STUDY ON THE EVOLUTION CHARACTERISTICS OF SOLITARY WAVES AND INTERNAL VORTICES IN LIQUID FILM FLOW ON INCLINED WALLS
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摘要: 研究了在重力作用下, 二维不可压黏性流体液膜沿倾斜壁面流动时, 其上孤立波及其内部涡的演化. 采用小参数摄动法与行波变换法, 首先推导出了非平整倾斜基底上液膜厚度的零阶和一阶的一般演化方程, 然后对该方程进行化简并采用Mathematica进行数值求解. 分析结果表明: 孤立波波形图中, 波前出现了一个毛细波, 而毛细波波谷处出现了完全开式涡; 通过对流量分析, 发现其与孤立波波形具有相同的变化趋势, 并且与波速呈正相关, 对于双峰与三峰孤立波, 前一波峰的流量比靠后的大; 随着波速增加到超过某一临界值时, 孤立波波峰内将出现涡流, 经过计算该临界波速与倾斜角呈正比关系, 对于双峰与三峰孤立波, 当波速继续增大, 靠后的波峰内也将出现涡流; 通过分析自由表面的速度分布得出: 该涡流的产生是自由表面的垂直速度在波前和波尾的速度梯度与大于波速的水平速度共同作用的结果, 波峰多更容易产生涡流; 通过分析在动坐标系下得到的迹线图, 发现该涡流面积也正比于波速且旋向为顺时针, 综合推断得出: 旋涡是在孤立波表面处开始形成的, 波峰内的流体沿壁面呈滚落状向下运动.Abstract: This paper investigates the evolution of solitary waves and internal vortices in a two-dimensional incompressible viscous liquid film flowing down an inclined wall under the influence of gravity. Using the method of small parameter perturbation and the traveling wave transformation, we first derive the general zeroth-order and first-order evolution equations for the film thickness on a non-smooth inclined substrate. These equations are then simplified and numerically solved using Mathematica. The analysis results show that in the waveform of the solitary wave, a capillary wave appears at the wavefront, and a fully open vortex forms at the trough of the capillary wave. Flow rate analysis indicates that it follows the same variation trend as the solitary wave profile and is positively correlated with the wave speed. For double-peak and triple-peak solitary waves, the flow rate of the preceding peak is greater than that of the subsequent peaks. When the wave speed increases beyond a certain critical value, vortices appear within the solitary wave peaks. Calculations show that this critical wave speed is directly proportional to the inclination angle. For double-peak and triple-peak solitary waves, further increase in wave speed results in vortices forming within the subsequent peaks as well. By analyzing the velocity distribution on the free surface, it is concluded that the generation of vortices is due to the combined effect of the velocity gradient of the vertical velocity at the wavefront and wave tail and the horizontal velocity exceeding the wave speed, with more wave peaks facilitating vortex formation. Analysis of the streaklines obtained in the moving coordinate system reveals that the vortex area is also proportional to the wave speed and rotates clockwise. Overall, it is inferred that the vortex starts to form at the surface of the solitary wave, with the fluid within the wave peaks rolling down along the wall.
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Keywords:
- solitary wave /
- liquid film /
- vortex /
- wave speed
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引 言
液膜流凭借其结构简单、传热传质系数高等优点, 在能源和化工等许多领域都有重要应用[1-3]. 自罗素发现孤立波以来, 到如今其发展已相当繁荣[4], 而液膜中的孤立波与涡对流动的演化与稳定性及对分析湍流形成具有重要意义, 因此受到许多学者们的关注与研究[5-8].
