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流体不稳定性是流体力学的一个基本问题, 通常发生在不同流体或者同种流体不同密度的界面. 常见的界面不稳定性有Reyleigh-Taylor不稳定性(RTI)[1,2]、Richtmyer-Meshkov不稳定性(RMI)[3,4]和Kelvin-Helmholz不稳定性(KHI)[5,6]. RTI发生在密度梯度和加速度方向相反的流体界面, 当冲击波通过两流体的界面时会产生RMI. RTI和RMI均垂直于界面方向, 而KHI则是由平行剪切作用于界面时引起的. 上述不稳定性在天体物理[7]、等离子体[8]、超流体[9]、磁流体[10]以及惯性约束聚变(ICF)[11-13]等领域中发挥着关键作用.
理解任何一种流体不稳定性不仅对该不稳定性本身很重要, 而且对其他流体不稳定性的理解也很重要. 事实上, KHI不仅会加剧非线性RTI和RMI的发展, 是界面蘑菇状结构演变的关键, 而且对由RTI和RMI产生的流动向湍流的过渡起着至关重要的作用[14,15].
KHI线性不稳定性增长的动力学分析是一个活跃的研究领域. 在过去的几十年里, 人们广泛研究了各种因素对KHI线性演化的影响. 这些因素包括压缩[16]、黏性[17]、表面张力[18]、热传导[19,20]、倾斜界面[21]等. 热传导现象是由物质中大量的分子热运动互相撞击, 从而使能量从物体的高温部分传至低温部分, 或由高温物体传给低温物体的过程. 热传导效应在KHI中不可忽视. 之前人们[19]在横截面不变的直管道中通过将热传导项加入到速度势中, 求解管道中上下流体界面的伯努利方程, 从而得到了界面热传导系数与上下流体的界面相对切向速度和KHI增长率的关系. 人们发现: 增加界面热传导系数会抑制上下流体的界面相对切向速度, 但会提高KHI的增长率[19,20]. 实际上, 管道的横截面和管道中的流体速度均会发生改变[22]. 管道横截面或流体速度的突然变化均会造成流体界面处的能量损失, 从而引起水头损失[22-24]. 水头损失一般分为沿程水头损失和局域水头损失, 而由管道横截面突变产生的水头损失为局部水头损失[22]. 另外, 流体流动中, 由于管道的横截面发生变化, 流体内部各粒子的动能和势能相互转换, 流体内部的热传导(这里主要是热导率)也会受到影响. 通过计算热传导对KHI的影响, 可以有效地设计日常通风供暖等流体管道.
目前, 人们对横截面变化的直管道中(或者水头损失存在下)热传导对流体不稳定的影响研究甚少. 本文将解析研究热传导对横截面不同的直管道中KHI的影响, 并与横截面相同的直管道结果相比较. 论文安排如下: 第2节给出模型; 第3节对结果进行分析和讨论; 第4节给出结论.
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假设一个水平放置的直管道, 如图1所示. 管道具有两种不同的截面
$ {A_1} $ 和$ {A_{\text{2}}} $ , 其中$ {A_1} $ 为窄管道的横截面,$ {A_{\text{2}}} $ 为宽管道的横截面, 这两部分的管径分别是$ {D_1} $ 和$ {D_2} $ . 管道中有两种不同的流体, 密度小的流体在上, 密度大的流体在下, 两种流体之间存在一个界面(y = 0). 假设上下流体密度、黏度、平均流速和温度分别为$ {\rho _{\text{u}}} $ 和$ {\rho _{\text{l}}} $ ,$ {\mu _{\text{u}}} $ 和$ {\mu _{\text{l}}} $ ,$ {U_{\text{u}}} $ 和$ {U_{\text{l}}} $ ,${T_{\text{u}}}$ 和${T_{\text{l}}}$ . 此外, 还假定流体不可压缩且两种流体之间没有旋转和摩擦, 即两种流体存在势流以及流体之间无耗散. 在流体流动过程中, 水头损失主要由管道横截面的突变引起.
图 1 两种不同横截面的直管道示意图
Figure 1. Schematic diagram of a straight pipe with two different cross sections.
