Al Flipo said:
I have made up, and installed about 40 muzzle brakes over the last couple of years (on target and hunting rifles) and I can honestly tell you, that there is very little science and math behind muzzle break design.
A piece of steel screwed on the end of the barrel with a large hole drilled from side to side (like the one you see in on a tank) is just as effective as any fancy design you see advertised with ridiculous claims.
There is a whole lot of math and science involved. But like anything, you can trial-and-error it until you get something that works. But math and science will make it better.
First of all, think of why a muzzle brake works. Similar to a compressed gas cylinder that becomes a torpedo (Mythbusters comes to mind), or the recoil felt from a fire hose when the first blast of water shoots out. A jet of fluid (gas or liquid) leaving a body exhibits a force in the opposite direction.
At the muzzle, the propellant gas is done its job of accelerating the bullet, and now provides the negative side effect of causing recoil. So a hole drilled into the side will shoot gasses out at 90 degrees. Since that means less gas is leaving straight forward, it will reduce recoil. If the gas is perfectly distributed, and the gas ports are perfectly located, then the force acting left and right on the muzzle will cancel each other out. So the better your machine, machinist, tool bits, and measuring equipment, the closer to perfect it will be. Poor workmanship will actually cause your muzzle to jump left or right, though these effects may be negligeable for say a hunter shooting a dear at 100m, it may be critical to a marksman shooting a gong at 800m, etc.
Now if that gas is ported backwards, it reduces recoil by reducing the amount of gas leaving forward, but also by providing a force in the opposite direction, further reducing recoil.
Knowing that recoil isn't completely eliminated, and knowing that a rifle/muzzle will 'jump' upward due to the hold of the rifle, some muzzle brakes have ports on top to provide some downward thrust to help reduce jump.
IIRC, the force caused by the jet of gas is related to Density of the gas, Area of the exit port NORMAL to the flow of gas, and the Velocity of the gas. This accounts for the size and directions of the ports. The direction of flow determines how much of this force is used against the recoil.
The gas will lose velocity while being redirected rearward, but proper design can limit losses. Surface roughness, turn radius, and shape/geometry all affect the flow velocity.
To further improve designs, the fluid flow needs to be modeled. This will require expensive software which is better left to the larger manufaturers.
So a simple design like AlFlipo mentioned may be suitable for a lot of people, both with respect to performance and cost. A calculated and well thought-out design could be made by someone with some Fluid Mechanic and Manufacturing/Machining background. And the best design would be made by an engineering team and computer software and C&C machinery. Though I suspect that many of the fancy-a$$ muzzle breaks on the market have you paying for looks, I have no doubt that some are well designed pieces.
I assume you were looking for info on the middle ground - so your next step would be to pick up a text book from a fluid mechanics course at a college or university. I have 'Fluid Mechanics 5ed by Frank M. White' and 'Fundamentals of Fluid Mechanics 5ed by Munson, Young, and Okiishi'. Either one will give you what you need.
TM