What did fins do on the back of cars?: they looked cool.
Fins did not improve the aerodynamics, at least as far as we know, because no one put these vintage cars in a wind tunnel to measure their drag.
I feel about flutes the same way, they just look cool. I am sure performance claims for fluting are minuscule, but fluting is designed to maximize profitability for the barrel maker.
Fluting the barrel does increase stiffness if the barrels are the same weight. However, you end up with a larger diameter barrel, and you really don’t know if the increased stiffness does anything on target.
When shooters are regularly firing weapons with barrels of this diameter,
maybe the increased stiffness makes a difference on target. Considering long range shooters are burning their barrels out from 1000 to 2000 rounds, is anyone going to bank roll extensive tests, shooting out multiple barrels, to prove the more expensive fluted barrels are better? Given all the variables in boring, reaming, chamber concentricity, and bullets (bullets are a huge contributor to inaccuracy, and a huge uncontrolled variable), how does anyone isolate fluting's contribution to anything?
I would be leery of fluting broach or button rifled barrels. A barrel maker told me removing material from the outside of a broach or button rifled barrel increases bore diameter. Cut rifling does not leave the same stresses in the barrel, so fluting a cut rifled barrel should do nothing.
The heat transfer argument. So what, who cares? Who is firing their match barrel at machine gun rates? Flutes in fact turned out to be expensive vestigial features on machine guns. Sure, in the non steady state heat transfer period to a steady state temperature, air is able to get closer to the bore in a fluted barrel, and the surface area was increased, but eventually with enough rounds the tube gets hot and stays hot. Machine gunners can tell us how many rounds it takes before a barrel is too hot to touch without mittens. I suspect it ain’t long. Heat conduction through metal is good, very good. Heat convection from metal to air is slow, very slow. And that is the bugger, the limitation in effective heat transfer is not surface area, but transferring that heat to churlish air molecules. Water cooling is better, and it takes longer for a water cooled barrel to reach white hot, but eventually even water cooling can not remove the heat out faster enough to prevent over heating.
The cost of fluting, the relative ineffectiveness of cooling and the frequent replacement period is why it is not cost effective to flute machine gun barrels.
Do flutes make a cooling difference in a 20 minute, 20 round record period? I doubt it makes a difference on target.
But they look cool, just like the fins on vintage vehicles.
Is it better to look good, than shoot good?