Adjoint sensitivity analysis of thermo-acoustic/hydrodynamic instabilities in turbulent combustion chambers

In this paper, we de ne a mathematically consistent set of thermo-acoustic equations via asymptotic multiple scale methods in the low-Mach number limit. The nal thermoacoustic equations consist of reacting low-Mach number (LMN) equations for hydrodynamic phenomena and acoustic (AC) equations. The two sets of equations are two-way coupled. The coupling terms depend on which multiple scales are used. We derive and discuss the coupling terms for three distinct limits: double-time-double-space (2T-2S); double-time-single-space (2T-1S); and single-time-double-space (1T-2S). We linearize the thermo-acoustic equations around the mean ow, which is obtained by time averaging Large-Eddy simulations. We show that only 1T-2S provides a two-way coupled linearized system. In the other limits, the coupling from the AC to the LMN is of higher order. We perform global direct and adjoint analysis to identify unstable modes and passivefeedback mechanisms to stabilize/lower the frequency of the oscillation. Preliminary results are shown for a dual-swirl gas turbine combustor and a simpli ed dump combustor