Addressing afterburner screech and pressure oscillations in supersonic engines involves several strategies to stabilize combustion and reduce unwanted oscillatory behaviors:

Passive Damping Mechanisms:

Acoustic Liners: Incorporating acoustic liners or resonators along the inner walls of the afterburner can absorb sound energy at screech frequencies, damping oscillations.

Baffles and Orifice Plates: Adding flow-restrictive elements, like baffles or orifice plates, within the afterburner can disrupt pressure waves, reducing their intensity and impact on the flame front.

Active Control Systems:

Feedback Control: Using sensors and actuators, an active feedback system can monitor oscillation frequencies in real-time and make adjustments to reduce them. By modulating fuel injection or airflow, active controls can counteract oscillations as they develop.

Adaptive Fuel Injection: Adjusting fuel injection patterns (e.g., pulsed or staged injection) to vary flame dynamics can also reduce pressure oscillations, with closed-loop systems adjusting the injection based on oscillation patterns.

Optimizing Fuel and Air Mixture:

Premixed Combustion: Modifying the fuel and air mixing process can stabilize combustion and reduce the risk of screech. Premixing before the afterburner reduces the risk of localized, high-intensity combustion events.

Tuned Fuel Nozzle Design: Designing fuel injectors and nozzles to distribute fuel evenly can help achieve a more stable flame, reducing localized hotspots that can trigger oscillations.

Flame Stabilizer Design:

Enhanced Flame Holders: Using flame holders designed to stabilize combustion in supersonic flows helps to mitigate oscillations by anchoring the flame in a controlled location, minimizing its tendency to fluctuate and create pressure waves.

V-Gutters or Bluff Bodies: These structures can enhance flame stability by creating low-velocity regions where the flame can anchor, reducing the likelihood of it being impacted by oscillations.

Afterburner Geometry Modification:

Variable Area Nozzles: Implementing a variable geometry nozzle allows the afterburner to adjust its exhaust area, helping to match the changing backpressure and flow conditions, which can minimize pressure oscillations.

Enhanced Duct Design: Using gradual transitions in duct geometry rather than abrupt changes helps reduce areas where shock waves or pressure waves could amplify, lessening screech issues.

Typically, to reduce or negate the noise induced by supersonic airflow from Jet exhaust, we would include a diffuser or "hush kit", to slow and direct the airflow. Surrounding the hotter core of air with a cooler, slower moving barrier of air is one of the primary reasons that large, high-bypass turbofan engines are so quiet as compared to pure turbo-jet engines.

There may be some ways to extract heat energy from the boundary layers of the hot jet exhaust stream, which would significantly lower the amplitude and frequency of sound produced. A corrugated perimeter "fence" at the nozzle boundary might work, especially if it is a fractal pattern at 2x or 3x scale. A Khovanski fractal snowflake would be your best bet.

Hope this helps!