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Interactions between second mode and low-frequency waves in a hypersonic boundary layer

Published online by Cambridge University Press:  12 May 2017

Xi Chen Open the ORCID record for Xi Chen [Opens in a new window] ,
Yiding Zhu  and
Cunbiao Lee Open the ORCID record for Cunbiao Lee [Opens in a new window]
Xi Chen
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, Advanced Aero Engine Collaborative Innovation Center, Peking University, Beijing 100871, PR China
Yiding Zhu
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, Advanced Aero Engine Collaborative Innovation Center, Peking University, Beijing 100871, PR China
Cunbiao Lee*
Affiliation:
State Key Laboratory for Turbulence and Complex Systems, Advanced Aero Engine Collaborative Innovation Center, Peking University, Beijing 100871, PR China
*
†Email address for correspondence: cblee@mech.pku.edu.cn
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Abstract

The stability of a hypersonic boundary layer on a flared cone was analysed for the same flow conditions as in earlier experiments (Zhang et al., Acta Mech. Sinica, vol. 29, 2013, pp. 48–53; Zhu et al., AIAA J., vol. 54, 2016, pp. 3039–3049). Three instabilities in the flared region, i.e. the first mode, the second mode and the Görtler mode, were identified using linear stability theory (LST). The nonlinear-parabolized stability equations (NPSE) were used in an extensive parametric study of the interactions between the second mode and the single low-frequency mode (the Görtler mode or the first mode). The analysis shows that waves with frequencies below 30 kHz are heavily amplified. These low-frequency disturbances evolve linearly at first and then abruptly transition to parametric resonance. The parametric resonance, which is well described by Floquet theory, can be either a combination resonance (for non-zero frequencies) or a fundamental resonance (for steady waves) of the secondary instability. Moreover, the resonance depends only on the saturated state of the second mode and is insensitive to the initial low-frequency mode profiles and the streamwise curvature, so this resonance is probably observable in boundary layers over straight cones. Analysis of the kinetic energy transfer further shows that the rapid growth of the low-frequency mode is due to the action of the Reynolds stresses. The same mechanism also describes the interactions between a second-mode wave and a pair of low-frequency waves. The only difference is that the fundamental and combination resonances can coexist. Qualitative agreement with the experimental results is achieved.

JFM classification

Boundary Layers: Boundary layer stability Instability: Parametric instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 820 , 10 June 2017 , pp. 693 - 735
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© 2017 Cambridge University Press 

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