Dynamic Stability and Multiscale Computation of 3D Incompressible Flows.

这位研究人员和他的同事研究了流体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。二是建立一个系统的多尺度模式来模拟三维N-S方程的长时间解。三维不可压缩Euler或Navier-Stokes方程的动力学稳定性对于我们理解流体的动力学稳定性和非线性涡旋拉伸的动态耗尽有着非常重要的作用。长期以来，许多专家认为非线性涡旋伸展项主要是一种失稳项，这可能导致三维Euler或Navier-Stokes方程在光滑初始条件下出现有限时间奇性。此外，通过对解在频率空间的重新参数化和具有多尺度相函数的嵌套多尺度展开，研究者提出了一种新的三维Navier-Stokes方程的多尺度模型。我们将进行仔细的数值实验，以验证多尺度模式与直接数值模拟的一致性，并利用所提出的多尺度模式研究湍流的统计特性。许多引人入胜的自然现象，如龙卷风、飓风、台风和海啸波，都由Navier-Stokes方程描述。了解Navier-Stokes方程的解行为，并发展有效的计算方法来模拟其解，对于提高国家技术水平和社会福祉具有巨大的影响。这项研究的进展可能会提高天气预报、环境变化和自然灾害预测的能力。这是物理学和科学界尚未解决的重大问题之一。系统的多尺度分析可能导致新一代多尺度计算方法来模拟湍流，这可能对整个科学和技术产生重大影响。这个项目的另一个影响是研究生和博士后研究员的参与。所提出的研究为数学分析、物理建模和数值模拟提供了坚实的培训。他们在这个项目中接受的跨学科培训对他们在数学和科学领域的职业生涯将非常重要。

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.