[Paper Review] A complex-valued resonance model for axisymmetric screech tones in supersonic jets

Matteo Mancinelli, Vincent Jaunet+2 more|arXiv (Cornell University)|Jan 1, 2021

Aerodynamics and Acoustics in Jet Flows50 references46 citations

TL;DR

This paper develops a complex-valued resonance model for axisymmetric screech tones in supersonic jets, incorporating finite shear-layer thickness and complex wavenumbers for both upstream and downstream waves. The model improves screech frequency prediction by accounting for wave growth rates and reflection coefficients, accurately capturing experimental data where simpler models fail.

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- A nd 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.

Motivation & Objective

• To improve screech tone prediction in under-expanded supersonic jets by extending a prior resonance model to include complex wavenumbers and finite shear-layer thickness.
• To investigate the effects of temperature and shear-layer thickness on screech tone resonance and wave dynamics.
• To identify reflection coefficients at upstream and downstream boundaries to close the resonance loop and enable accurate frequency prediction.
• To determine the conditions under which the resonance loop is amplified in time, a necessary condition for sustained screech.
• To validate the model against experimental data, showing improved agreement over previous models that neglect complex wave behavior and finite shear-layer effects.

Proposed method

• Formulates a linear-stability-based resonance model where both upstream and downstream waves have complex wavenumbers and frequencies.
• Incorporates finite shear-layer thickness in the jet-dynamics model, shown to be essential for accurate experimental agreement.
• Uses the resonance condition ΔkrLs + φ = 2pπ and amplification criterion eΔkiLs = |R1R2| to determine stable resonance frequencies.
• Employs a single analytical velocity profile with R/θR = 10 as an average value across axial distance and Mach number to simplify modeling.
• Identifies reflection coefficients R1 and R2 at the upstream and downstream reflection points using a combination of theoretical and experimental constraints.
• Validated against experimental data by comparing predicted screech frequencies and growth rates with measured values across varying jet Mach numbers and temperature ratios.

Experimental results

Research questions

• RQ1 How do finite shear-layer thickness and complex wavenumbers affect the accuracy of screech frequency prediction in supersonic jets?
• RQ2 What is the role of temperature variation in shifting branch and saddle points of guided jet modes, and how does this impact resonance conditions?
• RQ3 How do reflection coefficients at the upstream and downstream boundaries influence the amplification and stability of the resonance loop?
• RQ4 In what regions of the frequency-flow parameter space is the resonance loop time-amplified, and how does this relate to sustained screech tone generation?
• RQ5 To what extent does the complex-mode model improve agreement with experimental data compared to models assuming real wavenumbers and idealized shear layers?

Key findings

• Incorporating finite shear-layer thickness into the jet-dynamics model is essential for accurate prediction of experimental screech tones, as models assuming idealized or infinite-thickness shear layers fail to match data.
• The complex-mode resonance model identifies limited regions in the frequency-flow parameter space where the resonance loop is time-amplified, a necessary condition for sustained screech tone generation.
• Temperature reduction shifts branch and saddle points of guided jet modes (B(0,n) and S(0,n)) to higher frequencies, with a more pronounced effect on higher radial modes (n=2) than on the fundamental (n=1).
• The model shows that the resonance frequency is determined by the balance between phase condition (ΔkrLs + φ = 2pπ) and amplification (eΔkiLs = |R1R2|), with reflection coefficients R1 and R2 playing a critical role in stabilizing or destabilizing the loop.
• For jet Mach number Mj = 1.3, strong non-linear interaction between m = 1 and m = -1 azimuthal modes at St ≈ 0.34 generates the B screech mode in m = 0, confirmed by cross-bicoherence analysis showing unitary bicoherence levels at this frequency.
• The model’s prediction of screech frequency is consistent with the wave-interaction model (Tam & Tanna, 1982) when the phase condition p − φ*/Ns = 1 is satisfied, validating the equivalence of the two frameworks under proper parameter matching.