Is there a relation between the combustion chamber pressure, and either an increase or decrease in both high and/or low frequency combustion instabilities. If so, how has this influenced engine design, and does this relation differ between different engine cycles, eg. pressure fed, full flow, expander cycle, gas generator? Any graphs would be really great!
I'm not going to do this beautiful beast of a paper justice, so by all means, feel free to dig into it yourself. It actually looks pretty readable, although processing all the math and theory would take me about an hour per page. But luckily, the highlights are at the end:
For chugging (low frequency instability):
"4. The frequency of unstable oscillation increases as the steady state chamber pressure is increased (Figure 48)."
"8. When the chamber pressure is increased, the stability of the motor is improved.
- The low frequency type of instability is found to be the predominating type in liquid propellant rockets operating under low chamber pressures and low injection pressures. Under relatively high chamber pressures, high frequency modes of the acoustic type develop concurrently, and may become preponderant in some cases"
For screaming (high frequency instability):
The discussion and conclusion section doesn't explicitly link steady-state pressure to high frequency instability. Maybe it's mentioned elsewhere in the paper, or maybe the relationship is insignificant. The closest thing to an answer for your question that I could find is
"High frequency instability can be induced in a given motor by lowering the pressure in the propellant tank which results in a simultaneous decrease of the chamber pressure, the pressure drop across the injector, and the flow rate."
However, I interpreted your question as mostly focusing on steady-state chamber pressure, whereas that statement above sounds like changing pressure during operation. I could be misinterpreting, though.