Several years back I suggested a type of compensating nozzle that should be inexpensive to build and test. Unfortunately the ones I made for demonstration with compressed air were hit and miss as I didn’t have the theory quite right. Hit and miss is not good enough for serious companies, so I mostly dropped the idea for a while. I thought a few of my acquaintances in the business might do something with the idea for a while. Now I think the idea has been mostly forgotten as unworkable.
A few years back I did finally find the missing part of the concept and did nothing about it as I thought at the time that others had picked it up and moved on. Since I don’t think that has happened, I am going to repost the concept.
On the left is the engine with the notch showing on the right side. The notch allows the atmosphere to enter the bell to compensate for over expansion at lower altitudes. At higher altitudes and in vacuum the exhaust gradually uses the whole bell with some losses through the notch. This will allow a nozzle to be optimum at sea level when most are over expanded. It will also be nearly as good as a full diameter high expansion nozzle in regimes with the exhaust under expanded.
The missing ingredient in the prior concept was appreciation for the momentum of the exhaust at the notch site. The momentum, especially with the rounded notches that I was advocating before, would prevent the atmosphere from entering the notch in a controlled manner. The addition of a sharp edge at the notch to assure a clean break and a slight reverse on the notch edge to direct the exhaust inwards controls the momentum of the exhaust in a manner that allows the atmosphere to interact and provide pc/pa compensation at a range of back pressures.
A compensating nozzle allows lower pressure engines to operate more efficiently in a launch vehicle. They should allow a payload increase of 1-5% depending on the vehicle and the assumptions going in. For a VTVL that wants to operate at very low pressures in the landing phase, a compensating nozzle would be a very important upgrade, though the successes of Blue and SpaceX take some of the edge off that argument.
This is a public domain concept as I described it here years ago. So anyone that wants to see what I am talking about can build a quick and dirty nozzle to use with shop air. The ones have done were an air chuck and fiberglass. About $10.00 in materials. I know it works at 135 psi. Then you can try a higher pressure gas if it might be useful to you or someone you know.