Can weakly nonlinear theory explain Faraday wave patterns near onset?
Can weakly nonlinear theory explain Faraday wave patterns near onset?
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DOI：
10.1017/jfm.2015.388
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发表时间：
2015-08-01
影响因子：
3.7
通讯作者：
Rucklidge, A. M.
中科院分区：
文献类型：
--
作者：
Skeldon, A. C.;Rucklidge, A. M.
法拉第问题是一个重要的模式形成系统，它提供了一些中间地带的系统，其中初始不稳定性只涉及一个单一的模式，并在其中复杂性，然后从模式相互作用或二次分叉的结果，并在系统是高度湍流和许多空间和时间模式被激发的情况下。它是一个丰富的来源，新颖的模式和理论工作，旨在了解如何和为什么这样的模式发生。然而，将理论与实验联系起来是特别具有挑战性的：实验很难进行;感兴趣的参数范围（大盒子，中等粘度）沿着求解自由边界Navier-Stokes方程的技术困难使得问题的数值解很难;并且不稳定性导致整个不稳定波矢圆的事实提出了相当大的理论困难。原则上，弱非线性理论应该能够预测哪些模式在模式开始时是稳定的。在本文中，我们提出的第一个定量比较弱非线性理论的完整的Navier-Stokes方程和（以前出版的）实验结果的法拉第问题的多频强迫。我们证实，三波相互作用是复杂模式稳定的核心，但也强调了理论和实验之间的一些差异。这些都表明需要进一步的实验和理论工作，以充分调查的问题模式的双稳态和双临界/三临界点在确定分叉结构的作用。
The Faraday problem is an important pattern-forming system that provides some middle ground between systems where the initial instability involves just a single mode, and in which complexity then results from mode interactions or secondary bifurcations, and cases where a system is highly turbulent and many spatial and temporal modes are excited. It has been a rich source of novel patterns and of theoretical work aimed at understanding how and why such patterns occur. Yet it is particularly challenging to tie theory to experiment: the experiments are difficult to perform; the parameter regime of interest (large box, moderate viscosity) along with the technical difficulties of solving the free-boundary Navier-Stokes equations make numerical solution of the problem hard; and the fact that the instabilities result in an entire circle of unstable wavevectors presents considerable theoretical difficulties. In principle, weakly nonlinear theory should be able to predict which patterns are stable near pattern onset. In this paper we present the first quantitative comparison between weakly nonlinear theory of the full Navier-Stokes equations and (previously published) experimental results for the Faraday problem with multiple-frequency forcing. We confirm that three-wave interactions sit at the heart of why complex patterns are stabilised, but also highlight some discrepancies between theory and experiment. These suggest the need for further experimental and theoretical work to fully investigate the issues of pattern bistability and the role of bicritical/tricritical points in determining bifurcation structure.