上下流体的界面满足:
$ F(x,y,t) = y - \eta (x,t) = 0 {, } $ 式中函数
$F(x, y, t)$ 中; t为时间; η为解界面扰动振幅, 表示为$ \eta = {\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t) + {\text{c}}.{\text{c}}. $ 其中
$ {\eta _0} $ 为常数, c.c.代表复共轭. 这里假定管道壁面为光滑壁面. 对于上下均匀流体, 小的界面扰动振幅满足速度势[19,25]:$ {\phi _{\text{u}}} = - \frac{1}{k}\left( {\frac{{\alpha '}}{{{\rho _{\rm{u}}}}} - {\rm{i}}k{U_{\rm{u}}} - {\rm{i}}\omega } \right)\frac{{\cosh [k(y - {h_{\text{u}}})]}}{{\sinh (k{h_{\text{u}}})}}\eta + {\text{c}}.{\text{c}}{, } $ $ {\phi _{\text{l}}} = \frac{1}{k}\left( {\frac{{\alpha '}}{{{\rho _{\rm{l}}}}} - {\rm{i}}k{U_{\rm{l}}} - {\rm{i}}\omega } \right)\frac{{\cosh [k(y + {h_{\text{l}}})]}}{{\sinh (k{h_{\text{l}}})}}\eta + {\text{c}}.{\text{c}}{, } $ 式中, k, ω为扰动的波数和频率;
$\alpha '$ 为界面热传导系数, 定义为界面处的热流强度与单位长度的潜热之比[19];$ {h}_{\text{u}}({h}_{{\rm{l}}}) $ 为流体界面到上(下)壁面的距离.因为流体是无旋的, 所以速度势(3)式和(4)式满足拉普拉斯方程[26]:
$ \frac{{\partial u}}{{\partial x}} + \frac{{\partial v}}{{\partial y}} = \frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {y^2}}} = {\text{0}} . $ 流体界面的控制方程为
$ \frac{{\partial \eta }}{{\partial t}} + {U_{\rm{u}}}\frac{{\partial \eta }}{{\partial x}} = \frac{{\partial {\varphi _{\text{u}}}}}{{\partial y}} {, } \;\;y = {h_{\rm{u}}} {, } \;\frac{{\partial {\varphi _{\rm{u}}}}}{{\partial y}} = 0 {, } $ $ \frac{{\partial \eta }}{{\partial t}} + {U_{\text{l}}}\frac{{\partial \eta }}{{\partial x}} = \frac{{\partial {\varphi _{\text{l}}}}}{{\partial y}} {, }\; \;y = - {h_{\rm{l}}} {, } \;\frac{{\partial {\varphi _{\text{l}}}}}{{\partial y}} = 0 . $ 流体界面的动力学方程为
$ \begin{split} &{{\rho _{\rm{u}}}\left( {g\eta + \frac{{\partial {\varphi _{\rm{u}}}}}{{\partial t}} + {U_{\rm{u}}}\frac{{\partial {\varphi _{\rm{l}}}}}{{\partial x}} + g{h_{{\rm{u}},{\rm{loss}}}}} \right)}\\ & - {\rho _{\rm{l}}}\left( {g\eta + \frac{{\partial {\varphi _{\rm{l}}}}}{{\partial t}} + {U_{\rm{l}}}\frac{{\partial {\varphi _{\rm{l}}}}}{{\partial x}} + g{h_{{\rm{l}},{\rm{loss}}}}} \right) \\ &+ {P_{\rm{u}}} - {P_{\rm{l}}} = - \gamma \frac{{{\partial ^2}\eta }}{{\partial {x^2}}}, \end{split} $ 式中g为重力加速度, γ为界面张力系数. 上下流体之间的正压力为
$ {P_{\rm{u}}} - {P_{\text{l}}} $ , 可以表达为[20]$ \begin{split} & {P_{\rm{u}}} - {P_{\rm{l}}} \\ =\;& 2k\bigg[{\mu _{\rm{l}}}\left( {\frac{{\alpha '}}{{{\rho _{\rm{l}}}}} - {\rm{i}}k{U_{\rm{l}}} - {\rm{i}}\omega } \right)\cosh (k{h_{\rm{l}}}) \\ & \left. + {\mu _{\rm{u}}}\left( {\frac{{\alpha '}}{{{\rho _{\rm{u}}}}} - {\rm{i}}k{U_{\rm{u}}} - {\rm{i}}\omega } \right)\cosh (k{h_{\rm{u}}}) \right]{\eta _0} \\ & \times\exp ({\rm{i}}kx - {\rm{i}}\omega t). \end{split} $ 局域水头损失可表示为[22]
$ {h_{{\text{u}},{\text{loss}}}} = \frac{1}{{2g}}\zeta u_{{\text{u}},x}^2 {, } $ $ {h_{{\text{l}},{\text{loss}}}} = \frac{1}{{2g}}\zeta u_{{\text{l}},x}^2 {, } $ 式中水头损失系数ζ与管道两部分的截面比
