Flopping a straight, missed value on paired board OTR?

not on NL2, nor anywhere else

Is indeed a standard value bet on the river and if he shoves on you there is still room to fold, but that would be tough.

Cheers,
SDK1987

i think he have 2 hearts that why he call and check the river fore he have miss

I’m wondering what sizing would be optimal as a river bet?

I dont think the flop and turn need to be so big what sizing are you taking with the rest of your range ? i think id go more like .8 otf then .22 ott and .50 otr leaving room to fold if needs be although still tough.

As you block flush draws with the Ah villains range should be mostly QT KT TJ QJ KJ KQ and some worse kx Qx. The bare flush draws likely fold the turn so were targetting 1 pair and 2 pair hands for value so should make out bet size accordingly

Flop cbet is too big. You could consider slowplaying your hand holding the A♥:, it is the best slowplay candidate in your range (not needed though).
Turn is too big.
River check makes no sense. You should notice villain is capped. He doesn't have KK, JJ, QQ, KQ in his flatting range vs SB usually. 44 would definitely raise the turn. KJ is possible (but river is nasty for this hand).
You block missed flushdraws. You block weak straights.
QJ beats you now.
The problem here was betsizing throughout the hand. You can't really find a suitable size for the river when you bet pot OTF and OTT on a board where villain is totally capped and you are not. That said, river bet must be another big bet, although I don't think you'll get called by worse ever (unless he is a monster fish, which is why everyone advocates betting the river and I specifically advocate a big size). Maybe QT finds a call if not holding hearts, but would be a hero call most of the times (both OTT and OTR).

Sometimes, you could also try exploitative lines in NL2 against the typical NL2 player pool.

From OOP, you could push your made hand. To your opponents holding top pair or decent flush draws (in this case, only Q-high as you're holding the ace), it looks like that you're trying to push him out with a flush draw or just a decent pair and some draws. Pushing here instead of betting the pot looks more suspicious.

Or, you could bet smaller on turn, like 1/4 or 1/3 pot, and bet big on river. The aim here is to get raised on the turn. If raised on the turn, push. If called and raised on the river, just fold. This line is asking your opponent what you could do, instead of figuring it out yourself. One downside of this line is that you would be often called by weak flush on river.