Acoustic decoupling of longitudinal modes in generic combustion systems

Conditions are examined under which acoustic modes of a chamber filled with hot combustion products can be considered to be decoupled from the plenum acoustics supplying the fresh reactants through a feeding manifold. It is shown that this is controlled by a coupling index Ξ = (ρ_bc_b)/(ρ_uc_u)S₁/S₂ ? (T_u/T_b)^1/2(S₁/S₂), where T_u and T_b are the fresh and burned gases temperatures and S₂/S₁ is the expansion ratio between the chamber and injection unit cross sections. It is demonstrated that the acoustic response of a coupled system can be analyzed by considering the plenum and the chamber acoustics separately for small values of the coupling parameter Ξ. Longitudinal self-sustained combustion oscillations may then lock on (i) the plenum resonant frequencies, thus becoming independent of downstream modifications of the combustion chamber acoustics, or on (ii) the combustion chamber modes, thus becoming essentially indifferent to the plenum acoustics. The case of a plenum featuring a Helmholtz resonance is investigated in further detail when the chamber exhaust impedance is varied. Exact relations under which the plenum and the chamber modes are decoupled are derived when the chamber is open to atmospheric conditions or when it is equipped with a sonic nozzle. Predictions are compared to measurements for a generic system equipped with a swirl injector, a compact chamber and terminated by an open atmospheric pressure exhaust. It is shown that in this case, self-sustained longitudinal combustion-instabilities develop preferentially near the plenum mode frequencies and are weakly sensitive to modifications in the chamber geometry.