Okay, so the RANS equations use the idea that we can generate a statistical equation of motion from the Navier Stokes equations by using the idea that the velocity field is split into a mean field and a fluctuating field. I've had a great deal of trouble understanding this. From a purely physical perspective, there is one velocity field and, if turbulent, we are taught it contains these "eddy" like parts to it, which come in various shapes and sizes. So from the statistics definition, how are these eddies sort of "split up"? Is it the case that the mean velocity field is treated as purely laminar and the fluctuating part contains all these "eddies" or "turbulence" or is it that maybe large eddies are contained in the mean velocity field and the smaller eddies in the fluctuating field? I've heard a few people talk about the "fluctuating component" being the turbulence, but it confuses me because there is only one velocity field... As another question which has stemmed from this, what therefore, is the significance of the idea that the turbulence "extracts" its energy from the mean flow? We see the equations for the transport of kinetic energy for instance. How should I interpret this when there is, quite literally (physically), only one velocity field, i.e. one instantaneous value of velocity in time and space (assuming N-S is smooth and continuous for all time)? Sorry if its a weird question, its just been bugging me quite a lot, because a lot of papers I'm reading go off on a merry dance with the statistics, and its very difficult to interpret certain things.