From a single couple: Flow Left means that the beau walks straight ahead then does a Quarter Right as the belle follows and does a Quarter Right also. Finishes as a couple. Flow Right means that the belle walks forward and does a Quarter Left as the beau follows and does a Quarter Left also. Finishes as a couple.
NOTE: This seems to have been replaced for the most part by the Sweep a Quarter move.
Art T-Bow 1971
This definition is from The Handbook of Modern Square Dancing by Jay King.
It is provided here for informational and educational purposes only.