$ {{{A_1}} / {{A_2}}} $ 有关[27], 即$ \zeta = {(1 - {{{A_1}} / {{A_2}}})^2} $ ;$ {u_{{\text{u}}, x}} $ 和$ {u_{{\text{l}}, x}} $ 分别是上、下流体在产生水头损失时的瞬时速度.$ {u_{{\text{u}}, x}} $ 和$ {u_{{\text{l}}, x}} $ 可由速度势(3)式和(4)式的负梯度得到$ u_{{\rm{u}},x}^2 = {\left( {{\rm{i}}\frac{{\alpha '}}{{{\rho _{\rm{u}}}}} + k{U_{\rm{u}}} - \omega } \right)^2}{\coth ^2}(k{h_{\rm{u}}})\eta _0^2\exp 2({\rm{i}}kx - {\rm{i}}\omega t), $ $ u_{{\rm{l}},x}^2 = {\left( {{\rm{i}}\frac{{\alpha '}}{{{\rho _{\rm{l}}}}} + \omega - k{U_{\rm{l}}}} \right)^2}{\coth ^2}(k{h_{\rm{l}}})\eta _0^2\exp 2({\rm{i}}kx - {\rm{i}}\omega t). $ 将(3)式、(4)式和(9)式—(13)式代入(8)式, 可以得到
$ \begin{split} &- 2k[{\rho _{\text{u}}}{U_{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}{U_{\text{l}}}\coth (k{h_{\text{l}}})]\omega + \zeta {k^2}[{\rho _{\text{l}}}{U_{\text{l}}}{\coth ^2}(k{h_{\text{l}}}) - {\rho _{\text{u}}}{U_{\text{u}}}{\coth ^2}(k{h_{\text{u}}})]\omega {\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t)\\ & \times 4{\text{i}}{k^2}[{\mu _{\text{u}}}\coth (k{h_{\text{u}}}) + {\mu _{\text{l}}}\coth (k{h_{\text{l}}})]\omega + {\text{i}}\alpha '[\coth (k{h_{\text{u}}}) + \coth (k{h_{\text{l}}})]\omega + {\text{i}}\zeta k\alpha '[{\coth ^2}(k{h_{\text{u}}}) \\ &- {\coth ^2}(k{h_{\text{l}}})]\omega {\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t) + {k^2}[{\rho _{\rm{u}}}U_{\rm{u}}^2\coth (k{h_{\rm{u}}}) + {\rho _{\rm{l}}}U_{\rm{l}}^{\rm{2}}\coth (k{h_{\rm{l}}})] - 4{k^2}\alpha '\left[ {\frac{{{\mu _{\rm{l}}}}}{{{\rho _{\rm{l}}}}}\coth (k{h_{\rm{l}}}) + \frac{{{\mu _{\rm{u}}}}}{{{\rho _{\rm{u}}}}}\coth (k{h_{\rm{u}}})} \right]\\ & + \frac{1}{2}\zeta {k^3}[{\rho _{\text{u}}}U_{\text{u}}^2{\coth ^2}(k{h_{\rm{u}}}) - {\rho _{\text{l}}}U_{\text{l}}^{\text{2}}{\coth ^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t) + ({\rho _{\text{u}}} - {\rho _{\text{l}}})gk - \gamma {k^3} \\ & - \frac{1}{2}\zeta k{\alpha'^{2}}\left[ {\frac{{{{\coth }^2}(k{h_{\rm{u}}})}}{{{\rho _{\rm{u}}}}} - \frac{{{{\coth }^2}(k{h_{\rm{l}}})}}{{{\rho _{\rm{l}}}}}} \right]{\eta _0}\exp ({\rm{i}}kx - {\rm{i}}\omega t) - 4{\text{i}}{k^3}[{\mu _{\text{u}}}{U_{\text{u}}}\coth (k{h_{\text{u}}}) + {\mu _{\text{l}}}{U_{\text{l}}}\coth (k{h_{\text{l}}})] \\ &- {\text{i}}k\alpha '[{U_{\text{u}}}\coth (k{h_{\text{u}}}) + {U_{\text{l}}}\coth (k{h_{\text{l}}})] - {\rm i}\zeta {k^2}\alpha '[{U_{\text{u}}}{\coth ^2}(k{h_{\text{u}}}) - {U_{\text{l}}}{\coth ^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t) =0. \\[-12pt] \end{split}$ 通过(14)式, 可得到与ω有关的色散方程:
$ Y(\omega,k) = {a_0}{\omega ^2} + ({a_1} + {\text{i}}{b_1})\omega + ({a_2} + {\text{i}}{b_2}) = 0 . $ $ \begin{split} {a_0} = \; &{\rho _{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}\coth (k{h_{\text{l}}}) + \frac{1}{2}\zeta k[{\rho _{\text{u}}}{\coth ^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}{\coth ^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t), \\[-10pt]\end{split} $ $ {a_1} = - 2k[{\rho _{\text{u}}}{U_{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}{U_{\text{l}}}\coth (k{h_{\text{l}}})] + \zeta {k^2}[{\rho _{\text{l}}}{U_{\text{l}}}{\coth ^2}(k{h_{\text{l}}}) - {\rho _{\text{u}}}{U_{\text{u}}}{\coth ^2}(k{h_{\text{u}}})]{\eta _0} \exp ({\text{i}}kx - {\text{i}}\omega t) , $ $ \begin{split} {a_2} =\;& {k^2}[{\rho _{\rm{u}}}U_{\text{u}}^{\text{2}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}U_{\text{l}}^{\text{2}}\coth (k{h_{\text{l}}})] + \frac{1}{2}\zeta {k^3}[{\rho _{\text{u}}}U_{\text{u}}^{\text{2}}{\coth ^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}U_{\text{l}}^{\text{2}}{\coth ^2}(k{h_{\text{l}}})]{\eta _0} \exp ({\text{i}}kx - {\text{i}}\omega t)\\ & + ({\rho _{\text{u}}} - {\rho _{\text{l}}})gk - \gamma {k^3} - 4{k^2}\alpha '\left[ {\frac{{{\mu _{\rm{l}}}}}{{{\rho _{\rm{l}}}}}\coth (k{h_{\rm{l}}}) + \frac{{{\mu _{\rm{u}}}}}{{{\rho _{\rm{u}}}}}\coth (k{h_{\rm{u}}})} \right] - \frac{1}{2}\zeta k\alpha {'^2}\left[ {\frac{{{{\coth }^2}(k{h_{\rm{u}}})}}{{{\rho _{\rm{u}}}}} - \frac{{{{\coth }^2}(k{h_{\rm{l}}})}}{{{\rho _{\rm{l}}}}}} \right]{\eta _0}\\ &\times\exp ({\text{i}}kx - {\text{i}}\omega t) ,\\[-10pt]\end{split} $ $ \begin{split} {b_1} = \; & 4{k^2}[{\mu _{\text{u}}}\coth (k{h_{\text{u}}}) + {\mu _{\text{l}}}\coth (k{h_{\text{l}}})] + \alpha '[\coth (k{h_{\text{u}}}) + \coth (k{h_{\text{l}}})] \\ & + \zeta k\alpha '[{\coth ^2}(k{h_{\text{u}}}) - {\coth ^2}(k{h_{\text{l}}})] {\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t), \end{split} $ $ \begin{split} {b_2} =\; &- 4{k^3}[{\mu _{\text{u}}}{U_{\text{u}}}\coth (k{h_{\text{u}}}) + {\mu _{\text{l}}}{U_{\text{l}}}\coth (k{h_{\text{l}}})] - k\alpha '[{U_{\text{u}}}\coth (k{h_{\text{u}}}) + {U_{\text{l}}}\coth (k{h_{\text{l}}})] \\ & - \zeta {k^2}\alpha '[{U_{\text{u}}}{\coth ^2}(k{h_{\text{u}}}) - {U_{\text{l}}}{\coth ^2}(k{h_{\text{l}}})] {\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t). \end{split} $ 令
$ \omega = {\omega _{\text{R}}} + {\text{i}}{\omega _{\text{I}}} $ , 可得$\begin{split} &{a_0}(\omega _{\text{R}}^{\text{2}} - \omega _{\text{I}}^{\text{2}}) + ({a_1}\omega _{\text{R}} - {b_1}\omega _{\text{I}}) + 2{\text{i}}{a_0}\omega _{\text{R}}\omega _{\text{I}} + {\text{i}}{a_1}\omega _{\text{I}} + {\text{i}}{b_1}\omega _{\text{R}} + {a_2} + {\text{i}}{b_2} = 0 {, } \end{split}$ $ {a_0}(\omega _{\text{R}}^{\text{2}} - \omega _{\text{I}}^{\text{2}}) + ({a_1}\omega _{\text{R}}^{} - {b_1}\omega _{\text{I}}^{}) + {a_2} = 0 {, } $ $ 2{a_0}\omega _{\text{R}}^{}\omega _{\text{I}}^{} + {a_1}\omega _{\text{I}}^{} + {b_1}\omega _{\text{R}}^{} + {b_2} = 0 . $ $ {\omega _{\text{R}}} = - \frac{{{a_1}\omega _{\text{I}}^{} + {b_2}}}{{2{a_0}{\omega _{\text{I}}} + {b_1}}} {, } $ 再将
$ {\omega _{\text{R}}} $ 代入到方程(22)中, 就得到关于增长率$ {\omega _{\text{I}}} $ 的四次方程:$ {I_4}\omega _{\text{I}}^{\text{4}} + {I_3}\omega _{\text{I}}^{\text{3}} + {I_2}\omega _{\text{I}}^{\text{2}} + {I_1}\omega _{\text{I}}^{} + {I_0} = 0 . $ 这里
