I would think an open cylinder type rocket where the rocket fuel gets injected into the "hollow" cavity that air is going through would be a more effective than the rocket fuel just burning itself. Does anyone know of any links showing data from when they tested these types of rockets?
There are several issues with using ramjets on rockets:
The startup problem. Ramjets rely on the supersonic velocity of the vehicle to create the compression needed for the combustion chamber to operate. So you'll need something (traditional motor stage, air breathing engine, etc) to get the craft moving to the point the ramjets can actually function.
Ramjets have relatively low thrusts. They don't weigh much because they have no moving parts, but they don't produce a whole lot of thrust. So a rocket would need a large number of the engines to function.
Ramjets will eventually stop working as the speed gets higher. At some point, the duct upstream of the combustion chamber will not slow down the incoming air enough. This will make the air in the combustion chamber supersonic instead of subsonic. This is what a SCRAMJET engine is used for, but that's a slightly different configuration. So either the ramjet would need to have moving parts to change its geometry to handle subsonic and supersonic flow in the combustion chamber, or you would need to shut down the ramjets and open up some scramjets (which means the rocket would now have 3 different types of engines).
Ramjets, and scramjets, are air-breathing engines. For anything that gets very high in to atmosphere, there isn't much air left. So the engines won't work very well anymore. Traditional rockets carry both their fuel and oxidizer so they work anywhere.
All of that said, there is considerable research going on in the area. The USAF is actively funding research into them, and the Russian military has a pretty cool design for some self-guiding bombs that drop off of an ICBM and fly to their targets using scramjet engines.
There has been ongoing interest in air-augmented rockets which use ramjet technology.
The main technical problems are that:
(1) They are significantly heavier than regular rockets.
(2) The need for a large air intake makes it much more complicated to design a practical frame and carry a load.
(3) The ramjet would only be useful for about 20-30 seconds of the flight, so it would have to be ejected as a stage after that, further complicating the design.
Air augmented rockets may be a good way to get into space, but the design challenges have so far been too complex for anyone to solve... so far. One factor that may help spur the development of air augmented rockets is that carbon fiber materials may help make components that are lighter than traditional structural materials.