Collaborative Research: Theories and experiments on scalar mixing in chaotic flows

TangDMS-1212144 Ouellette DMS-1211952 研究人员在均匀化理论无效的极限上对标量扩散进行建模和量化。 当流中的标量由于物理或生物力而具有源或汇时，这种状态是典型的，例如污染物的释放或减轻，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。 因此，湍流各向同性和均匀性的经典假设不适用，数学上的挑战是适当地表示增强扩散（从分子运动）的相关流体运动的尺度更小或相当于流动的特征迫使规模。 研究人员的互补专业知识使得能够开发基于数学的模型，并通过数值和物理实验进行验证。 研究人员从不同的角度研究标量混合。 拓扑措施的背景搅拌，动力系统方法的基础上，用于模拟湍流扩散相干结构和测试。 此外，混合措施与源和汇的上限方法进行了研究。 总的来说，研究人员研究如何使用数学理论来解释从物理实验中观察到的不均匀，各向异性扩散过程。 与污染物（如墨西哥湾漏油和日本的放射性污染）以及与生物示踪物（如浮游生物大量繁殖）有关的输送和混合过程受到背景流搅动模式的极大影响。 这里提出的新的理论和实验研究奠定了基础，改进的数学模型和模拟标量混合的许多环境和地球物理过程，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训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，沿着本研究中使用的一些分析和计算工具，被融入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.