Characterizing Quasi-Secondary Instabilities in Compressible Flows Using the Structured Input-Output Framework

We extend the recently proposed structured input-output analysis framework to identify quasi-secondary instabilities in compressible flows. A key component of any structured input-output analysis is the uncertainty modeling, which arises due to a pseudo-linearization of the quadratic nonlinearity in the governing flow equations. The uncertainty element can be interpreted as a realization of a flow-field ‘frozen’ in time and the homogeneous spatial coordinates, which modifies elements of the base flow with which linear instabilities interact (i.e., quasi-secondary instabilities). We improve upon recently introduced methods for computing structured uncertainties that arise in the context of incompressible flows, focusing on straightforwardness and computational tractability. Results from our proposed analysis method qualitatively match prior results in the literature on incompressible Couette and channel flows. We then extend these methods for the analysis of compressible flows. Analysis of a compressible laminar Couette flow at subsonic, transonic, and supersonic conditions reveals that the spatial auto-correlation structures capture the Mach-number-dependence of the momentum and thermodynamic properties of a flow instability. For example, the specific volume auto-correlations amplify with an increase in the Mach number and thereby introduce a larger modification to the thermodynamic properties of the base flow. These findings are further corroborated by the structured forcing modes associated with the instability.