A novel model of unsteady fluctuations of supersonic mainstream influenced by a forward-facing-step-induced shock

To develop the noise reduction strategies required for constructing a quiet wind tunnel, this paper focuses on the critical mechanism of noise amplification in the supersonic mainstream across shock waves, which are often generated by uneven wall surfaces in the wind tunnel, such as a forward-facing step (FFS). Supported by datasets from experiments and Direct Numerical Simulations (DNS), a novel theoretical model has been proposed for predicting post-shock mainstream fluctuations based on the given signals upstream of the shock. This model successfully incorporates the unsteadiness of moving shock waves. It is derived by decomposing the shock wave relationships using Reynolds decomposition and reveals that pre-shock fluctuations of pressure, Mach number, and shock angle collectively contribute to the post-shock levels of pressure fluctuations. The conclusions obtained from the theoretical analysis have been verified experimentally. Assessments demonstrate that this model can predict the amplitude of post-shock pressure fluctuations with high accuracy, with an uncertainty level below 0.21%. Meanwhile, the frequency spectrum is also a critical parameter for characterizing pressure fluctuations. The characteristic frequencies of post-shock pressure fluctuations, as derived from pre-multiplied spectral analysis, show close agreement between the theoretical predictions and DNS results. This is crucial for accurately quantifying and suppressing unsteady wave disturbances in the design of quiet wind tunnels. The unsteady oscillation characteristics of the shock induced by the FFS, as well as the disturbance characteristics in the mainstream, were studied using high-frequency pressure sensors in the experiments. The experimental pre-multiplied spectra reveal the coexistence of two distinct peaks in the mainstream fluctuations, for the current micro-step case (h/δ = 0.058). One peak corresponds to a low-frequency signal at St = 0.016 (679.8 Hz), which is attributed to the unsteady motion of the shock wave. The other is a high-frequency peak at St = 0.38 × (15.75 kHz), in close agreement with DNS results, corresponding to disturbances in the boundary layer propagating into the mainstream along the Mach line.