A while back I was learning about doing high quality approximate fluid simulations. If you take the curl of the Navier-Stokes equation and quantize (perhaps not the right word), you can simulate a collection of vortex particles that generate the velocity vector field, which is then used to advect those particles to get the next simulation step. Constructing the field is expensive and (now I know what it's called) you can use the multipole method to approximate it, since clusters of vortex particles, from far enough away, behave like a single particle whose vorticity is the sum of the vorticities of the particles.

I could never get the boundary conditions right (it involves generating vortex particles to neutralize torques), but it was still a lot of fun. The benefit of this method is that energy can be conserved and fine scale structures remain mostly intact, which is important for turbulent flows.