关键词:统一气体动理学格式;范德瓦尔斯状态方程;相界面捕捉;麦克斯韦等面积律;液滴表面张力 Abstract The study of interfacial motion of gas-liquid phase is very important in science and engineering. Considering the non-equilibrium flow calculation, a unified gas-kinetic scheme for gas-liquid two phase interface capturing is presented in this paper. Since the free transport and particle collision are coupled to update the macroscopic variables and microscopic distribution functions, the unified gas-kinetic scheme can solve the non-equilibrium flow. The van der Waals (vdW) equation of state (EOS) is included to describe the coexistence of gas and liquid and the phase transition between them. Because of the characteristics of vdW EOS, the interface between gas and liquid can be captured naturally through condensation and evaporation processes. As a result, the new scheme can solve the gas-liquid two phase problems. Finally, the proposed method is used to obtain the numerical solution of Maxwell construction, which agrees well with the corresponding theoretical solution. Then, the Laplace’s theorem is verified by numerical calculation of the surface tension of the droplet corresponding to the van der Waals state equation. In addition, the collision of the two droplets is simulated, which proves the validity of the scheme further. However, due to the characteristics of the van der Waals equation of state, the constructed scheme is only applicable to the case where the liquid/gas two-phase density ratio is less than 5.
气液两相均有稳态、亚稳态与完全非稳态三种状态. 下面采用本文提出的方法首先计算 时气液两相密度处于不同初值下的数值解, 观察其在定常情况下是否能够回到麦克斯韦等面积律所描述的理论解. 而后得出不同温度下的气液共存密度, 绘制成气液共存曲线, 并与理论解进行对比. 对于UGKS而言, 计算域为 , 划分为 1 60 个均匀网格. X 与Y 方向均采用周期性边界条件. 初始速度为零, 密度分布为[36] 其中, 初始的气液密度分布 赋为麦克斯韦理论解, 为Y方向网格边长, 为初始的相界面厚度, 取为 . 为了避免表面张力对密度分布的影响, 初始的相界面是水平的. 例1. 初始值为麦克斯韦等面积律的理论解所对应的气液相密度, 即 . 如图2所示, 虚线为初始密度和压力分布, 实线便是数值计算所得到的结果, 其具体数值为 , , 与理论解的相对误差分别为 1.48%和0.14%. 其中, 液相密度与理论解相符较好, 而气相密度则偏差较大, 与文献[12]的结果是一致的. 这是由于气相密度较小, 故而液相发生很小的数值振荡便会引起气相的较大偏差. 此外, 在相界面附近, 压力发生了急剧变化, 这与物理实际是不相符的, 究其原因, 这是由于范德瓦尔斯方程并非真实的状态方程所导致的. 范德瓦尔斯方程在两相共存区与实际情况偏差较大, 其所描述的相变线并非实际的相变过程线. 相界面处的密度分布在两相之间, 由图1 可知, 其对应的压力分布或大于气液两相的共存压力, 或小于气液两相的共存压力, 故而会在相界面处出现急剧变化, 这与文献[36]中图1(b) 所示结果是一致的. 显示原图|下载原图ZIP|生成PPT 图2例1: 虚线为初始分布, 实线为麦克斯韦等面积律的数值解. -->Fig.2Case1: The dotted line is the initial distribution. The solid line is the numerical result of Maxwell Construction. -->
例2. 气液相初始密度均处于稳态, 具体值为 . 其数值结果如图3 所示. 不同于例1的是Y= 0.5处由液相蒸发为气相, 这应该是由于初始液相的压力远大于平衡压力, 如图3(b) 所示, 因此随着时间的演化, 液相通过相界面将压力扰动传到气相, 故而远离相界面的液相(Y= 0.5)压力便会急剧下降, 从而变为气相. 其压力与两侧的液体平衡, 如图3(b)中实线所示. 如果降低初始液相的压力, 该现象便会消失. 总之, 无论最终存在几个相界面, 气液两相的压力是相等的, 且收敛在理论解附近. 显示原图|下载原图ZIP|生成PPT 图3例2: 虚线为初始分布, 实线为麦克斯韦等面积律的数值解. -->Fig.3Case2: The dotted line is the initial distribution. The solid line is the numerical result of Maxwell Construction. -->
例3. 气液相初始密度均处于完全不稳定椭圆域, 即 , 并且 . 具体值为 . 其数值结果如图4所示. 可以看到, 当气液两相初始密度均处于完全不稳定椭圆域时, 会通过蒸发和液化实现相分离, 并最终稳定在理论解附近. 显示原图|下载原图ZIP|生成PPT 图4例3: 虚线为初始分布, 实线为麦克斯韦等面积律的数值解. -->Fig.4Case3: The dotted line is the initial distribution. The solid line is the numerical result of Maxwell Construction. -->
显示原图|下载原图ZIP|生成PPT 图8密度分布: (a)图7(f)中沿X方向中心线, (b)图7(f)中沿Y方向中心线. -->Fig.8The density distribution (a) along the central line of Fig.7 (f) in the X direction and (b) along the central line of Fig.7 (f) in the Y direction. -->
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