Dynamic Stability and Multiscale Computation of 3D Incompressible Flows.

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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.