The interplay between fluid dynamic instability and potentially singular behavior of the 3D Euler/Navier-Stokes equations

Tom Hou, Caltech

Abstract:

Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in mathematics and fluid dynamics. In this talk, we will present a class of potentially singular solutions of the 3D Euler and Navier-Stokes equations based on our recent numerical study. An interesting feature of these solutions is that their velocity fields produce a ``tornado'' like structure. Near the center of the "tornado'', the angular velocity develops a very sharp gradient and becomes almost discontinuous. As a result, the solution approaches to a vortex sheet like structure as time evolves. Near the center of the tornado, there is a strong nonlinear alignment in the vortex stretching term, and the solution becomes increasingly singular with a scaling consistent with a finite time blow-up. However, as the thickness of the vortex sheet becomes smaller and smaller, the Kelvin-Helmholtz instability of the fluid flow eventually kicks in and destroys such nonlinear alignment, leading to the subsequent development of turbulent flow. We will also discuss the possibility of adding a regular nonlinear forcing based on feedback control to maintain the dynamic stability of the vortex sheet structure. If this could be done, it may provide a way to produce a potentially highly unstable singular solution of the 3D Euler equation in a finite time.