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Absolute instability in shock-containing jets

Published online by Cambridge University Press:  08 November 2021

Petrônio A.S. Nogueira Open the ORCID record for Petrônio A.S. Nogueira [Opens in a new window] ,
Peter Jordan Open the ORCID record for Peter Jordan [Opens in a new window] ,
Vincent Jaunet Open the ORCID record for Vincent Jaunet [Opens in a new window] ,
André V.G. Cavalieri Open the ORCID record for André V.G. Cavalieri [Opens in a new window] ,
Aaron Towne Open the ORCID record for Aaron Towne [Opens in a new window]  and
Daniel Edgington-Mitchell Open the ORCID record for Daniel Edgington-Mitchell [Opens in a new window]
Petrônio A.S. Nogueira*
Affiliation:
Department of Mechanical and Aerospace Engineering, Laboratory for Turbulence Research in Aerospace and Combustion, Monash University, Clayton, VIC 3800, Australia
Peter Jordan
Affiliation:
Département Fluides, Thermique, Combustion, Institut PPrime, CNRS–Université de Poitiers–ENSMA, 86036 Poitiers, France
Vincent Jaunet
Affiliation:
Département Fluides, Thermique, Combustion, Institut PPrime, CNRS–Université de Poitiers–ENSMA, 86036 Poitiers, France
André V.G. Cavalieri
Affiliation:
Divisão de Engenharia Aeronáutica, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP 12228-900, Brazil
Aaron Towne
Affiliation:
Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109, USA
Daniel Edgington-Mitchell
Affiliation:
Department of Mechanical and Aerospace Engineering, Laboratory for Turbulence Research in Aerospace and Combustion, Monash University, Clayton, VIC 3800, Australia
*
†Email address for correspondence: petronio.nogueira@monash.edu
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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.

JFM classification

Acoustics: Jet noise Compressible Flows: Supersonic flow Instability: Absolute/convective instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 930 , 10 January 2022 , A10
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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