I have never before noticed this feature of the little fluted piece below the crown but lately have seen it on many cases. Can someone explain it's function?

The knurled ring below the crown is used to adjust the pressure on the packing under the crown. The whole assembly is called a dust-proof crown, a feature commonly found on swing-ring cases. That was the whole premise of swing-ring cases, making a watch as dust-proof as possible, by eliminating the case back/frame threaded joint and using a dust-proof crown.

Here are some prior topics that go into more detail about it:

Prior dust-proof crown topics . . .

Thanks Ed, had I remembered the word knurled I would have found it.

Lorne,

Just out of curiosity, are the examples you posted swing-ring cases?