2D vs. 3D Turbulence
Turbulence “… the last unsolved problem of classical physics … ”
Feynman
Turbulence is much different in 3D.
The main reason is that vortices stretch other vortices in 3D whereas they only move others around without changing
strength in 2D.
In 2D there is no tendency towards higher frequencies as there is in 3D.
Exact solution in 2D for finite set of vortices
In 2D you can write a simple program that treats a finite number of vortices which are each characterized by a location and scalar strength.
Each vortex contributes to the motion of each other vortex.
The motion is at right angles to the intervening vector and proportional to the strength of the vortex causing the motion and the reciprocal of the distance.
You can do this in a finite box or circle if you invent the right image vortices on the other side of the boundary so as to make the fluid velocity at the edge parallel to the edge.
You can also do this on a sphere if the sum of the vortex strengths is zero.
These are exact solutions to the incompressible flow equations.
This is a non-disipative system.
In this model the curl of the velocity is zero except at the vortices where it in infinite.
You can model any incompressible 2D flow with a large number of vortices.
You merely scatter them with a density proportional to the local average of the curl of the modeled flow.
There are real turbulence wizards, but I have already said most of what I know.
There is a lot of lore out there that defies clean math.
GP or Expert Systems might do well.