A numerical study on parasitic capillary waves using unsteady conformal mapping
A numerical study on parasitic capillary waves using unsteady conformal mapping
复制标题
使用非定常共形映射对寄生毛细波进行数值研究
DOI：
10.1016/j.jcp.2016.10.015
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
W. Choi
中科院分区：
文献类型：
--
作者：
S. Murashige;W. Choi
本文描述了完全非线性计算的寄生毛细波的非定常运动出现在前端的陡峭的重力波前进的无限深的水，在无旋平面流的框架内。作为广泛使用的混合欧拉-拉格朗日（MEL）时间更新边界积分方法的替代方案，我们重点介绍了一种基于非定常保角映射的数值方法，以下将其称为非定常速端图变换（UHT）方法。在该方法中，我们求解非线性发展方程，以找到一个非定常保角映射在一个复杂的平面上的流动域映射到单位圆盘上，而自由表面是固定的单位圆。本工作的目的是比较的UHT方法和MEL方法，并找到一个更有效的方法来计算寄生毛细波。通过线性稳定性分析，发现两种方法的一个关键区别在于奇异积分中余切函数的核，而UHT方法可以避免由此带来的数值不稳定性.数值算例表明，UHT方法比MEL方法更适用于寄生毛细波和毛细支配波.特别地，在这些示例中，UHT方法不需要人工技术（诸如滤波）来控制数值误差。此外，这两种方法之间的另一个主要区别是观察在自由表面上的样本点的聚类属性，取决于波的恢复力（重力或表面张力）。
This paper describes fully nonlinear computation of unsteady motion of parasitic capillary waves that appear on the front face of steep gravity waves progressing on water of infinite depth, within the framework of irrotational plane flow. As an alternative to the widely-used boundary integral method with mixed-Eulerian–Lagrangian (MEL) time updating, we focus on a numerical method based on unsteady conformal mapping, which will be hereafter referred to as the unsteady hodograph transformation (UHT) method. In this method, we solve the nonlinear evolution equations to find an unsteady conformal map in a complex plane with which the flow domain is mapped onto the unit disk while the free surface is fixed on the unit circle. The aim of this work is to compare the UHT method with the MEL method and find a more efficient method to compute parasitic capillary waves. From linear stability analysis, it is found that a critical difference between these two methods arises from the kernel of cotangent function in singular integrals, and the UHT method can avoid some numerical instability due to it. Numerical examples demonstrate that the UHT method is more suitable than the MEL method for not only parasitic capillary waves, but also capillary dominated waves. In particular, the UHT method requires no artificial techniques, such as filtering, to control numerical errors, in these examples. In addition, another major difference between the two methods is observed in terms of the clustering property of sample points on the free surface, depending on the restoring force of waves (gravity or surface tension).