$ {I_0} = {a_0}b_2^2 - {a_1}{b_1}{b_2} + {a_2}b_1^2 {, } $ $ {I_1} = 4{a_0}{a_2}{b_1} - b_1^3 - a_1^2{b_1} {, } $ $ {I_2} = 4a_0^2{a_2} - 5{a_0}b_1^2 - {a_0}a_1^2 , $ $ {I_{\text{3}}} = - 8a_0^2{b_1}, $ $ {I_4} = - 4a_0^3 . $ KHI是由于扰动的增长率大于零时引起的. 如果增长率小于零, 扰动将随时间快速衰减. 然而, 当增长率大于零时, 扰动随时间呈指数增长[18]. 因此只有当增长率为零时界面稳定. KHI的增长率由扰动频率的虚部直接表征. 令
$ {\omega _{\rm{I}}} = 0 $ (即稳定界面), 并将其代入(25)式, 于是有$ {a_0}b_2^2 - {a_1}{b_1}{b_2} + {a_2}b_1^2 = {\text{0}} . $ 将(16)式—(20)式代入(31)式, 可以得到不同传热系数下, 上下流体界面相对切向速度与波数之间的关系, 见(A1)式, 其中
$ V = {U_{\rm{u}}} - {U_{\rm{l}}} $ 是上下流体的相对切向速度.在考虑KHI的增长率时, 由于有黏和无黏情况下得到的结果几乎是一样的[21], 因此令
${\mu _{\text{u}}} = $ $ {\mu _{\text{l}}} = 0$ , 并且将(16)式—(20)式代入(25)式中, 就可以得到不同传热强度下, KHI的增长率随波数的变化关系, 见(A2)式. -
为了定量分析热传导效应, 取扰动振幅
$ \eta = $ $ 0.03 $ m, 管径$ {D_1} = 0.05 $ m, 其他参数取自参考文献[25]:$ \gamma = 0.06 $ N/m,${\rho _{\text{u}}} = 1.2$ kg/m3,${\rho _{\text{l}}} = $ $ 1000$ kg/m3,$ {\mu _{\rm{u}}} = 0.00018 $ Pa·s,$ {\mu _{\text{l}}} = 0.01 $ Pa·s, hu = 0.025 m, hl = 0.025 m. 界面热传导系数$\alpha ' = \alpha\; \times $ $ {10^3}$ kg/(m3·s)的取值参照文献[19], 这里α是无量纲界面热传导系数. 此外, 由文献[22]可以看出, 在水头损失系数等于0和0.2时, 能够明显得到管道横截面的变化对KHI的影响效果. 因此, 图2给出了$ \zeta = 0, \;0.2 $ 时, 不同界面热传导系数下上下流体的界面相对切向速度与波数的关系曲线. 从图2可以发现: 上下流体的界面相对切向速度V随波数k的增加而增加, 当波数达到一定值的时候, 界面相对切向速度达到最大, 尔后随着波数的继续增加相对切向速度反而会降低. 再者, 我们发现, 当$ \zeta = 0 $ 时, 即不存在水头损失(或管道横截面不发生变化)时, 相对切向速度随着波数增加, 最终趋于一致. 而当$ \zeta = 0.2 $ 且相对切向速度开始随波数增大而减小时, 小的界面热传导系数会比大的界面热传导系数对相对切向速度的抑制更大. 当$ \zeta = 0.2 $ 时, 界面相对切向速度要低于$ \zeta = 0 $ 时的相对切向速度. 这是由于当流体流经横截面变化的管道时能量会损失, 从而导致较低的相对切向速度.
图 2 上下流体的界面相对切向速度与波数的关系
Figure 2. Wavenumber dependence of the relative tangential velocity at the interface between upper and lower fluids.
图3给出了
$ \zeta = 0, \;0.2 $ 时, 不同界面热传导系数下KHI的增长率随波数的关系曲线, 其中参数取自参考文献[19]:$ \gamma = {\text{0}}.{\text{06}} $ N/m,$ {\rho _{\text{u}}} = 1.2 $ kg/m3,$ {\rho _{\text{l}}} = {\text{0}}.001 \;{\text{kg}}/{{\text{m}}^{\text{3}}} $ ,$ {h_{\text{u}}} = 0.{\text{0}}25 $ m,$ {h_{\text{l}}} = 0.{\text{0}}25 $ m. 我们发现, 当$ \zeta = 0 $ 时, 增长率随波数的增长先增大后减小并最终趋于不变. 这是因为界面动能随波数增加而增大, 导致增长率增大, 当波数接近临界值$ {k_{\text{c}}} $ 时, KHI被破坏, 增长率随波数的增加而急剧减小, 趋近于零[19,22]. 当$ \zeta = 0.2 $ 时, 即存在水头损失(或管道横截面发生变化)时, KHI的增长率要大于不存在水头损失时的增长率, 这是由于在管道横截面变化时, 能量损耗的一部分转化为内能, 促进界面热传导对KHI增长率的影响. 比较不同热传导系数的曲线, 可以明白这一点.
图 3 KHI的增长率与波数的关系
Figure 3. Wavenumber dependence of the growth rate of KHI.
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本文建立了一个横截面变化的直管道模型, 并分析了热传导对在该管道中流动的上下两种流体KHI的影响. 结果表明, 在管道横截面变化的情况下增加界面热传导系数会抑制界面相对切向速度, 与文献[19, 20]中报道的横截面不变的直管道情况一致; 与之不同的是, 当波数到达一定值, 相对切向速度开始随着波数的增加而减小, 并且小的界面热传导系数在波数较大的情况下对相对切向速度的抑制效果更大. 另外, 增加界面热传导系数会促进KHI的增长率, 这与文献[19]中报道的横截面不变的直管道的情况一致, 但是相对于横截面不变的直管道, 热传导对KHI增长率的促进效果更为明显.
值得指出的是, 本文没有考虑不同横截面处流体的密度、黏度以及平均流速的不同对结果的影响. 考虑上述不同可能会对结果造成较大影响. 另外, 尽管本文的结论是由圆形横截面的管道得出, 但是也适用于非圆形横截面的直管道. 本文的结果对实际管道中流体不稳定性的研究以及管道的通风设计和供暖等工程领域具有一定的参考价值.
