EFFECTS OF NUMERICAL TREATMENT OF VISCOUS AND SURFACE TENSION FORCES ON PREDICTED INTERFACE MOTION-猫眼课题宝

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2012

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K. Hayashi;A. Tomiyama

粘性和表面张力的处理界面运动的预测的影响进行了研究。用虚流体法计算了压力和表面张力法向分量。粘性和切向表面张力的处理在两个不同的方法：一个是涂抹出界面的方法，另一个是鬼流体方法（GFM）。用这些方法模拟了线性剪切流、单个振荡液滴和正弦波。主要结论如下：（1）GFM能很好地预测界面附近两相的速度梯度，而涂抹界面法的预测误差较大;（2）涂抹界面法高估了液滴形状振荡的粘性阻尼;（3）GFM和涂抹界面法都能定性地预测表面活性剂对表面张力波的阻尼，而后一种方法易于在数值上不稳定。水平集方法（Sussman等人，1994）已被广泛应用于模拟气泡、液滴、界面波等多相流，而在原有的水平集方法中，界面被建模为有限厚度的涂抹界面。因此，表面张力被计算为一个体力，它也涂抹的压力分布，并在界面附近造成一个大的寄生电流。用于水平集方法的尖锐界面方法，其被称为鬼流体方法（Kang等人，2000），由于该方法能准确处理界面处物理量的跳跃，近年来引起了人们的广泛关注。大多数最近的水平集方法利用鬼流体方法GFM来处理物理量（即压力、密度等）的跳跃（Tanguy等人，2007; Teigen等人，2010; Bjorklund，2009），而粘性应力通常通过涂抹界面法（Gibou等人，2007; Tong & Wang，2007; Yang & Stern，2007; Desjardins等人，2008; Hayashi & Tomiyama，2012 a，2012 b）。这主要是因为虚流体法在粘性应力下的数值实现比涂抹界面法复杂得多。然而，当粘性应力的跳跃条件包括由表面活性剂的存在引起的Marangoni力时，界面处的表面张力和粘性应力的准确评估是非常重要的，所述Marangoni力显著地衰减表面张力波。因此，清洁和污染界面的运动的模拟使用上述方法进行，即，的幽灵流体和涂抹出界面的方法，以证明它是如何重要的是准确评估的粘性和表面张力在界面上获得良好的数值预测的界面运动。数值方法水平集方法通过求解以下水平集方程来跟踪界面（Sussman等人，1994年）：0 = φ + φ V t（1）其中t是时间，V是速度，φ是水平集函数。界面由零水平集φ = 0表示。界面的单位法线和曲率由下式给出：

Effects of treatment of viscous and surface tension forces on predictions of interface motion are investigated. The pressure and the normal component of surface tension force are calculated by using a ghost fluid method. The viscous and tangential surface tension forces are treated in two different ways: one is a smeared-out interface method and the other is the ghost fluid method (GFM). Linear shear flows, a single oscillating drop and a sinusoidal wave are simulated by using these methods. The main conclusions are as follows: (1) the velocity gradients in the two-phases near the interface are well predicted by using GFM, whereas the smeared-out interface method gives large errors, (2) the viscous damping of drop shape oscillation is overestimated by the smeared-out interface method, (3) both GFM and smeared-out interface method can qualitatively predict the damping of surface tension wave due to surfactant, whereas the latter method is apt to be numerically unstable. INTRODUCTION Level set methods (Sussman et al., 1994) have been widely used for simulating multiphase flows such as bubbles, drops, interfacial waves and so on. The interface is modeled as a finite-thickness smeared-out interface in the original level set method. Therefore the surface tension force is calculated as a body force, by which the distribution of pressure is also smeared and a large spurious current is caused in the vicinity of interface. A sharp interface method for level set methods, which is known as a ghost fluid method (Kang et al., 2000), recently attracts much attention since this method can accurately deal with the jumps of physical quantities at the interface. Most of recent level set methods utilize the ghost fluid method, GFM, to deal with the jumps of physical quantities, i.e. pressure, density and so on (Tanguy et al., 2007; Teigen et al., 2010; Bjorklund, 2009), whereas the viscous stress is often treated by means of the smeared-out interface method (Gibou et al., 2007; Tong & Wang, 2007; Yang & Stern, 2007; Desjardins et al., 2008; Hayashi & Tomiyama, 2012a, 2012b). This is mainly because the numerical implementation of the ghost fluid method in the viscous stress is more complicated compared with the smeared-out interface method. However accurate evaluation of the surface tension force and viscous stress at the interface is of great importance when the jump condition for the viscous stress includes the Marangoni force caused by the presence of surfactant, which drastically damps the surface tension wave. Simulations of the motions of clean and contaminated interfaces are, therefore, carried out using the abovementioned methods, i.e., the ghost fluid and smeared-out interface methods, to demonstrate how it is important to accurately evaluate the viscous and surface tension forces at the interface for obtaining good numerical predictions of the interface motion. NUMERICAL METHOD Level Set Method The interface is tracked by solving the following level set equation (Sussman et al., 1994): 0 = φ ∇ ⋅ + ∂ φ ∂ V t (1) where t is the time, V the velocity, and φ the level set function. The interface is represented by the zero-level set, φ = 0. The unit normal to the interface and the curvature are given by

EFFECTS OF NUMERICAL TREATMENT OF VISCOUS AND SURFACE TENSION FORCES ON PREDICTED INTERFACE MOTION

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