Mathematical Methods for Chaotic Advection in Three-Dimensional Fluid Flows

三维流体流动中混沌平流的数学方法

• 批准号: 9803555
• 负责人: Igor Mezic
• 金额: $ 7.5万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2001-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803555&HistoricalAwards=false
• 关键词: Mathematical; Methods; Chaotic; Advection; Three

DMS 9803555Mathematical Methods for Chaotic Advection in Three-dimensional FluidFlowsPI: Igor MezicThe goal of the proposed research is to extend the existing theory ofchaotic advection and mixing in two-dimensional, time-dependent flows.In particular, we wish to study three-dimensional incompressible steadyand unsteady and two- and three-dimensional compressible flows usingthe methods of geometric theory of dynamical systems. We will alsostudy effects of reaction and diffusion on the motion of particles inthese flows, properties of chaotic advection in general, and particularmodels of great relevance in engineering applications: flows betweenconcentric and eccentric rotating cylinders. These problems will beaddressed starting from our recent developments in geometric theory ofthree-dimensional divergence-free vector fields. The issues ofcantori, resonances and lobe dynamics in volume-preserving maps andflows will be addressed through theoretical analysis and computersimulation. Connection with experiments on mixing in three-dimensionalflows will be made. Effects of inertia and viscosity on chaotic mixingwill be studied via asymptotic, large Reynolds number analysis inconjunction with the transport theory of dynamical systems. This willallow for discussion of the change of mixing properties of laminar,incompressible, viscous flows with the change of the Reynolds number.Further, we will study the effects of molecular diffusion on chaoticadvection based on the perturbative multiple-scales method combinedwith ergodic theory, and through numerical simulations. We willinvestigate a general relationship between the pure advection problem(possibly with chaotic mixing) and the full advection-diffusionproblem. We will study reaction-diffusion-advection equations throughthe combination of tools mentioned before. We shall pursue nonlinearstability analysis in search of effects of chaotic advection onstability. The particular examples we will be studying are flowsbetween concentric rotating cylinders. Compressible flows receivedvirtually no attention in chaotic advection studies. We propose toremedy this situation by pursuing a basic study of this problem.Simple model flows will be identified starting with a compressiblevortex flow in a cylindrical container. Comparison with the behavior ofincompressible flows will be pursued.The above study is useful in a variety of technological contexts. Thereis a recent surge of interest in micro- and nano-scale technology thatposes a number of mathematical challenges. For example, flows indevices such as large mixers and combustion chambers are designed tomix well by moving the flow in a turbulent regime and thus causingenhanced mixing by rapid random movements of fluid in the flow. This isnot possible in microscopic devices. The process of mixing needs to beunderstood much better in order to design microscopic mixers,combustion chambers etc. necessary as the building block ofmicroscopic processing devices and microengines. The mixing process insuch devices is typically three-dimensional. Thus, the design ofmicroscopic mixing devices will benefit from the fundamental study ofthree-dimensional mixing processes outlined above. In addition, theperformance of macroscopic devices is challenged by new requirementson the levels of environmental chemical pollution (NO_x) and soundpollution. The improvement in the design of these devices will be basedon a better understanding of the underlying mixing processes. The abovestudy will provide some of the crucial concepts for such design byunraveling the fundamentals of mixing in three-dimensional flows andusing these concepts to study the effect of mixing on combustion andnoise production.

DMS 9803555三维流体中混沌平流的数学方法PI：Igor Mezic本研究的目的是扩展现有的二维、含时流动中混沌平流和混合的理论，特别是我们希望用动力系统几何理论的方法来研究三维不可压缩定常、非定常以及二维和三维可压缩流动。我们还将研究反应和扩散对这些流动中颗粒运动的影响，混沌平流的一般性质，以及在工程应用中非常相关的特殊模型：同心和偏心旋转圆柱之间的流动。这些问题将从我们在三维无散度矢量场几何理论方面的最新发展开始讨论。将通过理论分析和计算机模拟来解决保体积映射和流动中的坎托利、共振和波瓣动力学问题。将与三维流动中的混合实验相联系。结合动力系统的输运理论，通过渐近的大雷诺数分析，研究惯性和粘性对混沌混合的影响。这将有助于讨论层流、不可压缩、粘性流动的混合性质随雷诺数的变化。此外，我们将基于摄动多尺度方法并结合遍历理论，通过数值模拟来研究分子扩散对混沌平流的影响。我们将研究纯平流问题(可能有混沌混合)和完全平流-扩散问题之间的一般关系。我们将通过前面提到的工具组合来研究反应-扩散-平流方程。我们将进行非线性稳定性分析，以寻找混沌平流对稳定性的影响。我们将要研究的特例是同心旋转圆柱体之间的流动。可压缩流动在混沌平流研究中几乎没有受到关注。我们建议通过对这一问题的基础研究来纠正这种情况。简单的模型流动将从圆柱形容器中的可压缩涡流开始识别。与不可压缩流动的行为进行比较。上述研究在各种技术背景下都是有用的。最近，人们对微纳尺度技术的兴趣激增，这带来了许多数学挑战。例如，大型混合器和燃烧室等流动装置通过在湍流区域中移动流动而设计得很好，从而通过流动中流体的快速随机运动来增强混合。这在微型设备中是不可能的。需要更好地了解混合过程，以便设计微型混合器、燃烧室等，作为微型处理设备和微型发动机的组成部分。这类设备中的混合过程通常是三维的。因此，微观混合装置的设计将受益于上述三维混合过程的基础研究。此外，对环境化学污染(NO_X)和声污染水平的新要求也对宏观装置的性能提出了挑战。这些设备的设计改进将建立在对基本混合过程的更好理解的基础上。上述研究将通过揭示三维流动中混合的基本原理，并利用这些概念来研究混合对燃烧和噪声产生的影响，从而为这种设计提供一些关键的概念。