Collaborative Research: Theories and Experiments on Scalar Mixing in Chaotic Flows

研究者在均质化理论失效的极限下对标量扩散进行了建模和量化。当流中的标量由于物理或生物力量（如污染物的释放或减少、出生和死亡过程、生物物种中的捕食者-猎物相互作用或许多其他因素）而具有源或汇时，这种状态是典型的。因此，湍流各向同性和均匀性的经典假设不适用，数学上的挑战是在较小或与流动的特征强迫尺度相当的尺度上，适当地表示相关流体运动增强的扩散（来自分子运动）。研究人员的互补专业知识使开发基于数学的模型能够通过数值和物理实验进行验证。研究者们从不同的角度研究标量混合。基于动力系统方法，对背景搅拌的拓扑测量方法进行了应用和试验，以模拟具有相干结构的湍流扩散。此外，用上界方法研究了与源和汇相关的混合措施。总的来说，研究人员研究如何使用和适应数学理论来解释从物理实验中观察到的非均匀、各向异性扩散过程。与污染物有关的运输和混合过程，如墨西哥湾漏油和日本的放射性污染，以及与生物示踪剂有关的过程，如浮游生物大量繁殖，都受到背景流中搅拌模式的极大影响。本文提出的新的理论和实验研究为改进许多环境和地球物理过程的标量混合数学模型和模拟奠定了基础；特别是，它们可以用来解决现实的环境问题。其他广泛的影响包括培养一名研究生和一名博士后研究员。此外，模拟的实验和可视化，以及本研究中使用的一些分析和计算工具，都融入了研究人员的推广工作，以吸引高中生对数学和物理的兴趣。该项目由化学、生物工程、环境和运输系统部门以及数学科学部门支持。

TangDMS-1212144OuelletteDMS-1211952 The investigators model and quantify scalar diffusion in the limit where homogenization theory is invalid. This regime is typical when the scalar in the flow has sources or sinks due to physical or biological forces, such as release or mitigation of contaminants, birth and death processes, predator-prey interactions in biological species, or many others. As a result, the classical assumptions of turbulent isotropy and homogeneity do not apply, and the mathematical challenge is to properly represent enhanced diffusion (from molecular motion) by the associated fluid motion at scales smaller or comparable to the characteristic forcing scale of the flow. The complementary expertise of the investigators enables the development of mathematically based models that are verified with numerical and physical experiments. The investigators study scalar mixing from a variety of angles. Topological measures on the background stirring, based on dynamical systems methods, are used and tested for modeling turbulent diffusion with coherent structures. Additionally, mixing measures associated with sources and sinks are studied with an upper bound approach. Overall, the investigators study how mathematical theory can be used and adapted to explain inhomogeneous, anisotropic diffusion processes observable from physical experiments. Transport and mixing processes associated with pollutants, such as the Gulf oil spill and radioactive contamination in Japan, and with biological tracers, such as plankton blooms, are greatly affected by stirring patterns in the background flow. The novel theoretical and experimental studies proposed here lay the foundation for improved mathematical models and simulations of scalar mixing for many environmental and geophysical processes; in particular, they can be used to address realistic environmental problems. Other broad impacts include training of a graduate student and a postdoctoral researcher. Moreover, the experiments and visualizations from simulations, along with some analytical and computational tools used in this research, are blended into the investigators' outreach efforts to engage high school students' interest in mathematics and physics. The project is supported by the Division of Chemical, Bioengineering, Environmental, and Transport Systems and by the Division of Mathematical Sciences.