角, 其对空化泡溃灭演化产生的影响在本节展开讨论. 选取空化泡内外初始压差$ \Delta P_{}^{} = 0.008 $, $ 0.009 $和$ 0.01 $ mu·lu−1·ts−2共3种压力, 无量纲温度$ k = 0.6 $, 初始空化泡半径$ r_0^{} = 70 $lu, 圆心位置$ (x_0^{},y_0^{}) = (250,450) $, 探究不同壁面润湿性对空化泡溃灭过程中溃灭形态、溃灭时间、最大速度和最大溃灭压力的影响.
通过调整流体与壁面间的相互作用强度$ G_w^{} $来调整壁面的润湿性, 模拟得到在3种不同空化泡内外初始压差下, 不同壁面润湿性条件下附壁空化泡的溃灭形态如图11所示. 在不同壁面润湿性的条件下, 空化泡溃灭的形式保持一致, 均从底部凹陷并于中间断裂, 形成溃灭. 附壁空化泡的三相接触点在压差及流固间非平衡作用力下沿着壁面移动, 对于疏水壁面, 流固作用力使气相在壁面横向铺展, 与压差作用方向相反, 且该作用大于压差作用, 使气相与壁面的接触面积增大; 对于亲水壁面, 流固作用力使气相横向收缩, 且与压差作用方向相同, 使气相与壁面的接触面积减小.
图12展示了在不同压力条件下, 不同壁面润湿性条件下附壁空化泡的溃灭时间. 在$ \Delta P_{}^{} = 0.008 $, $ 0.009 $和$ 0.01 $ mu·lu−1·ts−2 三种压力条件下, 空化泡的平均溃灭时间为902, 826和769 ts, 当空化泡内外压差增加时, 空化泡界面的收缩速率加快, 从而缩短了空化泡达到溃灭状态的时间. 当空化泡内外压差保持一致时, 空泡溃灭时间随着流体与壁面间的相互作用强度$ G_w^{} $的增加而减小, 当壁面的疏水性增强时, 空化泡在壁面的铺展能力增强, 横向拉伸速率增大, 加速了纵向凹陷进程, 从而缩短了泡壁与顶部壁面撞击的时间. 由此可以说明, 提升壁面的疏水性缩短了空化泡溃灭的时间.
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图 12 不同压力条件下空化泡溃灭时间随壁面润湿性的变化Figure 12. Variation of cavitation bubble collapse time with wall wettability under different pressure不同压力条件下空化泡溃灭最大压力和最大速度随壁面润湿性的变化规律如图13和图14所示. 在$ \Delta P = 0.008 $, $ 0.009 $和$ 0.01 $ mu·lu−1·ts−2三种压力条件下, 空化泡的最大溃灭压力均值分别为0.087, 0.129和0.168 mu·lu−1·ts−2, 最大溃灭速度均值分别为0.35, 0.40和0.44 lu·ts−1. 随着空化泡内外压差的增加, 空化泡收缩速率增加, 溃灭时相界面形变更大, 能量积蓄更为集中, 从而使空化泡溃灭的最大压力和最大速度增加, 溃灭强度提升. 当空化泡内外压差保持一致, 壁面润湿性由亲水朝着疏水方向变化时, 空化泡溃灭的最大压力呈现先降低后升高的趋势, 在强疏水性和强亲水性壁面的最大溃灭压力较高, 中性壁面处较低, 强疏水性和强亲水性壁面的空化泡在流固作用力下相界面变形程度较中性壁面更高, 从而使空化泡溃灭的最大压力提升; 空化泡溃灭的最大速度随着壁面疏水性的增强逐渐增加, 随着壁面疏水性的增加, 空化泡中部形成凹陷通道逐渐变窄, 导致微射流更为集中, 空化泡的最大溃灭速度随之增大.
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图 13 不同压力条件下空化泡溃灭最大压力随壁面润湿性的变化Figure 13. Variation of the maximum collapse pressure with wall wettability under different pressure![]()
图 14 不同压力条件下空化泡溃灭最大速度随壁面润湿性的变化Figure 14. Variation of the maximum collapse velocity with wall wettability under different pressure5. 结 论
基于格子Boltzmann方法, 使用改进的多松弛伪势模型, 对附壁空化泡的平衡接触角进行了分析, 并对附壁空化泡的溃灭过程进行了模拟. 进一步, 研究了在不同压力条件下壁面润湿性对空化泡溃灭过程的影响规律, 结论如下:
(1)附壁空化泡的平衡接触角$ \theta _w^{} $与流体与壁面间的相互作用强度$ G_w^{} $呈线性相关, 随着$ G_w^{} $的增大, $ \theta _w^{} $逐渐增大, 壁面的润湿能力逐渐变弱;
(2)不同壁面润湿性条件下, 空化泡溃灭形式保持一致, 均从底部凹陷并于中间断裂, 形成溃灭. 随着空化泡内外压差的增加, 空化泡溃灭时间将缩短, 最大溃灭压力和最大溃灭速度将增加, 提升了溃灭强度. 空化泡内外压差不变, 壁面润湿性由亲水逐渐变为疏水时, 空化泡溃灭的最大压力先降低后升高, 而空化泡溃灭的最大速度逐渐变大.
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图 4 初始时刻附壁空化泡密度分布
Figure 4. Initialized density distribution of attached cavitation bubble
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图 6 $ \theta _w^{} $随$ G_w^{} $的变化
Figure 6. Variation of $ \theta _w^{} $with$ G_w^{} $
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图 8 初始时刻附壁空化泡密度分布
Figure 8. Initialized density distribution of attached cavitation bubble
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图 9 空化泡溃灭模拟结果与实验结果的对比
Figure 9. Comparison between simulation results and experimental results of cavitation bubble collapse
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图 12 不同压力条件下空化泡溃灭时间随壁面润湿性的变化
Figure 12. Variation of cavitation bubble collapse time with wall wettability under different pressure
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图 13 不同压力条件下空化泡溃灭最大压力随壁面润湿性的变化
Figure 13. Variation of the maximum collapse pressure with wall wettability under different pressure
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