Dynamic Stability and Multiscale Computation of 3D Incompressible Flows.

这位研究者和他的同事研究了流体动力学中的两个基本问题。第一个问题是三维不可压缩流动的动力稳定性。第二部分是建立一个系统的多尺度模式，模拟三维Navier-Stokes方程的长时间解。三维不可压Euler或Navier-Stokes方程的动力学稳定性对于我们理解流体动力学稳定性和非线性涡拉伸的动力学损耗有着非常重要的作用。长期以来，许多学者认为非线性涡拉伸项主要是一个失稳项，它可能导致具有光滑初始条件的三维Euler或Navier-Stokes方程的有限时间奇异性，本文提出了一种利用奇异支撑的各向异性尺度和局部解结构来研究流体流动动力稳定性的新策略.此外，研究者提出了一个新的多尺度模型的三维Navier-Stokes方程通过使用在频率空间和嵌套的多尺度扩展与多尺度相函数的解决方案的重新参数化。我们将进行细致的数值实验，以验证多尺度模型与直接数值模拟的对比，并使用所提出的多尺度模型研究湍流的统计特性。许多迷人的自然现象，如龙卷风、飓风、台风和海啸波都由Navier-Stokes方程控制。了解Navier-Stokes方程的解的行为和开发有效的计算方法来模拟它们的解，对提高国家技术和社会福祉有着巨大的影响。拟议研究的进展可能会提高天气预报，研究环境变化，对涡拉伸的动态稳定性和动态损耗的研究，对不可压流动的大时间行为有重要的启示。这是物理学和科学中的一个主要开放问题。系统的多尺度分析可能会导致新一代的多尺度计算方法来模拟湍流，并可能对整个科学和技术产生重大影响。该项目的另一个影响将是研究生和博士后研究员的参与。该研究为数学分析、物理建模和数值模拟提供了坚实的基础。他们在这个项目中接受的跨学科培训将对数学和科学的职业生涯非常重要。

The investigator and his colleagues study two fundamental problems in fluid dynamics. The first one is the dynamic stability property of 3D incompressible flows. The second one is to derive a systematic multiscale model to simulate the long time solution of the 3D Navier-Stokes equations. The dynamic stability property of the 3D incompressible Euler or Navier-Stokes equations plays a very important role in our understanding of the fluid dynamic stability and dynamic depletion of nonlinear vortex stretching. For a long time, many experts had believed that the nonlinear vortex stretching term is mostly a destabilizing term, which may lead to a finite time singularity of the 3D Euler or Navier-Stokes equations with a smooth initial condition.In this proposal, the investigator proposes a new strategy to study the dynamic stability of fluid flows by exploiting the anisotropic scaling of the singular support and the local solution structure. Furthermore, the investigator proposes a new multiscale model for the 3D Navier-Stokes equations by using a reparameterization of the solution in the frequency space and a nested multiscale expansion with a multiscale phase function. Careful numerical experiments will be performed to validate the multiscale model against direct numerical simulations and study the statistical properties of turbulent flows using the proposed multiscale model.Many fascinating natural phenomena such as tornadoes, hurricanes, typhoons, and tsunami waves are governed by the Navier-Stokes equations. The understanding of the solution behavior of the Navier-Stokes equations and the development of efficient computational methods to simulate their solutions have a tremendous impact in improving the national technology and for the well-being of the society.The advances in the proposed research could potentially improve the ability in weather forecasting, studying environmental change, and in predicting natural disasters.The proposed study on the dynamic stability and dynamic depletion of vortex stretching could lead to important insights on the large time behavior of the incompressible flows. This is one of the major open problems in physics and science. A systematic multiscale analysis could lead to a new generation of multiscale computational method to simulate turbulent flows, with potential for great impact throughout science and technology. An additional impact of this project will be the involvement of graduate students and postdoctoral fellows. The proposed research provides a solid training in mathematical analysis, physical modeling and numerical simulation. The interdisciplinary training they receive in this project will be very important for careers in mathematics and science.