20世纪中期, Kapitza父子[9]的开创性实验, 最早在沿倾斜壁面下降的液膜中观察到了孤立波, 吸引了后来的许多学者[10]投入相关的研究中来. 液膜流动受到多种因素的影响, 如壁面形状、流速、表面张力、雷诺数、倾斜角和温度等, 这些因素相互作用导致了流动现象的复杂性, 现有的理论模型难以将全部因素都囊括在内并求解, 进行理论分析往往要对影响因素进行取舍. Benjamin[11]和Yih[12]最早使用线性稳定性分析研究了受重力驱动的液膜沿倾斜平板下降过程, 证明了表面波的波长远大于液膜厚度, 为长波近似方法提供了理论基础. 当雷诺数较小(一般指$Re < 10$), 同时韦伯数较大(1000左右)时, 也即液膜惯性力与黏性力在同一数量级, 厚度很薄而表面张力较大, 就适用长波近似方法. Benney[13]首先推导了适用于小雷诺数的倾斜平板的液膜长波近似方程, 在其假设中忽略了表面张力的影响. Gjevik[14]将表面张力考虑在内, 推导了自己的方程并进行了稳定性分析, 但其结果对强非线性部分的分析不够准确. 后来又有许多学者[15-17]根据Benney的思想推导了各自的演化方程并进行了弱非线性分析. 对于中等雷诺数, 长波近似方程得到的某些结果与实验相差达到45倍[18], 为此Shkadov等[19]最先提出了新的积分边界层法, 许多学者通过它研究二维液膜中的非线性现象. Chang等[20]利用它研究了孤立波的相互作用, 计算结果与实验非常相似. 但Kliakhandler[18]通过分析色散关系, 发现在不是长波的波数范围, Benney长波近似方程和Shkadov积分边界层方程与真实色散关系有很大差异. 后来Joo等[21]和Lin[17]对它们进行了绝对/对流不稳定性分析, 结合实验发现两种方程只在特定的参数区域是有效准确的. Wang等[22]与Chu等[23]分别使用修正辅助方程法和Riccati方程展开法研究了薄膜铁电材料中的孤立波, 得到了控制方程的孤子解. 刚傲等[24]则利用小波变换探究涡的结构与孤立波波高之间的关系, 与实验印证得很好.
液膜流动中的非线性效应容易导致三维不稳定性, 同时液膜的现实尺度很小($1 \sim 2 \text{ mm}$), 需要精密的测量技术和设备, 如高速摄像机和电容式探针等[25], 受到技术和经济因素的限制, 因此通过实验研究液膜流动相对较少, 王千等[26]通过实验研究了孤立波与淹没平板相互作用, 为三维波面成像提供了新的测量手段. Zeng等[27]研究了波纹板的结构参数对孤立波的作用机制和液膜厚度的轴向分布的影响. 随着计算机算力的不断提升, CFD数值模拟逐渐应用到液膜研究中来, Miyara[28]计算了垂直壁面降液膜流动中孤立波的流动与传热, 发现在孤立波波峰内出现了涡流, 认为其对传热产生了积极影响. Yu[29]的研究结果表明孤立波内的涡流对传热分布有很大影响, 随雷诺数增加, 孤立波中的传热由热传导为主转到热对流为主. 马学虎等[30]研究了溴化锂溶液沿平板下降液膜流动中波的演化, 对波内动力学特性与可能的强化传热传质机理进行了探讨. Kunugi等[31]对比了是否有人为扰动的情况, 发现强化传热的不只有孤立波内的涡, 还有孤立波和毛细波之间的小涡. Vlachogiannis等[32]使用荧光成像法研究了水和水-甘油溶液沿斜壁面流动中的非线性波演化, 观察到了孤立波相互作用. 后来Malamataris等[33]通过DNS验证了他们的观察, 事实上这也是第一次通过直接求解完整的N-S方程来研究液膜上的孤立波相互作用. 由于需要考虑多相流理论、自由表面形态以及复杂的动力学特性, 液膜的数值模拟大都是通过简化模型进行的, Chakraborty等[34]使用DNS计算了高惯性下的液膜流中的孤立波并将结果与低维模型相对比, 结果发现只有Ruyer-Quil和Manneville的四方程模型能够得到令人满意的结果.
本文将针对倾斜壁面上液膜中的孤立波及其内部涡进行研究, 为不失一般性, 首先对非平整基底上的流动进行分析, 引入合理的假设来推导演化方程, 然后使用小参数摄动法进行简化求解, 并详细分析液膜上的孤立波及其内部涡的演化过程.
