Dynamic Stability and Multiscale Computation of 3D Incompressible Flows.

研究者和他的同事们研究流体动力学中的两个基本问题。首先是三维不可压缩流动的动态稳定性。二是建立系统的多尺度模型来模拟三维Navier-Stokes方程的长时间解。三维不可压缩Euler或Navier-Stokes方程的动态稳定性对我们理解非线性涡旋拉伸的流体动力稳定性和动力耗竭具有非常重要的意义。长期以来，许多专家认为非线性涡旋拉伸项大多是一个不稳定项，这可能导致具有光滑初始条件的三维Euler或Navier-Stokes方程出现有限时间奇异。在本文中，研究者提出了一种利用奇异支撑和局部解结构的各向异性尺度来研究流体流动动态稳定性的新策略。此外，研究人员提出了一种新的三维Navier-Stokes方程的多尺度模型，该模型采用频率空间解的再参数化和多尺度相函数的嵌套多尺度展开。将进行仔细的数值实验来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、台风和海啸波都是由纳维-斯托克斯方程控制的。对Navier-Stokes方程解行为的理解以及模拟其解的有效计算方法的发展对提高国家技术水平和社会福祉具有巨大的影响。拟议研究的进展可能会提高天气预报、研究环境变化和预测自然灾害的能力。对涡旋拉伸的动态稳定性和动态耗竭的研究可以对不可压缩流动的大时间行为有重要的认识。这是物理学和科学中主要的开放性问题之一。系统的多尺度分析可能导致新一代的多尺度计算方法来模拟湍流，在整个科学技术领域具有巨大的影响潜力。这个项目的另一个影响是研究生和博士后的参与。本研究提供了扎实的数学分析、物理建模和数值模拟训练。他们在这个项目中接受的跨学科训练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.