Spectral Energy of High-Speed Over-Expanded Nozzle Flows at Different Pressure Ratios

This paper addresses the long-standing question of understanding the origin and evolution of low-frequency unsteadiness interactions associated with shock waves impinging on a turbulent boundary layer in transonic flow (Mach: 1.1 to 1.3 ). To that end, high-speed experiments in a blowdown open-channel wind tunnel have been performed across a convergent–divergent nozzle for different expansion ratios ( PR = 1.44, 1.6, and 1.81). Quantitative evaluation of the underlying spectral energy content has been obtained by processing time-resolved pressure transducer data and Schlieren images using the following spectral analysis methods: Fast Fourier Transform (FFT), Continuous Wavelet Transform (CWT), as well as coherence and time-lag evaluations. The images demonstrated the presence of increased normal shock-wave impact for PR = 1.44, whereas the latter were linked with increased oblique λ -foot impact. Hence, significant disparities associated with the overall stability, location, and amplitude of the shock waves, as well as quantitative assertions related to spectral energy segregation, have been inferred. A subsequent detailed spectral analysis revealed the presence of multiple discrete frequency peaks (magnitude and frequency of the peaks increasing with PR ), with the lower peaks linked with large-scale shock-wave interactions and higher peaks associated with shear-layer instabilities and turbulence. Wavelet transform using the Morlet function illustrates the presence of varying intermittency, modulation in the temporal and frequency scales for different spectral events, and a pseudo-periodic spectral energy pulsation alternating between two frequency-specific events. Spectral analysis of the pixel densities related to different regions, called spatial FFT, highlights the increased influence of the feedback mechanism and coupled turbulence interactions for higher PR . Collation of the subsequent coherence analysis with the previous results underscores that lower PR is linked with shock-separation dynamics being tightly coupled, whereas at higher PR values, global instabilities, vortex shedding, and high-frequency shear-layer effects govern the overall interactions, redistributing the spectral energy across a wider spectral range. Complementing these experiments, time-resolved numerical simulations based on a transient 3D RANS framework were performed. The simulations successfully reproduced the main features of the shock motion, including the downstream migration of the mean position, the reduction in oscillation amplitude with increasing PR , and the division of the spectra into distinct frequency regions. This confirms that the adopted 3D RANS approach provides a suitable predictive framework for capturing the essential unsteady dynamics of shock–boundary layer interactions across both temporal and spatial scales. This novel combination of synchronized Schlieren imaging with pressure transducer data, followed by application of advanced spectral analysis techniques, FFT, CWT, spatial FFT, coherence analysis, and numerical evaluations, linked image-derived propagation and coherence results directly to wall pressure dynamics, providing critical insights into how PR variation governs the spectral energy content and shock-wave oscillation behavior for nozzles. Thus, for low PR flows dominated by normal shock structure, global instability of the separation zone governs the overall oscillations, whereas higher PR , linked with dominant λ -foot structure, demonstrates increased feedback from the shear-layer oscillations, separation region breathing, as well as global instabilities. It is envisaged that epistemic understanding related to the spectral dynamics of low-frequency oscillations at different PR values derived from this study could be useful for future nozzle design modifications aimed at achieving optimal nozzle performance. The study could further assist the implementation of appropriate flow control strategies to alleviate these instabilities and improve thrust performance.