1. 数学描写
1.1 物理模型
如图1所示, 在重力驱动下, 二维黏性不可压流体液膜沿倾角为$\theta $的非平整基底向下流动. 为了后续的推导能够成立, 这里假设基底的3阶导数$ {S_{xxx}} $存在且为一有限值, 建立图中所示的笛卡尔坐标系, $u$和$v$分别是 $x$ 和 $y$ 方向上的速度, 基底函数为$y = S(x)$, ${h_0}$ 和 ${u_0}$分别为Nusselt液膜厚度和界面速度, 其中${u_0} = gh_0^2\sin \theta /(2 v)$, 液膜的厚度为$h$, 液膜和基底的总高度为$H(x,t) = S(x) + h(x,t)$, ${l_0}$与 ${d_0}$分别代表基底的特征长度和特征高度.
1.2 控制方程及边界条件
控制方程组及边界条件[35]为
$$ \nabla \cdot {\boldsymbol{\nu}} = 0 $$ (1) $$ \frac{{\partial u}}{{\partial t}} + ({\boldsymbol{\nu}} \cdot \nabla )u = - \frac{1}{\rho }\frac{{\partial p}}{{\partial x}} + \nu {\nabla ^2}u + g\sin \theta $$ (2) $$ \frac{{\partial v}}{{\partial t}} + ({\boldsymbol{\nu}} \cdot \nabla )v = - \frac{1}{\rho }\frac{{\partial p}}{{\partial y}} + \nu {\nabla ^2}v - g\cos \theta $$ (3) 式中, ${\boldsymbol{\nu}} = (u,v),\nabla ,{\nabla ^2},t,\;\rho ,\nu ,p,g$分别表示流体的速度、Hamilton算子、Laplace算子、时间、密度、运动黏度、压强及重力加速度. 边界条件如下
$$ u = v = 0,\quad y = S(x) $$ (4) $$ \left( {1 - H_x^2} \right)\left( {{u_y} + {v_x}} \right) + 2{H_x}\left( {{v_y} - {u_x}} \right) = 0,\quad y = H(x,t) $$ (5) $$\begin{split} &p - {p_0} = \frac{{2\rho v}}{{1 + H_x^2}}\left[\left({v_y} - {H_x}{u_y}\right) - {H_x}\left({v_x} - {H_x}{u_x}\right)\right] -\\ &\qquad \frac{{\sigma {H_{xx}}}}{{{{\left( {1 + H_x^2} \right)}^{3/2}}}}\end{split} $$ (6) $$ {H_t} + u{H_x} = v,\quad y = H(x,t) $$ (7) 其中, $p$与${p_0}$分别是大气和流体压强, $p = \bar p + p'$, $\bar p = {p_0} + \rho g\cos \theta \left( {{h_0} - y} \right)$是压力的稳态解, $p'$是压力的扰动, 之后计算会用到4个无量纲数, 分别为雷诺数$Re = gh_0^3\sin \theta /(2{v^2})$, 韦伯数$We = \sigma \Big/\left(\rho gh_0^2\right)$, 液膜参数${\alpha _1} = {h_0}/{l_0}$, 基底参数 ${\alpha _2} = {d_0}/{l_0}$, 且本文假定: $Re \sim 1,We \sim \alpha _1^{ - 2}$(实验使用的多数流体), ${\alpha _1} \sim {\alpha _2} \ll 1$. 引入流函数$\psi (x,y,t)$(即$u = \partial \psi /\partial y,v = - \partial \psi /\partial x$)及如下所示无量纲量
$$\left. \begin{split} & x = {l_0}{x^*},\quad y = {h_0}{y^*},\quad t = \frac{{{l_0}}}{{{u_0}}}{t^*},\quad \psi = {u_0}{h_0}{\psi ^*} \\ & p = \rho g{h_0}\sin \theta {p^*},\quad H = {h_0}{H^*},\quad S = {h_0}{S^*} \end{split}\right\} $$ (8) 则控制方程组及边界条件可表示为如下形式(省略了*与')