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上下流体界面相对切向速度与波数之间的关系满足以下方程
$: $ $ \begin{split} &\Big\{ {\rho _{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}\coth (k{h_{\text{l}}}) + \frac{1}{2}\zeta k[{\rho _{\text{u}}}{{\coth }^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}{{\coth }^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t) \Big\} \\ & \times \Big\{ - k\alpha '\left[ {\left( {V + {U_{\text{l}}}} \right)\coth \left( {k{h_{\text{u}}}} \right) + {U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - 4{k^3}\left[ {{\mu _{\text{u}}}\left( {V + {U_{\text{l}}}} \right)\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}{U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] \\ &- \zeta {k^2}\alpha '\left[ {\left( {V + {U_{\text{l}}}} \right){{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {U_{\text{l}}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \Big\}^2 \\ & - \Big\{ - 2k\left[ {{\rho _{\text{u}}}\left( {V + {U_{\text{l}}}} \right)\coth \left( {k{h_{\text{u}}}} \right) + {\rho _{\text{l}}}{U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - \zeta {k^2}\left[ {{\rho _{\text{u}}}\left( {V + {U_{\text{l}}}} \right){{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {\rho _{\text{l}}}{U_{\text{l}}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \Big\} \\ & \times \Big\{ \alpha '\left[ {\coth \left( {k{h_{\text{u}}}} \right) + \coth \left( {k{h_{\text{l}}}} \right)} \right] + 4{k^2}\left[ {{\mu _{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] + \zeta k\alpha '\left[ {{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \Big\} \\ & \times \Big\{ - k\alpha '\left[ {\left( {V + {U_{\text{l}}}} \right)\coth \left( {k{h_{\text{u}}}} \right) + {U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - 4{k^3}\left[ {{\mu _{\text{u}}}\left( {V + {U_{\text{l}}}} \right)\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}{U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] \\ &- \zeta {k^2}\alpha '\left[ {\left( {V + {U_{\text{l}}}} \right){{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {U_{\text{l}}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \Big\} \\ & + \bigg\{ gk\left( {{\rho _{\text{u}}} - {\rho _{\text{l}}}} \right) - \gamma {k^3} + {k^2}\left[ {{\rho _{\text{u}}}{{\left( {V + {U_{\text{l}}}} \right)}^2}\coth \left( {k{h_{\text{u}}}} \right) + {\rho _{\text{l}}}U_{\text{l}}^{\text{2}}\coth \left( {k{h_{\text{l}}}} \right)} \right] \\ & - 4{k^2}\alpha '\left[ {\frac{{{\mu _{\text{l}}}}}{{{\rho _{\text{l}}}}}\coth \left( {k{h_{\text{l}}}} \right) + \frac{{{\mu _{\text{u}}}}}{{{\rho _{\text{u}}}}}\coth \left( {k{h_{\text{u}}}} \right)} \right] - \frac{1}{2}\zeta k\alpha {'^2}\left[ {\frac{{{{\coth }^2}\left( {k{h_{\text{u}}}} \right)}}{{{\rho _{\text{u}}}}} - \frac{{{{\coth }^2}\left( {k{h_{\text{l}}}} \right)}}{{{\rho _{\text{l}}}}}} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \\ & + \frac{1}{2}\zeta {k^3}\left[ {{\rho _{\text{u}}}{{\left( {V + {U_{\text{l}}}} \right)}^2}{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {\rho _{\text{l}}}U_{\text{l}}^{\text{2}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \bigg\} \\ & \times \Big\{ \alpha '\left[ {\coth \left( {k{h_{\text{u}}}} \right) + \coth \left( {k{h_{\text{l}}}} \right)} \right] + 4{k^2}\left[ {{\mu _{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] \\ & + \zeta k\alpha '\left[ {{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {{\coth }^2}\left( {k{h_{\rm{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \Big\}^2 = 0 . \end{split}\tag{A1} $ KHI的增长率随波数的变化关系满足:
$ \begin{split} &- 4{\left\{ {{\rho _{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}\coth (k{h_{\text{l}}}) + \frac{1}{2}\zeta k[{\rho _{\text{u}}}{{\coth }^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}{{\coth }^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t)} \right\}^3}\omega _{\text{I}}^{\text{4}} \\ & - 8{\left\{ {{\rho _{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}\coth (k{h_{\text{l}}}) + \frac{1}{2}\zeta k[{\rho _{\text{u}}}{{\coth }^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}{{\coth }^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t)} \right\}^2} \\ & \times \big\{ \alpha '\left[ {\coth \left( {k{h_{\text{u}}}} \right) + \coth \left( {k{h_{\text{l}}}} \right)} \right] + 4{k^2}\left[ {{\mu _{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] + \zeta k\alpha '\left[ {{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \big\}\omega _{\text{I}}^3 \\ & + \Big(4{\left\{ {{\rho _{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}\coth (k{h_{\text{l}}}) + \frac{1}{2}\zeta k[{\rho _{\text{u}}}{{\coth }^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}{{\coth }^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t)} \right\}^2} \\ & \times \bigg\{ gk\left( {{\rho _{\text{u}}} - {\rho _{\text{l}}}} \right) - \gamma {k^3} + {k^2}\left[ {{\rho _{\text{u}}}U_{\text{u}}^{\text{2}}\coth \left( {k{h_{\text{u}}}} \right) + {\rho _{\text{l}}}U_{\text{l}}^{\text{2}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - 4{k^2}\alpha '\left[ {\frac{{{\mu _{\text{l}}}}}{{{\rho _{\text{l}}}}}\coth \left( {k{h_{\text{l}}}} \right) + \frac{{{\mu _{\text{u}}}}}{{{\rho _{\text{u}}}}}\coth \left( {k{h_{\text{u}}}} \right)} \right] - \frac{1}{2}\zeta k\alpha {'^2}\\ &\times\left[ {\frac{{{{\coth }^2}\left( {k{h_{\text{u}}}} \right)}}{{{\rho _{\text{u}}}}} - \frac{{{{\coth }^2}\left( {k{h_{\text{l}}}} \right)}}{{{\rho _{\text{l}}}}}} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) + \frac{1}{2}\zeta {k^3}\left[ {{\rho _{\text{u}}}U_{\text{u}}^{\text{2}}{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {\rho _{\rm{l}}}U_{\text{l}}^{\text{2}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \bigg\} \\ & - 5\left\{ {{\rho _{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}\coth (k{h_{\text{l}}}) + \frac{1}{2}\zeta k[{\rho _{\text{u}}}{{\coth }^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}{{\coth }^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t)} \right\}\\ & \times \big\{ \alpha '\left[ {\coth \left( {k{h_{\text{u}}}} \right) + \coth \left( {k{h_{\text{l}}}} \right)} \right] + 4{k^2}\left[ {{\mu _{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] + \zeta k\alpha '\left[ {{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \big\}^2 \end{split} $ $ \begin{split} & - \left\{ {{\rho _{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}\coth (k{h_{\text{l}}}) + \frac{1}{2}\zeta k[{\rho _{\text{u}}}{{\coth }^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}{{\coth }^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t)} \right\} \\ & \times {\left\{ { - 2k\left[ {{\rho _{\text{u}}}{U_{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\rho _{\text{l}}}{U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - \zeta {k^2}\left[ {{\rho _{\text{u}}}{U_{\text{u}}}{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {\rho _{\text{l}}}{U_{\text{l}}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right)} \right\}^2}\bigg)\omega _{\text{I}}^{\text{2}} \\ & + \bigg(4\left\{ {{\rho _{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}\coth (k{h_{\text{l}}}) + \frac{1}{2}\zeta k[{\rho _{\text{u}}}{{\coth }^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}{{\coth }^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t)} \right\} \\ & \times \bigg\{ gk\left( {{\rho _{\text{u}}} - {\rho _{\text{l}}}} \right) - \gamma {k^3} + {k^2}\left[ {{\rho _{\text{u}}}U_{\text{u}}^2\coth \left( {k{h_{\text{u}}}} \right) + {\rho _{\text{l}}}U_{\text{l}}^{\text{2}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - 4{k^2}\alpha '\left[ {\frac{{{\mu _{\text{l}}}}}{{{\rho _{\text{l}}}}}\coth \left( {k{h_{\text{l}}}} \right) + \frac{{{\mu _{\text{u}}}}}{{{\rho _{\text{u}}}}}\coth \left( {k{h_{\text{u}}}} \right)} \right]\\ &- \frac{1}{2}\zeta k\alpha {'^2}\left[ {\frac{{{{\coth }^2}\left( {k{h_{\text{u}}}} \right)}}{{{\rho _{\text{u}}}}} - \frac{{{{\coth }^2}\left( {k{h_{\text{l}}}} \right)}}{{{\rho _{\text{l}}}}}} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right)+ \frac{1}{2}\zeta {k^3}\left[ {{\rho _{\rm{u}}}U_{\text{u}}^{\text{2}}{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {\rho _{\text{l}}}U_{\text{l}}^{\text{2}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \bigg\} \\ & \times \big\{ \alpha '\left[ {\coth \left( {k{h_{\text{u}}}} \right) + \coth \left( {k{h_{\text{l}}}} \right)} \right] + 4{k^2}\left[ {{\mu _{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] + \zeta k\alpha '\left[ {{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \big\} \\ & - \big\{ \alpha '\left[ {\coth \left( {k{h_{\text{u}}}} \right) + \coth \left( {k{h_{\text{l}}}} \right)} \right] + 4{k^2}\left[ {{\mu _{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] + \zeta k\alpha '\left[ {{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \big\}^3 \\ & - \big\{ - 2k\left[ {{\rho _{\rm{u}}}{U_{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\rho _{\text{l}}}{U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - \zeta {k^2}\left[ {{\rho _{\text{u}}}{U_{\text{u}}}{{\coth }^2}\left( {k{h_{\rm{u}}}} \right) - {\rho _{\text{l}}}{U_{\text{l}}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \big\}^2 \\ & \times \big\{ \alpha '\left[ {\coth \left( {k{h_{\text{u}}}} \right) + \coth \left( {k{h_{\text{l}}}} \right)} \right] + 4{k^2}\left[ {{\mu _{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] + \zeta k\alpha '\left[ {{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \big\}\bigg){\omega _{\text{I}}} \\ & + \left\{ {{\rho _{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}\coth (k{h_{\text{l}}}) + \frac{1}{2}\zeta k[{\rho _{\text{u}}}{{\coth }^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}{{\coth }^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t)} \right\} \end{split} $ $ \;\;\begin{split} &+ \left\{ {{\rho _{\text{u}}}\coth (k{h_{\text{u}}}) + {\rho _{\text{l}}}\coth (k{h_{\text{l}}}) + \frac{1}{2}\zeta k[{\rho _{\text{u}}}{{\coth }^2}(k{h_{\text{u}}}) - {\rho _{\text{l}}}{{\coth }^2}(k{h_{\text{l}}})]{\eta _0}\exp ({\text{i}}kx - {\text{i}}\omega t)} \right\} \\ & \times \big\{ - k\alpha '\left[ {{U_{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - 4{k^3}\left[ {{\mu _{u} }{U_{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}{U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] \\ &- \zeta {k^2}\alpha '\left[ {{U_{\text{u}}}{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {U_{\text{l}}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \big\}^2 \\ & - \left\{ { - 2k\left[ {{\rho _{\text{u}}}{U_{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\rho _{\text{l}}}{U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - \zeta {k^2}\left[ {{\rho _{\text{u}}}{U_{\text{u}}}{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {\rho _{\text{l}}}{U_{\text{l}}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right)} \right\} \\ & \times \big\{ \alpha '\left[ {\coth \left( {k{h_{\text{u}}}} \right) + \coth \left( {k{h_{\text{l}}}} \right)} \right] + 4{k^2}\left[ {{\mu _{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] + \zeta k\alpha '\left[ {{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \big\} \\ & \times \big\{ - k\alpha '\left[ {{U_{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - 4{k^3}\left[ {{\mu _{\text{u}}}{U_{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}{U_{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] \\ &- \zeta {k^2}\alpha '\left[ {{U_{\text{u}}}{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {U_{\text{l}}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \big\} \\ & + \bigg\{ gk\left( {{\rho _{\text{u}}} - {\rho _{\text{l}}}} \right) - \gamma {k^3} + {k^2}\left[ {{\rho _{\text{u}}}U_{\text{u}}^{\text{2}}\coth \left( {k{h_{u} }} \right) + {\rho _{\text{l}}}U_{\text{l}}^{\text{2}}\coth \left( {k{h_{\text{l}}}} \right)} \right] - 4{k^2}\alpha '\left[ {\frac{{{\mu _{\text{l}}}}}{{{\rho _{\text{l}}}}}\coth \left( {k{h_{\text{l}}}} \right) + \frac{{{\mu _{\text{u}}}}}{{{\rho _{\text{u}}}}}\coth \left( {k{h_{\text{u}}}} \right)} \right] \\ &- \frac{1}{2}\zeta k\alpha {'^2}\left[ {\frac{{{{\coth }^2}\left( {k{h_{\text{u}}}} \right)}}{{{\rho _{\text{u}}}}} - \frac{{{{\coth }^2}\left( {k{h_{\text{l}}}} \right)}}{{{\rho _{\text{l}}}}}} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) + \frac{1}{2}\zeta {k^3}\left[ {{\rho _{\text{u}}}U_{\text{u}}^2{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {\rho _{\text{l}}}U_{\text{l}}^{\text{2}}{{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right] {\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \bigg\} \\ & \times \big\{ \alpha '\left[ {\coth \left( {k{h_{\text{u}}}} \right) + \coth \left( {k{h_{\text{l}}}} \right)} \right] + 4{k^2}\left[ {{\mu _{\text{u}}}\coth \left( {k{h_{\text{u}}}} \right) + {\mu _{\text{l}}}\coth \left( {k{h_{\text{l}}}} \right)} \right] + \zeta k\alpha '\left[ {{{\coth }^2}\left( {k{h_{\text{u}}}} \right) - {{\coth }^2}\left( {k{h_{\text{l}}}} \right)} \right]{\eta _0}\exp \left( {{\text{i}}kx - {\text{i}}\omega t} \right) \big\}^2 = 0. \end{split}\tag{A2} $
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Kelvin-Helmholtz 不稳定性(KHI)增长的动力学分析是一个活跃的研究领域. 本文解析研究了流体在横截面不同的直管道中流动时, 热传导对KHI的影响. 结果表明: 管道中上下流体的界面相对切向速度会随着波数的增加先增加后减小, 并且小的界面热传导系数导致相对切向速度随波数的减小更多, 不同于横截面相同的直管道结果. 另外, 热传导会提高KHI的增长率, 与横截面相同的直管道一致. 研究结果可以为实际管道中流体不稳定性的分析以及管道的通风设计和供暖等工程研究提供一定参考.
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关键词:
- Kelvin-Helmholtz不稳定性 /
- 热传导 /
- 界面热传导系数
We investigate analytically the effect of thermal conduction on the Kelvin-Helmholtz instability (KHI) in a straight pipe with different cross-sections. The results show that the relative tangential velocity of the interface between the upper and lower fluid in the pipe first increases and then decreases with the increase of the wave number. Furthermore, the smaller coefficient of interfacial heat conduction causes the relative tangential velocity to decrease considerably with the increase of the wave number, which is different from the behavior of the straight pipeline with the same cross-section. In addition, the heat conduction increases the growth rate of KHI, which is in accordance with the scenario of straight pipeline with the same cross-section.-
Keywords:
- Kelvin-Helmholtz instability /
- thermal conduction /
- coefficient of interfacial heat conduction
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图 1 两种不同横截面的直管道示意图
Fig. 1. Schematic diagram of a straight pipe with two different cross sections.
图 2 上下流体的界面相对切向速度与波数的关系
Fig. 2. Wavenumber dependence of the relative tangential velocity at the interface between upper and lower fluids.
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