Ok, I'm still not entirely clear on your goal, but let's see:
First, your opening (in your pic) would more than likely be blocked by the material of the barb, so you'll have to adjust it, or use a bigger barb (5/8?).
The main problem with jet inpingement is that it requires a lot of pressure, so a good pump. Because of the type of pumps we use, we're seriously limited in the size of the nozzle we can use.
Your arc may have the equivalent opening as a 1/4 round hole, and your pump may not be powerfull enough to make a jet that will be fast enough.
Now, for what I didn't get into... Why on earth would you use an odd shape like that? If you want to keep your design secret, that's fine, but you might consider using multiple smaller jets. It's an option.
The hyperbolic curve would be impossible to implement on that curved opening: it would cut into the barb's thread. You might consider using a much larger barb, so that there's room for a nice curved entry point. Maybe 1 inch? (add a reducer). I've used 1 1/2
I'll run the numbers, with the information that you gave me.