$$\begin{split} & {\psi _{yyyy}} = {\alpha _1}Re({\psi _{yyt}} + {\psi _y}{\psi _{xyy}} - {\psi _x}{\psi _{yyy}}) - 2\alpha _1^2{\psi _{xxyy}} + \\ &\qquad \alpha _1^3Re({\psi _{xxt}} + {\psi _y}{\psi _{xxx}} - {\psi _x}{\psi _{xxy}}) - \alpha _1^4{\psi _{xxxx}}\end{split} $$ (9) $$\begin{split} & {\psi _{yyy}} = - 2 + 2{\alpha _1}{p_x} - \alpha _1^2{\psi _{xxy}} + \\ &\qquad {\alpha _1}Re({\psi _{yt}} + {\psi _y}{\psi _{xy}} - {\psi _x}{\psi _{yy}}) \end{split} $$ (10) $$ {\psi _x} = 0,\quad y = S $$ (11) $$ {\psi _y} = 0,\quad y = S $$ (12) $$ \left( {{\psi _{yy}} - \alpha _1^2{\psi _{xx}}} \right)\left( {1 - \alpha _1^2H_x^2} \right) - 4\alpha _1^2{H_x}{\psi _{xy}} = 0,\quad y = H $$ (13) $$\begin{split} & - \frac{{We\alpha _1^2\csc \theta {H_{xx}}}}{{{{\left( {1 + \alpha _1^2H_x^2} \right)}^{3/2}}}} + (H - S - 1)\cot \theta - p - \\ &\qquad {\alpha _1}{\psi _{xy}}\frac{{1 - \alpha _1^2H_x^2}}{{1 + \alpha _1^2H_x^2}} - \frac{{4\alpha _1^3{\psi _{xy}}H_x^2}}{{1 - \alpha _1^4H_x^4}} = 0,\quad y = H \end{split}$$ (14) $$ {H_t} + {\psi _y}{H_x} = - {\psi _x},\quad y = H $$ (15) 1.3 摄动法近似求解
将$\psi $和$p$依${\alpha _1}$展开有
$$ \psi = {\psi ^{(0)}} + {\alpha _1}{\psi ^{(1)}} + \alpha _1^2{\psi ^{(2)}} + \cdots $$ (16) $$ p = {p^{(0)}} + {\alpha _1}{p^{(1)}} + \alpha _1^2{p^{(2)}} + \cdots $$ (17) 将式(16)和式(17)代入式(9)$ \sim $式(15), 可以求得流函数的零阶分量与一阶分量分别为
$$ {\psi ^{(0)}} = - \frac{1}{3}{(y - S)^2}(y - S - 3h) $$ (18) $$ \begin{split} & {\psi ^{(1)}} = \frac{1}{{30}}Reh{h_x}{(y - S)^5} - \frac{1}{6}Re{h^2}{h_x}{(y - S)^4} - \\ &\qquad \frac{1}{3}\left[ We\alpha _1^2\csc \theta {H_{xxx}} - \left( {{H_x} - {S _x}} \right)\cot \theta \right]{(y - S)^3} + \\ &\qquad \left[ \frac{2}{3}Re{h^4}{h_x} + We\alpha _1^2\csc \theta h{H_{xxx}} - \left( {{H_x} - {S _x}} \right)\cot \theta h \right]\cdot\\ &\qquad {(y - S)^2}\end{split} $$ (19) 流函数的更高阶分量导致的误差很小[36], 因此本文将流函数取到一阶, 将式(18)和式(19)代入式(16), 得非平整基底上液膜的流函数. 同时式(15)可改写成[15]
$$ {h_t} + {\left( {{{\left. \psi \right|}_{y = H}}} \right)_x} = 0 $$ (20) 将所得流函数代入上式, 得非平整基底上液膜厚度一般方程
$$ \begin{split} & {h_t} + \left\{\frac{2}{3}{h^3} + {\alpha _1}\left[\frac{8}{{15}}Re{h^6}{h_x} - \frac{2}{3}\cot \theta {h^3}{h_x} +\right.\right. \\ &\qquad \left.\left.\frac{2}{3}We\alpha _1^2\csc \theta {h^3}({h_{xxx}} + {S_{xxx}})\right]\right\}_x = 0\end{split} $$ (21) 若忽略基底与倾斜角, 即令$S = 0$且$\theta = {90^\circ }$时, 方程(21)与日本学者仲矢長恣所得到的液膜竖直壁面流动的表面波方程是一致的.
