This is a Roto-Zip tool placed in a cradle with the fluting bit protruding through a dowel.
The dowel acts as a depth gauge and allows you to follow the curve of the piece.
A Dremel type tool can also be used but the Roto-Zip allows for a 1/4" collet.

This picture shows a flat bed mounted in the tool rest.
Stops can be clamped in place if the beginning and end points of the fluting are critical.

A couple of points;

Make sure the tip of the tool is level with the lathe's centerline.
(Unless of course, you are trying for a different effect.)
I do that by aligning the tip with the point of the tailstock.
This tool can be used along the face of a piece also.
If your spindle is only 24 points, and you would like 48 grooves,
simply loosen the jaws of your chuck and rotate the piece half a space and re-tighten.
Not so easy if you're using a faceplate.

I now have a PowerMatic lathe and had to build an indexer. See Tips, then Odds and Ends for how I did that.