The Role of Global Hydrodynamic Mode Receptivity in Vortex-Acoustic Lock-On

In unstable reacting flows, 'lock-on' is defined as the transition of the frequency of dominant thermoacoustic oscillations f_d from the natural acoustic frequency f_a to that of the vortex shedding f_v as a parameter is varied, and the persistence of f_d at f_v with further change in parameter. We perform a linear global direct and adjoint stability analysis of reacting flow past a backward-facing step to quantify the role of its hydrodynamics and justify why lock-on occurs. Base flow profiles are acquired from an experiment where lock-on is observed as the inlet flow velocity is increased. Two flow rates are selected for the stability analysis - 1000lpm (F1) and 2500lpm (F2), corresponding to states before and after lock-on. We show that both flows are globally unstable, with F2 having a larger growth rate. These agree with spectral peaks from the experimental data. The receptivity of the modes to forcing depends on the adjoint mode shapes and detuning f_v-f_a. Both flows have a finite detuning, but the adjoint mode has a more distributed high amplitude region for F2 compared to F1, implying F2 is more receptive to acoustic forcing. Both these points imply that, despite a finite detuning, a highly receptive globally unstable hydrodynamic mode can act as the driver for lock-on.