Local Stability Analysis and Eigenvalue Sensitivity of Reacting, Bluff Body Wakes

This paper presents an experimental and theoretical investigation of high Reynolds number, low density reacting wakes near a hydrodynamic Hopf bifurcation. This configuration is applicable to the wake flows that are commonly used to stabilize flames in high velocity flows. First, an experimental study is conducted to measure the limit cycle oscillation of this reacting bluff body wake. The experiment is repeated while independently varying the bluff body lip velocity and the density ratio across the flame. In all cases, the wake exhibits a sinuous oscillation. Linear stability analysis is performed on the measured time-averaged velocity and density fields. In the first stage of this analysis, a local, spatio-temporal stability analysis is performed on the measured, time averaged velocity and density fields. The stability analysis results are compared to the experimental measurement, and demonstrate that the local stability analysis correctly captures the influence of the lip velocity and density ratio parameters on the sinuous mode. In the second stage of this analysis, the linear direct and adjoint global modes are estimated by combining the local results. The sensitivity of the eigenvalue to changes in intrinsic feedback mechanisms is found by combining the direct and adjoint global modes. This is referred to as the eigenvalue sensitivity throughout the paper for reasons of brevity. The predicted global mode frequency is consistently within 10 % of the measured value, and the linear global mode shape closely resembles the measured nonlinear oscillations. The adjoint global mode reveals that the oscillation is strongly sensitive to open loop forcing in the shear layers. The eigenvalue sensitivity identifies a wavemaker in the recirculation zone of the wake. A parametric study shows that these regions change little when the density ratio and lip velocity change. In the third stage of the analysis, the stability analysis is repeated for the varicose hydrodynamic mode. Although not physically observed in this unforced flow, the varicose mode can lock into longitudinal acoustic waves and cause thermoacoustic oscillations to occur. This paper shows that the local stability analysis successfully predicts the global hydrodynamic stability characteristics of this flow and shows that experimental data can be post-processed with this method in order to identify the wavemaker regions and the regions that are most sensitive to external forcing, for example from acoustic waves.