2. 液膜厚度方程的孤立波解
孤立波的波长无限长, 波数接近于0[4], 当考虑基底为平板时, 也即$S = 0$, 式(21)简化为
$$ \begin{split} & {h_t} + \left[\frac{2}{3}{h^3} + {\alpha _1}\left(\frac{8}{{15}}Re{h^6}{h_x} - \frac{2}{3}\cot \theta {h^3}{h_x} +\right.\right. \\ &\qquad \left.\left.\frac{2}{3}We\alpha _1^2\csc \theta {h^3}{h_{xxx}}\right)\right]_x = 0\end{split} $$ (22) 为了寻找其解, 采用如下的行波变换
$$ h = h(x,t) = h(\xi ),\quad \xi = x - ct $$ (23) 其中, $c$为波速, 将式(23)代入方程(22), 然后进行一次积分得
$$ \begin{split} & - ch + \frac{2}{3}{h^3} + {\alpha _1}\left(\frac{8}{{15}}Re{h^6}\frac{{{\mathrm{d}}h}}{{{\mathrm{d}}\xi }} - \right. \\ &\qquad\left.\frac{2}{3}\cot \theta {h^3}\frac{{{\mathrm{d}}h}}{{{\mathrm{d}}\xi }} + \frac{2}{3}We\alpha _1^2\csc \theta {h^3}\frac{{{{\mathrm{d}}^3}h}}{{{\mathrm{d}}{\xi ^3}}}\right) = A\end{split} $$ (24) 其中, $A$为积分常数. 在此假定方程满足以下边界条件
$$ h \to k,\quad \xi \to \pm \infty $$ (25) 其中, $k$为稳定状态下的液膜厚度, 是一个定值, 可以解得$A = - ck + (2/3){k^3}$, 对方程(24)作进一步的变量代换
$$ \eta = {\varepsilon ^{ - 1}}\xi ,\quad \varepsilon = {\alpha _1}W{e^{1/3}} $$ (26) 于是式(23)和式(24)变为
$$\begin{split} & - ch + \frac{2}{3}{h^3} + \frac{8}{{15}}\frac{{Re{h^6}}}{{W{e^{1/3}}}}\frac{{{\mathrm{d}}h}}{{{\mathrm{d}}\eta }} - \frac{2}{3}\frac{{\cot \theta {h^3}}}{{W{e^{1/3}}}}\frac{{{\mathrm{d}}h}}{{{\mathrm{d}}\eta }} + \\ &\qquad \frac{2}{3}\csc \theta {h^3}\frac{{{{\mathrm{d}}^3}h}}{{{\mathrm{d}}{\eta ^3}}} = - ck + \frac{2}{3}{k^3} \end{split} $$ (27) $$ h \to k,\quad \eta \to \pm \infty $$ (28) 式(27)和边界条件式(28)可以看成一个特征值问题, 为了解出该特征值, 首先计算$h$在无穷远处的情况. 由式(25)可知
$$ h = k + {h^\prime },\quad {h^\prime } < < k $$ (29) 将其代入方程(27), 略去二阶及以上的小量得
$$ \frac{{{{\mathrm{d}}^3}{h^\prime }}}{{{\mathrm{d}}{\eta ^3}}} + r\frac{{{\mathrm{d}}{h^\prime }}}{{{\mathrm{d}}\eta }} + s{h^\prime } = 0 $$ (30) 其中
$$ r = \left( {\frac{4}{5}\frac{{Re{k^3}}}{{W{e^{1/3}}}} - \frac{{\cot \theta }}{{W{e^{1/3}}}}} \right)\sin \theta ,s = \left( {\frac{3}{k} - \frac{{3c}}{{2{k^3}}}} \right)\sin \theta $$ (31) 易知$s > 0$, 方程(30)为三阶齐次线性微分方程, 其解具有如下形式
$$ {h^\prime } = \exp (\lambda \eta ) $$ (32) 将式(32)代入方程(30)可求出$\lambda $的值为
