With Navier Stokes in the news, sharing a 2010 SIGGRAPH paper I wrote about a clever solution that was artistically useful. These are "eigenflows". Each are stable solutions to Navier Stokes on a particular domain. If you set up one of these velocity fields, they will persist (or decay exponentially with viscosity), but won't veer off into turbulence, unless perturbed. It's an elegant formulation that has a bunch of interesting properties compared to grid based methods.
I used it to make some cool real time animations.