Simulating Nonlinear Faraday Waves on a Cylinder
Simulating Nonlinear Faraday Waves on a Cylinder
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模拟圆柱体上的非线性法拉第波
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发表时间：
2018
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通讯作者：
S. Qadeer
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作者：
S. Qadeer
作者：Qadeer，Saad|顾问：Wilkening，Jon A|翻译后摘要：在1831年，迈克尔法拉第观察到的振动流体的表面上的驻波的形成。随后的实验揭示了存在丰富的模式状态，可以通过改变振动的频率和振幅来访问，并激发了流体力学和模式形成的大量研究。这些包括线性分析，以确定发病的模式，弱非线性的研究，以了解模式选择，和动力系统的方法来研究模式竞争和混乱的条件。最近，已经有一些工作对各种三维几何形状的数值模拟。然而，这些方法具有较低的精度，使他们不适合非线性regimes.We提出了一种新的技术，快速和准确的模拟非线性法拉第波在圆柱。从粘性势流模型出发，我们将变换场展开推广到这种几何形式，以找到拉普拉斯方程的高度非局部Dirichlet-to-Neumann算子（DNO）。发展了一种基于Zernike多项式的快速计算体势的谱方法。我们证明了这些多项式的单位圆盘上的代表功能的有效性，也表明DNO算法具有频谱精度，不像一个方法的基础上贝塞尔函数。自由表面的发展方程的时间求解使用Picard迭代进行左Radau积分。结果与线性化问题预测的不稳定阈值和表面图案完全一致。非线性模拟再现实验观察到的几个定性特征。此外，通过使人们能够在各种非线性机制之间切换，该技术允许精确确定触发各种实验观察的机制。
Author(s): Qadeer, Saad | Advisor(s): Wilkening, Jon A | Abstract: In 1831, Michael Faraday observed the formation of standing waves on the surface of a vibrating fluid body. Subsequent experiments have revealed the existence of a rich tapestry of patterned states that can be accessed by varying the frequency and amplitude of the vibration and have spurred vast research in hydrodynamics and pattern formation. These include linear analyses to determine the conditions for the onset of the patterns, weakly nonlinear studies to understand pattern selection, and dynamical systems approaches to study mode competition and chaos. Recently, there has been some work towards numerical simulations in various three-dimensional geometries. These methods however possess low orders of accuracy, making them unsuitable for nonlinear regimes.We present a new technique for fast and accurate simulations of nonlinear Faraday waves in a cylinder. Beginning from a viscous potential flow model, we generalize the Transformed Field Expansion to this geometry for finding the highly non-local Dirichlet-to-Neumann operator (DNO) for the Laplace equation. A spectral method relying on Zernike polynomials is developed to rapidly compute the bulk potential. We prove the effectiveness of representing functions on the unit disc in terms of these polynomials and also show that the DNO algorithm possesses spectral accuracy, unlike a method based on Bessel functions. The free surface evolution equations are solved in time using Picard iterations carried out by left-Radau quadrature. The results are in perfect agreement with the instability thresholds and surface patterns predicted for the linearized problem. The nonlinear simulations reproduce several qualitative features observed experimentally. In addition, by enabling one to switch between various nonlinear regimes, the technique allows a precise determination of the mechanisms triggering various experimental observations.