Another stack exchange answer odds I'll hit my flush says that if you start with a flush draw, the odds of hitting on either the turn OR the river are about 35%:
Hitting on turn: 9/47 = 19.15%
Hitting on river (if you didn't hit on turn): 9/46 = 19.57%
Hitting on either turn or river: 19.15% + (19.57% * (1-0.1915)) = 35%
But what about if you started with just 2 hearts in your hand and one heart on the flop? In this case you have to get a heart on BOTH the turn AND the river.