Hey Grant,
The reason afterburners use flame holders is because the turbulent flame speed of kerosene (other fuels very similar) is at best case just below 4 m/s. G.A. Richards and Arthur Lefebvre measured the turbulent flame speeds when they were at Purdue University in the 1980's. The results are shown in a paper, "Turbulent Flame Speeds of Hydrocarbon Fuel Droplets in Air", Combustion and Flame 78, pp. 299-307. Note that their results are for air, while your application is injecting into combustion exhaust (termed "vitiated air"), so you can expect to get turbulent flame speeds below the near-4 mm/s mark that they measured.
So with the flame only being able to propagate in the fuel air mix at 4 m/s or less, and the mean gas speed in the jet exhaust being measured in the hundreds of m/s, you can see how the flame would just be blown out...just like a candle when you blow on it. So, by using a device such as a bluff body to create a region of very low flow velocity in it's wake, you give the flame a point with which to stabilize, just like cupping your hand around a candle in the wind. Reducing the mean exhaust velocity is always a good thing when adding heat to a internal flow like a combustor or a afterburner (see this link:
jetandturbineowners.proboards.com/post/8741 ), as the common approach to increase the AB diameter is a good way to go too.
You would want to place the injector upstream and inline with the flameholder because you want the injected fuel to have time to evaporate and get pulled into the recirculation region behind the flameholder...this will give you the best likelihood of achieving stable combustion. The smaller the droplet size that you can introduce into the exhaust stream, the better the mixing and the quicker the evaporation of the droplets will be...so an atomizer might help you get a shorter afterburner length, but being in the exhaust stream might make it coke up....worth a try though.
The other big consideration is the nozzle exit area. Since you are adding heat to the exhaust jet, you will be reducing it's density...ideal gas equation,
P = density * Rspecific * T
rearranged to --->
density = P / (Rspecific * T)
...we know that the conservation of mass tells us that mass flow will be constant...so,
mass flow = density * velocity * area
rearranging,
velocity = mass flow / (area * density)
From this equation you can see that if the density drops and the mass flow and area stay the same, the velocity has to go up....great, that is what we want.
But, it takes pressure to push that amount of mass out of the exhaust nozzle and we only have a certain amount of available pressure as dictated by the compressor performance and the various losses along the way such as the turbine pressure drop, combustor pressure losses, etc. In addition, there is an upper limit on what we can push out of the nozzle because of the compressibilty effects of the gas flow. When the nozzle throat reaches the speed of sound (relative to the local gas conditions, not the outside air), we can't push anymore mass through and the flow is said to be "choked". Usually, the nozzle is designed to run choked because that gives the highest exit velocity and thus the largest momentum increase to the flow.
Nozzles are designed often in terms of nozzle choking flow rate parameter,
w6*SQRT(T6)/(AN * P6) (where the nozzle is the 6th element in the flowpath, thus the 6, w6 = nozzle mass flow, AN = nozzle area)
So you can see, that to maintain a similar nozzle flow parameter as a means to not upset the existing nozzle pressure ratio and system mass flow, if you increase the nozzle inlet temperature, you must increase the nozzle flow area. This is why turbojet engines with afterburner systems have variable area exhaust nozzles.
On the bigger picture, the nozzle pressure ratio affects the turbine pressure ratio which affects the compressor pressure ratio and that is where the problem gets worse. If your nozzle is not sized correctly, the use of the afterburner can cause the compressor to surge because your compressor operating line is shifted up, closer to the surge line. If you don't have enough margin or your nozzle is too restrictive, you will actually push the compressor operating line into the surge region and cause surge.
Older jet fighters would experience this in combat situations when they would slam the throttle into full AB too quickly, the engine would surge and often require dropping to lower altitudes and shutting down the engine to clear the surge. Once restarted, the engine would operate normally. There were hydro-mechanical devices added to delay the AB light-off because the pilots seemed to complain about having to restart the engines in combat situations for some reason...hehee!
I look forward to seeing your progress!
~ Chris