Responses to disturbance of supersonic shear layer: Input-output analysis

We investigate the perturbation dynamics in a supersonic shear layer using a combination of large-eddy simulations (LES) and linear-operator-based input-output analysis. The flow consists of two streams - a main stream (Mach 1.23) and a bypass stream (Mach 1.0) - separated by a splitter plate of nonnegligible thickness. We employ spectral proper orthogonal decomposition to identify the most energetic coherent structures and bispectral mode decomposition to explore the nonlinear energy cascade within the turbulent shear-layer flow. Structures at the dominant frequency are also obtained from a resolvent analysis of the mean flow. We observe higher gain at the dominant frequency in resolvent analysis, indicating the dominance of Kelvin-Helmholtz (KH) instability as the primary disturbance energy-amplification mechanism. To focus on realizable actuator placement locations, we further conduct an input-output analysis by restricting a state variable and spatial location of an input and output. Various combinations of inputs and output indicate that the splitter plate trailing surface is the most sensitive location for introducing a perturbation. Upper and lower surface inputs are less influential in modulating wavepackets in the shear layer but introduce pressure instability waves in the main and bypass streams, respectively. The analysis reveals that the phase speed of pressure waves depends on the state variable and input location combination. For all combinations, the KH instability plays a key role in amplification, which reduces significantly as the input location is moved upstream relative to the splitter plate trailing edge. Furthermore, two-dimensional nonlinear simulations with unsteady input at the upper surface of the splitter plate show remarkable similarities between pressure modes obtained through dynamic mode decomposition and those predicted from linear input-output analysis at a given frequency. This study emphasizes the strength of linear analysis and demonstrates that predicted coherent structures remain active in highly nonlinear turbulent flow. The insights gained from the input-output analysis can be further leveraged to formulate practical flow control strategies.