Nonlinear Wave Resonances in Continuous Media

连续介质中的非线性波谐振

• 批准号: RGPIN-2014-05401
• 负责人: Amundsen, David
• 金额: $ 1.02万
• 依托单位: Carleton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567147
• 关键词: Nonlinear; Wave; Resonances; Continuous; Media

Resonances are a fundamental aspect of physical phenomena in settings ranging from fluid mechanics to acoustics to optics. They are generally characterized by a constructive interaction between external forcing and natural internal frequencies. While the immediate mechanism is well understood, the long term effects of such interactions can be quite varied and depend on a multitude of factors. A robust literature has emerged over recent decades to model and investigate resonant phenomena, but numerous fundamental questions remain. Given the distinct mathematical features which arise when dispersive effects are present, this proposal has two primary and parallel themes, one where dispersive effects are present (e.g. nonlinear sloshing) and one where they are not (e.g. acoustic resonators). The focus of this proposal will be on resonances in gas and fluid settings as these represent both tractable and broadly applicable regimes. For the non-dispersive case, based on the conservation principles embodied by the Euler Equations for inviscid flow, various systems will be modelled and studied. Of primary interest will be how the qualitative nature of the long term response depends on the underlying geometric and material (density) profiles. It is well known that in some cases (e.g. shock tubes) a discontinuous response arises, while in other cases (e.g. spherical resonators) the profile, while amplified, remains continuous. The nature of the transition between these cases is not well understood. A key objective of the present proposal will be to address this fundamental question by studying and characterising the transition regime linking them. This will involve detailed analysis of the solution structure and asymptotic scalings, leading to further insights into the underlying mechanisms. Given that each regime requires a distinct mathematical treatment, this will also involve development and extension of asymptotic techniques in order to establish this connection. Once these fundamental questions are resolved, the models will be extended to include additional effects such as control and combustion, with immediate application to design strategies in various industrial settings such as engines, pumps and compressors. On the dispersive side, the primary focus of the present proposal will be on nonlinear sloshing of shallow fluids. Prior work and subsequent generalizations provided an asymptotic framework for the steady state periodic response in the limit that dispersive effects are small. However experiments have shown that while one-dimensional effects do persist in two and three dimensional settings, there are significant limitations. Therefore the first objective of the present study will be to extend the previous results to higher dimensions with a view towards understanding of resonant response in more general geometric settings. This in turn will involve further refinement of the layer based asymptotic methodology with particular attention to the underlying bifurcation structure and potential break-down near bifurcation points. Based on these fundamental investigations, the final objective will be to extend this work to allow for more generalized pitching and surging motions, with important applications, for example, to the design and operation of cargo vessels and tankers. Both the dispersive and non-dispersive aspects will involve a blend of modelling, analysis and numerical simulation. They will afford and necessitate abundant opportunity for student involvement ranging from senior undergraduate to doctoral and postdoctoral levels. They will also not only provide fundamental insights into the underlying resonant mechanisms and outcomes, but also the opportunity for direct application in an array of industrial settings.

共振是物理现象的一个基本方面，从流体力学到声学再到光学。它们通常以外部强迫和内部固有频率之间的建设性相互作用为特征。虽然直接的机制很好理解，但这种相互作用的长期影响可能是多种多样的，并取决于多种因素。近几十年来出现了大量的文献来模拟和研究共振现象，但仍然存在许多基本问题。考虑到色散效应存在时出现的独特数学特征，本提案有两个主要和平行的主题，一个是色散效应存在的地方（例如非线性晃动），另一个是不存在的地方（例如声学谐振器）。本提案的重点将放在气体和流体环境中的共振上，因为它们既易于处理，又广泛适用。对于非色散情况，基于欧拉方程所体现的守恒原理，将对各种系统进行建模和研究。主要的兴趣将是长期反应的定性性质如何取决于潜在的几何和材料（密度）剖面。众所周知，在某些情况下（例如激波管）会产生不连续的响应，而在其他情况下（例如球形谐振器），虽然放大了，但剖面仍然是连续的。这些情况之间的过渡的性质还没有得到很好的理解。本建议的一个关键目标将是通过研究和描述将两者联系起来的过渡制度来解决这一基本问题。这将涉及对解结构和渐近缩放的详细分析，从而进一步深入了解底层机制。考虑到每一种状态都需要不同的数学处理，这也将涉及到渐近技术的发展和扩展，以建立这种联系。一旦这些基本问题得到解决，这些模型将扩展到包括控制和燃烧等额外影响，并立即应用于各种工业环境（如发动机、泵和压缩机）的设计策略。在色散方面，本建议的主要焦点将放在浅层流体的非线性晃动上。先前的工作和随后的推广为色散效应很小的极限下的稳态周期响应提供了一个渐近框架。然而，实验表明，虽然一维效果在二维和三维环境中确实存在，但存在明显的局限性。因此，本研究的第一个目标是将以前的结果扩展到更高的维度，以期理解更一般的几何设置中的共振响应。这反过来将涉及进一步改进基于层的渐近方法，特别注意潜在的分岔结构和分岔点附近的潜在故障。基于这些基础研究，最终目标将是扩展这项工作，以允许更广泛的俯仰和浪涌运动，并具有重要的应用，例如货船和油轮的设计和操作。色散和非色散方面都将涉及建模、分析和数值模拟的混合。他们将为学生提供丰富的参与机会，从高级本科到博士和博士后水平。它们不仅将提供对潜在共振机制和结果的基本见解，而且还将提供在一系列工业环境中直接应用的机会。