Applied Math Seminar
Fall Quarter 2008
Friday October 17, Special time! 4:30p.m.Tom Hou
Applied and Computational Mathematics
Caltech
On the stabilizing effect of convection in 3D incompressible flows.
We investigate the stabilizing effect of convection in 3D incompressible Euler and Navier-Stokes equations. The convection term is the main source of nonlinearity for these equations. It is often considered destabilizing although it conserves energy due to the incompressibility condition. Here we reveal a surprising nonlinear stabilizing effect that the convection term plays in regularizing the solution. We demonstrate this by constructing a new 3D model which is derived from axisymmetric Navier-Stokes equations with swirl using a set of new variables. The only difference between our 3D model and the reformulated Navier-Stokes equations is that we neglect the convection term in the model. If we add the convection term back to the model, we will recover the full Navier-Stokes equations. This model preserves almost all the properties of the full 3D Euler or Navier-Stokes equations. However, the 3D model has a completely different behavior from the full Navier-Stokes equations. We will present convincing numerical evidence which seems to support that the 3D model develop a potential finite time singularity. We will also analyze the mechanism that leads to these singular events in the new 3D model and how the convection term in the full Euler and Navier-Stokes equations destroys such a mechanism, thus preventing the singularity from forming in a finite time.
Seminar website: http://math.stanford.edu/~applmath/