radius is half diameter. A 2 by 6 is more like 1.5 x 5.5 so a 5.5 diameter would work perfect or 2 3 inch diameter pipes. in all honesty it would be easier to make a double barrel and it will probably hold up a lot longer.
Lumber is measured before it is planed and sanded etc. So a 2x6 measures 2" by 6" before it is planed and shit. After its planed its gonna be around 1.5" by 5.5", don't believe me? Go measure on at home depot.
Even if the 2x6 really was 2" by 6", the centers of two 3" radius pipes would be 6 inches apart. That leaves no room to drill in some screws. The screws would have no wood to bite into. Making the rail really sketchy and unstable. If you're summer setup is built like that, sucks man. Sorry I didn't make a quick paint drawing. I'm on my phone.