$$ {\lambda _1} = \alpha ,\quad {\lambda _{2,3}} = - \frac{1}{2}\alpha \pm ({\mathrm{i}}\sqrt 3 /2)\beta $$ (33) 其中, $\alpha = {\left( {{J_ + }} \right)^{1/3}} + {\left( {{J_ - }} \right)^{1/3}}$, $ \beta = {\left( {{J_ + }} \right)^{1/3}} - {\left( {{J_ - }} \right)^{1/3}} $, $ {J_ \pm } = - s/2 \pm {[{\left( {s/2} \right)^2} + {(r/3)^3}]^{1/2}} $. 方程(30)具有两个解, 第一个解对应于上游区域
$$ {h_u} = {C_1}\exp (\alpha \eta ) $$ (34) 其中, ${C_1}$为常数. 第二个解对应于下游区域
$$ {h_d} = {C_2}\exp \left( { - \frac{1}{2}\alpha \eta } \right)\cos \left(\sqrt 3 \beta \eta/2 + \gamma \right) $$ (35) 其中, ${C_2}$为常数, $\gamma $是相移.
现在利用Mathematica对方程(27)进行数值求解, 取$We = 1000$, 可以计算出孤立波的波形图, 如图2所示. 通过观察与对比得出: 首先, 孤立波的类型可以根据波形进行区分, 第一种孤立波有一个波峰, 第二种有两个; 其次, 在两个和3个波峰的情况, 前面的波峰最高, 然后逐渐降低, 随着波速的增加, 波峰的高度也随之增加. 倾角越大, 孤立波越“窄”; 最后, 所有孤立波有一个共同点, 在最高的波峰前存在一个毛细波, 在最后的波后有一个斜坡, 中间的非线性区域与二者光滑连接, 这与式(34)和式(35)的计算一致. 从图2中还能够看出$\Delta \eta \sim 1$. 表示在$\eta $方向上$h$有显著变化的距离, 由式(25)可知$\Delta \xi \sim {\alpha _1}W{e^{1/3}}$, 于是有: ${l_0}\Delta \xi /{h_0} \sim W{e^{1/3}}$, 而${l_0}\Delta \xi $表示孤立波对应于物理坐标的增量, 即, 若以${h_0}$为参照, 孤立波与$W{e^{1/3}}$同阶.
将液膜的流量${q_i} = \psi (h)$代入式(19), 然后利用式(27)进行简化有
$$ {q_i} = c(h - 1) + \frac{2}{3} $$ (36) 也即流量${q_i}$正比于$h$, 二者具有相同的性质, ${q_i}$可以看作是$h$的放大和平移, 适用于液膜厚度的结论也适用于流量, 稳定状态下$h = 1$, ${q_i}$ = $2/3$. 如图3所示, 涡流的出现使得流量急剧增加; ${q_i}$随$c$的增加而增加; 对于双峰与三峰孤立波, 靠前的波峰的${q_i}$大于靠后的, ${q_{i{\mathrm{max}}}}$对应于${h_{{\mathrm{max}}}}$.
由于对非平整基底液膜厚度方程进行了简化, 对应的流函数也得到简化, 将式(23)和式(26)代入该流函数有
$$ \begin{split} &\psi = \frac{1}{30}\frac{Re}{W{e}^{1/3}}h{h}_{\eta }{y}^{5}-\frac{1}{6}\frac{Re}{W{e}^{1/3}}{h}^{2}{h}_{\eta }{y}^{4}-\\ &\qquad \frac{1}{3}\left(1 + \mathrm{csc}\theta {h}_{\eta \eta \eta }-\frac{\mathrm{cot}\theta }{W{e}^{1/3}}{h}_{\eta }\right){y}^{3} + \Bigg(h +\\ &\qquad \frac{2}{3}\frac{Re}{W{e}^{1/3}}{h}^{4}{h}_{\eta } + \mathrm{csc}\theta h{h}_{\eta \eta \eta }-\frac{\mathrm{cot}\theta }{W{e}^{1/3}}h{h}_{\eta }\Bigg){y}^{2}\end{split} $$ (37) 根据流线的定义
$$ \psi (\eta ,y) = {\text{const.}} $$ (38) 通过求解, 以60°为例, 可以得到孤立波的流线图, 如图4所示.
