Whether the 3D incompressible Euler equation can develop a finite time
singualrity from smooth initial data has been an outstanding open problem.
It has been believed that a finite singularity of the 3D Euler equation
could be the onset of turbulence. Here we review some existing computational
and theoretical work on possible finite blow-up of the 3D Euler equation.
Further, we show that there is a sharp relationship between the geometric
properties of the vortex filament and the maximum vortex stretching. By
exploring this geometric property of the vorticity field, we have obtained
a global existence of the 3D incompressible Euler equation provided that
the normalized unit vorticity vector has certain mild regularity property
in a very localized region containing the maximum vorticity. Our assumption
on the local geometric regularity of the vorticity field is consistent
with recent numerical experiments. Further, we discuss how viscosity may
help preventing singularity formation for the 3D Navier-Stokes equation,
and present a new result on the global existence of the viscous Boussinesq
equation.

Turbulent mixing of multi-phase flows is of great interests in many research fields. In this talk, I will present results from turbulence simulations of two-phase flows. We use a hybrid particle level set method (Enright et al., 2003) to capture the interface in the turbulent flows. Based on the phase-field model, we propose a simple way to calculate the surface tension force from the re-initialized level set function. >From direct numerical simulations, we find quantities such as the number of drops and the total circumference of the drops to scale with the surface tension at the statistical equilibrium state. In addition we report evidence for self-similar probability distribution of drop size in turbulent mixing. If time allows, I will also present an algorithm to enforce volume conservation in level set methods using ``help" from volume of fluid methods.

This is a collaboration with J. Ferziger, N. Mansour, F. Ham and M. Herrmann.