A complex-valued resonance model for axisymmetric screech tones in supersonic jets

Abstract

We model the resonance mechanism underpinning generation of A1 and A2 screech tones in an under-expanded supersonic jet. Starting from the resonance model recently proposed by Mancinelli et al. (Exp. Fluids, vol. 60, 2019, p. 22), where the upstream-travelling wave is a neutrally stable guided-jet mode, we here present a more complete linear-stability-based model for screech prediction. We study temperature and shear-layer thickness effects and show that, in order to accurately describe the experimental data, the effect of the finite thickness of the shear layer must be incorporated in the jet-dynamics model. We then present an improved resonance model for screech-frequency predictions in which both downstream- and upstream-travelling waves may have a complex wavenumber and frequency. This resonance model requires knowledge of the reflection coefficients at the upstream and downstream locations of the resonance loop. We explore the effect of the reflection coefficients on the resonance model and propose an approach for their identification. The complex-mode model identifies limited regions of frequency–flow parameter space for which the resonance loop is amplified in time, a necessary condition for the resonance to be sustained. This model provides an improved description of the experimental measurements.

A unifying theory of jet screech

“…It is worth emphasizing, however, that there need not be a sharp peak in the shock spectra to underpin resonance; all wavenumbers will transfer energy to other wavenumbers via triadic interaction with the KH wavepacket. This transfer is a necessary, but insufficient component of jet screech; there must be a neutral or weakly damped G-JM at this wavenumber and frequency to then transport this energy back upstream (Gojon et al 2018;Gojon, Gutmark & Mihaescu 2019;Mancinelli et al 2021), and in turn a sufficiently unstable KH wavepacket, etc. In essence, these triadic interactions simply form one necessary component of the overall global stability of the system (Nogueira et al 2022b).…”

“…The effective source region was held at n = 5 for both cases, on the basis of a subjective observation of apparent sources in the pressure fields of the numerical dataset. An extension of this approach can be found in the model of Mancinelli et al (2021), which reduces the empiricism underpinning the approach of Gao & Li (2010) by calculating wave properties via stability theory. This model achieves excellent agreement with experimental data for both the A1 and A2 modes with the variation of only one model parameter, which is once again the number of concurrent upstream-and downstream-propagating waves.…”

“…To return to the question of mode staging, without any evidence that a change in the nature of the upstream wave is responsible, a clear picture of mechanism is still lacking. While models such as that presented in Mancinelli et al (2021) can accommodate modal staging, they do so by changing the concurrent number of waves in the resonance loop; why this might occur remains unclear.…”

“…Central to the model of Mancinelli et al. (2021) is the idea that the upstream-propagating wave is not a free-stream acoustic wave, but a guided-jet mode (G-JM) first described in Tam & Hu (1989). That this G-JM could close the resonance loop in screech was first proposed in Shen & Tam (2002), who suggested that a switch between a free-stream acoustic wave and the G-JM could explain the A1–A2 mode staging.…”

“…While models such as that presented in Mancinelli et al. (2021) can accommodate modal staging, they do so by changing the concurrent number of waves in the resonance loop; why this might occur remains unclear.…”