To simply answer your question..... none. Some LG class rifles used fluting, but it's the std fluting used to save weight only.
Even in HG class to where you fire normally 5-8 sigthers then 10 rds for your record string in 30-45 secs, barrels don't get that hot externally that quickly for me to warrant the expense of this type of fluting. The only time I've ever had barrel heat trouble in competition was in 2000 during the NBRSA championship match to where you fire 3 relays back-to-back with the same gun. My 338 has 2" Lilja barrel on it and it was hot then. Had a rig to pour ice water down the barrel and had ice cold towels wrapped around the outside of it. That still wasn't enough. So special fluting does increase your surface area and it does cool quicker, but does it cool quickly enough during the practical application in which the rifle is used..... I'm not sold on that a this point in time.
In regard to harmonics.... The biggest thing long barrels have to overcome upon firing (and this is based on tests done and monitored on an oscilliscope(spelling?)) is the sagging moment and getting rid of the up/down whiping of the barrel. This circular fluting only weakens the barrel in this aspect and compounds the problem. But I would be curious to see the harmonic patterns produced by this barrel though. As an engineer I never rule anything out without hard data to go by. And I'm only going by first thoughts and common sense right now. And we all know bubble bees can't fly also.
Thanks for the very good reply. I agree with all of your statements. I like the "look" of standard fluting, but as a practical appliction in regards to accuracy, I think the myriad of flutings available is hyperbole. I don't want to be disrespectful, but in my opinion it's just another way for various barrel manufacturers to "stand-out" among the the others available as a different marketing scheme. Just build an exceptional barrel at a fair price and you'll have more business than you know what to do with.
Again, these are just my opinions.
Learn from others mistakes, you won't live long enough to make them all yourself
The fluting remindes me of older machine gun barrels.(I also think they got it on 20mm and 40mm guns in WW II} I would think they would work better than regular fluting because they used them in machine guns where you fire alot more rounds. For rapid cooling of rifle barrels I noticed that JP Enterprise makes a heat sink finned aluminum block for AR-15 barrels. I recall reading their ad, that it cools faster than a fluted barrel and is lighter. Does anybody have experience with it? I was thinking of trying that radial fluted barrel for my AR-15 when my Krueger barrel is worn out.