I hope to create a better version of this video soon where we can show what happens to the line a bit more. But, this should give you a good idea of how to sink the fly by using currents.
To make the fly sink, cast upstream from plunges or channels where the water gets faster, then drop the rod tip to relax line into the plunge or channel which will carry the fly down. Often the line will stop for a few seconds at the plunge, sometimes it will start spinning around, and then it will start moving downstream. Aim to keep the line tight as it moves downstream.