2D vs 3D simulation
 

Can anyone tell me what is different between 2D and 3D simulation? (about math, not complexity)

For large reynolds numbers, there is no turbulence due to absence of vortex tube stretching in 2D (i.e. there is no turbulence in 3D sence there, but an inverse cascade). The 2D equation for vorticity is just dw/dt=0, that is why 3D simulation is needed (in absence of turbulence moelling, which is often doubtful) to get more or less meaningful results in CFD.

2D simulation is just an idealization of 3D simulation. Consider a flow over a delta wing. Usually, the delta wing taper towards wing tip. If one makes several 2D cuts along spanwise direction, the flow field for each of these cuts will look different. Now, consider a rectangular wing that does not taper towards the tips and has same cross-section all along the span. If the spanwise length is "long enough", and one makes similar 2D cuts along spanwise direction again, the flow field for each of the cuts (not very near the wing tip) will look the same. Ideally, such a flow can be simulated as a 2D flow, because the flow field is independent of one of the directions (in this case spanwise direction).

You should imagine the third dimension of your 2D simulation as infinite. All the properties are constant over this third dimension (dudz = 0, d2udz2=0 dpdz=0 enz..).