Absolute instability in shock-containing jets

Abstract

We present an analysis of the linear stability characteristics of shock-containing jets. The flow is linearised around a spatially periodic mean, which acts as a surrogate for a mean flow with a shock-cell structure, leading to a set of partial differential equations with periodic coefficients in space. Disturbances are written using the Floquet ansatz and Fourier modes in the streamwise direction, leading to an eigenvalue problem for the Floquet exponent. The characteristics of the solution are directly compared with the locally parallel case, and some of the features are similar. The inclusion of periodicity induces minor changes in the growth rate and phase velocity of the relevant modes for small shock amplitudes. On the other hand, the eigenfunctions are now subject to modulation related to the periodicity of the flow. Analysis of the spatio-temporal growth rates led to the identification of a saddle point between the Kelvin–Helmholtz mode and the guided jet mode, characterising an absolute instability mechanism. Frequencies and mode shapes related to the saddle points for two conditions (associated with axisymmetric and helical modes) are compared with screech frequencies and the most energetic coherent structures of screeching jets, resulting in a good agreement for both. The analysis shows that a periodic shock-cell structure has an impulse response that grows upstream, leading to oscillator behaviour. The results suggest that screech can occur in the absence of a nozzle, and that the upstream reflection condition is not essential for screech frequency selection. Connections to previous models are also discussed.

A unifying theory of jet screech

“…Further, in a consideration of twin jets, which can exhibit a range of different coupling symmetries, it has been demonstrated that only those symmetries where the G-JM is propagative are observed in experimental data . Thus the suggestion of Shen & Tam (2002) that the G-JM might be a component of jet screech appears to have been prescient; however, they suggested that it might be active only in certain mode stages, while the present evidence suggests that the G-JM is actually the upstream component of all screech stages (Nogueira et al 2022b). The G-JM only exists for a finite band of frequencies, and screech tones are observed to fall within this band, and be sharply cut off at its edges.…”

“…The result here, however, is underpinned by a fundamentally different concept. Nogueira et al (2022b) demonstrated that screech could be predicted simply through a consideration of mean-flow periodicity, without a requirement that this periodicity arose from shock cells. The present analysis supports the requirement of periodicity (implicit in the use of wavenumber to characterize the flow), but the interpretation of suboptimal peaks in the axial wavenumber spectrum is not necessarily an intuitive one.…”

“…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).…”

“…As examples of the former, both Raman (1997) and Mercier et al (2017) localize the screech source to a single shock cell, for rectangular and axisymmetric jets, respectively. However, in both cases the localization was based on an a priori assumption that screech did indeed arise at a single 'effective' source, which as Raman (1997) notes is not incompatible with How then to reconcile the success of linear models based only on the wave-like nature of the flow (Nogueira et al 2022b) with these clear observations of highly nonlinear processes such as shock leakage producing high-amplitude tones? In this paper we have provided evidence that screech mode selection is underpinned by triadic interactions between the KH wavepacket and local quasi-periodicity in the shock structures.…”

“…The ‘weakest-link’ model in results directly from the triadic interaction ; it is an approximation to the absolute instability mechanism that has been demonstrated to underpin screech (Nogueira et al. 2022 b ). This would suggest that the theoretical underpinning of the weakest-link model is fundamentally correct (despite the original conception of the model involving free-stream acoustic waves rather than G-JMs).…”