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图 3 孤立波流量图($\theta = {60^\circ }$)Figure 3. Solitary wave instantaneous flow diagram ($\theta = {60^\circ }$)3. 孤立波中涡的演化研究
3.1 涡流的成因分析
一直以来, 孤立波中的涡流被认为是波动强化传热传质的机理之一. 在图4中可以发现, 首先, 在毛细波的波谷处, 存在一个完全开式涡, 该回流区强化了流体与壁面之间的传热, 这与文献[37-38]的计算结果一致. 其次, 通过对比单峰与双峰孤立波的流线图可以发现, 在波速等于5和6时(图4(e)与图4(f)), 在最高的波峰内也出现了涡流, 此处流场十分不均, 左侧流线密集, 流速较快, 右侧相对稀疏, 流速较慢, 因此涡流方向为顺时针, 此结论也能与后文中的迹线图相印证. 在图4(h)与图4(i)中, 第二个波峰内也出现了明显的涡流, 但其尺寸要比最高峰的小, 也即波速越大, 波峰越高, 越容易出现涡流. 最后, 波峰数越多, 其涡流的数量也越多, 即: 多峰孤立波更有利于传热传质.
速度在$x$和$y$方向上的分量分别为$u$和$v$, 其中$u = {\psi _y},v = - {\psi _\eta }$, 将式(36)代入其中, 然后令$y = h$, 可得自由表面处的速度分布${u_f}$和${v_f}$
$$\begin{split} & {u_f} = {h^2} + \frac{5}{6}\frac{{Re}}{{W{e^{1/3}}}}{h^5}{h_\eta } - \frac{{\cot \theta }}{{W{e^{1/3}}}}{h^2}{h_\eta } +\\ &\qquad \csc \theta {h^2}{h_{\eta \eta \eta }} \end{split} $$ (39) $$ \begin{split} & {v_f} = - c{h_\eta } + {h^2}{h_\eta } + \frac{5}{6}\frac{{Re}}{{W{e^{1/3}}}}{h^5}h_\eta ^2 - \\ &\qquad \frac{{\cot \theta }}{{W{e^{1/3}}}}{h^2}h_\eta ^2 + \csc \theta {h^2}{h_\eta }{h_{\eta \eta \eta }} \end{split} $$ (40) 这里式(40)的推导使用了式(27). 数值求解式(39)和式(40), 得${u_f}$和${v_f}$的分布如图5所示.
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图 5 液膜自由表面的速度分布图(第一种孤立波)Figure 5. Velocity distribution diagram on the free surface of the liquid film (the first solitary wave)图5(a)说明$c = 4$时, ${u_f}$始终比$c$小, 当$c$增大到5和6时, ${u_f}$的最大值${u_{f{\mathrm{max}}}}$已经大于$c$了, 可见${u_{f{\mathrm{max}}}}$随$c$的增大逐渐接近甚至超过$c$, 存在一个临界值$c_s^{(1)}$对应的${u_{f{\mathrm{max}}}} = c$, 这说明流体沿壁面方向的运动速度能够大于孤立波的传播速度. 图5(b)说明$c = 4$时, ${v_f}$仅一个波峰, 当$c$增大到5和6时, ${v_f}$存在两个波峰, 高峰在前, 低峰在后, 二者之间的波谷处${v_f} = 0$, 此时${v_f}$才可能达到甚至超过$c$, 之后${v_f}$迅速增大. 孤立波内部的涡流正是在${v_f}$在波前和波尾的速度梯度与大于$c$的${v_f}$的共同作用下产生的.
表1列出了3种孤立波在3种倾斜角下的${c_s}$, 通过对比可知: ${c_s}$随$\theta $的增大而增大, 第一种孤立波的${c_s}$最大, 第二种次之, 第三种最小, 也即第三种孤立波更易产生涡流.
