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Instability of shock train behaviour with incident shocks

Published online by Cambridge University Press:  01 December 2020

Nan Li ,
Juntao Chang Open the ORCID record for Juntao Chang [Opens in a new window] ,
Kejing Xu ,
Daren Yu  and
Wen Bao
Nan Li
Affiliation:
School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, PR China
Juntao Chang*
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin150001, PR China
Kejing Xu
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin150001, PR China
Daren Yu
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin150001, PR China
Wen Bao
Affiliation:
School of Energy Science and Engineering, Harbin Institute of Technology, Harbin150001, PR China
*
†Email address for correspondence: changjuntao@hit.edu.cn
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Abstract

In a back pressured duct with incident shocks, the shock train exhibits violent oscillations or even a rapid movement when it passes through a shock-wave–boundary-layer interaction (SWBLI) region. In this study, the dynamics of a shock train system was investigated. Linear stability analysis was used to identify the underlying cause of the unstable behaviour. Results from the eigenvalue analysis indicated that as the shock train enters the SWBLI region, the divergent vibration, which is the outcome of a Hopf bifurcation, emerges. An analysis based on the feedback mechanism identified a criterion for this instability, i.e. the sign of the gradient of the maximal pressure that the boundary layer can sustain. Different unstable motions were also investigated according to the condition of the non-existence of a limit cycle. These motions were associated with the speed of the shock train and the configurations of the flow parameter gradients. It was shown in the controllability matrix that the rapid movement is uncontrollable, which indicates that there is a low correlation between the shock train motion and the flap actuator in the SWBLI region. However, for the remaining part of the unstable motion, a fast-response actuator is required. According to the observability analysis, the shock train movement contributes more to the variation in the pressure behind the first separation shock than the backpressure further downstream, which confirms that monitoring the pressure change along the tunnel is a better method for shock train detection rather than a polynomial model using the backpressure.

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Compressible Flows: Shock waves Boundary Layers: Boundary layer separation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 907 , 25 January 2021 , A40
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Li et al. supplementary movie 1

The entire movement of the shock train at Mach 1.85 shown in figure 2.

Download Li et al. supplementary movie 1(Video)
Video 8.6 MB

Li et al. supplementary movie 2

The entire movement of the shock train at Mach 2.70 shown in figure 2.

Download Li et al. supplementary movie 2(Video)
Video 9 MB