Either windmilling or at idle power, as when the plane is on the ground, the fan is still turning slowly enough to see the moving spiral (on the order of a few hundred RPM at most). The core rotor, which you can’t see, will be at something more like 10,000 RPM, though.
How are the front fan, and the core rotor coupled? Is it a fairly linear link of some variety?
As far as I know, they aren’t. The last stages of the compressor and the first stages of the turbine (i.e. those rotating parts closest to the combustion chamber) are on a single shaft and turn at the highest rpm. That shaft is actually hollow. Running down the center of that is another shaft, which turns slower and connects the lower pressure compressor and turbine stages. On a three-shaft turbofan, the outer two are hollow and there’s a third shaft running down the middle of both. It connects the fan (at the very front of the engine) with the very last turbine (at the back).
Is that about right, aerodave?
That’s pretty much it, Robot Arm. The fan on a turbofan may be driven by the lower spool of a 2-shaft engine, or the lowest spool of a three-shaft engine. Whether an engine has two or three shafts is a design choice that depends on a variety of factors.
But no matter what, there’s not a direct, mechanical coupling between the high-speed and low-speed shafts. That is, they are basically free to spin at independent speeds. But in practice, there will be a predictable (but not perfectly fixed or linear) relationship between them. That’s because the turbine geometry dictates what percentage of the flow’s energy will be bled off by each turbine stage. But this relationship between one shaft’s speed and another’s is also influenced by factors like the load on the engine, air speed, etc. Military engines often have variable-area exhaust nozzles, which alter the backpressure on the engine, and this has a lot of influence on the relationship between the fan and core speeds.