表 1 临界波速数据Table 1. Critical wave speed data$\theta $ $c_s^{(1)}$ $c_s^{(2)}$ $c_s^{(3)}$ ${30^\circ }$ 4.36 3.68 3.47 ${60^\circ }$ 4.40 3.71 3.49 ${90^\circ }$ 4.43 3.72 3.50 3.2 波速对涡流的影响分析
对于非定常流动, 迹线与流线并不重合, 在动坐标系下能更好地观察涡流, 为此引入$\zeta $和$\tau $代替$y$和$t$
$$ \zeta = {\varepsilon ^{ - 1}}y,\quad \tau = {\varepsilon ^{ - 1}}t $$ (41) 于是有
$$ \frac{{{\mathrm{d}}\eta }}{{{\mathrm{d}}\tau }} = u - c,\quad \frac{{{\mathrm{d}}\zeta }}{{{\mathrm{d}}\tau }} = v $$ (42) 其中, $u$和$v$由上节给出的公式计算, 计算结果如图6所示.
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图 6 孤立波迹线图 $(\theta = {90^\circ })$Figure 6. Solitary waveform chart $(\theta = {90^\circ })$从图6可知, 当$c$ < ${c_s}$时, 波峰内无闭合曲线, 即不会产生涡流. 当$c$ > ${c_s}$时, 由式(42)知动坐标系的平动速度为$c$, 波峰内有闭合曲线且其面积随$c$的提高而增大, 最外侧曲线与自由表面重合且曲线旋向为顺时针, 可以推断: 旋涡是在自由表面$ {u_f} = 0 $处开始形成的, 同时有一个由孤立波导致的质量输运, 波峰内的流体沿壁面向下的运动过程呈滚落状. 图6(f)说明对于第二种孤立波, $c = 6$时, 第二个波峰内也出现了旋涡, 该涡流的尺寸相对较小, 旋向同为顺时针.
4. 结 论
本文利用小参数摄动法及行波变换法, 首先得到了液膜沿非平整倾斜基底流动的一般演化方程, 然后详细研究了基底为平板时其上孤立波及其内部涡的演化, 主要结论如下.
(1)在孤立波的前方存在一个毛细波, 此处出现了完全开式涡, 能够强化流体与基底的传热传质, 随着$c$的增大, 孤立波内也逐渐开始出现涡流且波峰越多, 涡流数量越多, 流量是波形$h$的线性函数, 有着相同的变化规律.
(2)当孤立波以${h_0}$为参照时, 孤立波与$W{e^{1/3}}$同阶, 本文中不同种类的孤立波以波峰数量区分, 波峰的高度与波速成正比, 倾斜角$\theta $越大, 孤立波波形越“窄”.
(3)产生涡流的必要条件是波速达到临界值${c_s}$, 此时自由表面的垂直速度${v_f}$在波前和波尾的速度梯度与大于$c$的水平速度${u_f}$共同作用产生了涡流, 可以推测旋涡在自由表面${u_f} = 0$处开始形成, 波峰内的流体向下运动呈滚落状. ${c_s}$的值正比于$\theta $, 涡流的面积正比于$c$且旋向为顺时针.
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图 3 孤立波流量图($\theta = {60^\circ }$)
Figure 3. Solitary wave instantaneous flow diagram ($\theta = {60^\circ }$)
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图 5 液膜自由表面的速度分布图(第一种孤立波)
Figure 5. Velocity distribution diagram on the free surface of the liquid film (the first solitary wave)
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图 6 孤立波迹线图 $(\theta = {90^\circ })$
Figure 6. Solitary waveform chart $(\theta = {90^\circ })$
表 1 临界波速数据
Table 1 Critical wave speed data
$\theta $ $c_s^{(1)}$ $c_s^{(2)}$ $c_s^{(3)}$ ${30^\circ }$ 4.36 3.68 3.47 ${60^\circ }$ 4.40 3.71 3.49 ${90^\circ }$ 4.43 3.72 3.50 
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