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Absolute instability in shock-containing jets
Fluid Dynamics · Submitted 23 Jun 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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 to 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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present an analysis of the linear stability characteristics of shock-containing jets. | contribution 1.00 | 0.94 | 0.03 | asserted 0.87 |
| s1 | 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. | method 1.00 | 0.88 | 0.02 | asserted 1.00 |
| s2 | Disturbances are written using the Floquet ansatz and Fourier modes in the streamwise direction, leading to an eigenvalue problem for the Floquet exponent. | method 1.00 | 0.91 | 0.02 | asserted 1.00 |
| s3 | The characteristics of the solution are directly compared to the locally parallel case, and some of the features are similar. | result 0.94 | 0.86 | 0.03 | asserted 0.96 |
| s4 | The inclusion of periodicity induces minor changes in the growth rate and phase velocity of the relevant modes for small shock amplitudes. | result 1.00 | 0.91 | 0.04 | asserted 1.00 |
| s5 | On the other hand, the eigenfunctions are now subject to modulation related to the periodicity of the flow. | result 0.99 | 0.83 | 0.04 | asserted 1.00 |
| s6 | 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. | result 0.99 | 0.96 | 0.47 | asserted 1.00 |
| s7 | 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. | result 0.98 | 0.92 | 0.19 | asserted 0.95 |
| s8 | The analysis shows that a periodic shock-cell structure has an impulse response that grows upstream, leading to oscillator behaviour. | result 0.88 | 0.90 | 0.24 | asserted 1.00 |
| s9 | 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. | interpretation 0.99 | 0.80 | 0.44 | hedged possibility 1.00 |
| s10 | Connections to previous models are also discussed. | background 0.47 | 0.48 | 0.02 | unclear 0.71 |
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The paper
Original 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 to 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.
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How the literature uses it
2 arXiv papers cite this in 17 sentences
Papers citing it · 13 sentences without a confident label
Builds on it 1 paper
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This formulation builds upon the spatially periodic framework developed in [ref] ; here, the effect of the mean flow development can be directly estimated, providing a low-cost method for predicting the overall characteristics of the KH mode in these jets.
Wavepacket modulation in shock-containing jets · Introduction · 0.91
Uses its method or tool 1 paper
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The operators $Ã$, $𝐁̃$ are an extension of the locally parallel operators $𝐀$, $𝐁$ to the periodic case, and they can be found in [ref] .
Closure mechanism of the A1 and A2 modes in jet screech · Spatially periodic linear stability analysis · 0.87
Compares with it 1 paper
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Figure [ref] shows sample eigenspectra for the present flow at $St=0.4$ and $0.7$, where similar features to [ref] are observed.
Wavepacket modulation in shock-containing jets · Model problem: underexpanded $M_j=1.086$ jet · 0.96
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