  Axisymmetric simulations of a liquid rocket engine are performed using a
delayed detached-eddy-simulation (DDES) turbulence model with the Compressible
Flamelet Progress Variable (CFPV) combustion model. Three different pressure
instability domains are simulated: completely unstable, semi-stable, and fully
stable. The different instability domains are found by varying the combustion
chamber and oxidizer post length. Laminar flamelet solutions with a detailed
chemical mechanism are examined. The $\beta$ Probability Density Function (PDF)
for the mixture fraction and Dirac $\delta$ PDF for both the pressure and the
progress variable are used. A coupling mechanism between the Heat Release Rate
(HRR) and the pressure in an unstable cycle is demonstrated. Local extinction
and reignition is investigated for all the instability domains using the full
S-curve approach. A monotonic decrease in the amount of local extinctions and
reignitions occurs when pressure oscillation amplitude becomes smaller. The
flame index is used to distinguish between the premixed and non-premixed
burning mode in different stability domains. An additional simulation of the
unstable pressure oscillation case using only the stable flamelet burning
branch of the S-curve is performed. Better agreement with experiments in terms
of pressure oscillation amplitude is found when the full S-curve is used.
