I’m not sure where the name came from, but in my opinion it doesn’t look like a flying movement more like barrel hugging.  The dumbbell fly can be performed in many variations, depending on how you want to build your chest.  First there is the angle you want on the bench.  If you are looking to improve your upper chest, then an incline of 35 to 45 degrees is necessary, over 45 degrees you will work more of the shoulders.  To work the belly of the chest and to get the outer dimples to indent then flat bench is key, and to hit the lower chest, any decline lower than flat bench will do that.  Next is to focus on the fly movement.  Many lifters tend to go to heavy and end up doing a press instead.  When performing a fly you want to visualize yourself hugging a barrel or punching bag and squeezing.  I like to start at the top touching the end of my dumbbells, then lowering out as far as I can stretch until my arms are parallel with the floor.  Don’t go past that point or you could injure your pectoral or rotator cuff.  Keeping your elbows slightly bent then raising them up until you touch the dumbbells again.  There are many variations that you can use for gripping or turning the dumbbells to the hit the chest differently also. 

Normal- the normal fly positon begins with your palms facing each other, and then lowering your arms with slightly bent elbows to a parallel position with the floor.  Then raise them up in a semi-circular motion (not pressing) as if you are holding a punching bag until you touch the dumbbells together.
Parallel-The movement is the same as normal, except you start by touching the ends of the dumbbells together where your palms are facing down towards your feet.  Lower until you are parallel to floor then in a semi-circular motion with elbows slightly bent squeeze the chest to bring them back to touch together again at the top.
Supinating-The movement is also the same as the previous two but the different is that your start at the top like a parallel fly touching the end of the dumbbells together, then as you lower you twist the dumbbells into a normal starting position where palms are facing each other than when you raise them back up twist until they are touching at the ends again.
Tips- Instead of dumbbells, you can also use cable machines with handles, Bowflex, TRX, or Pek Decks to simulate the same exercises, and if you don’t have dumbbells at home don’t be afraid to fill up some water jugs (8lbs).

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