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Effect of cone rotation on the nonlinear evolution of Mack modes in supersonic boundary layers

Published online by Cambridge University Press:  12 September 2023

Runjie Song Open the ORCID record for Runjie Song [Opens in a new window] ,
Ming Dong Open the ORCID record for Ming Dong [Opens in a new window]  and
Lei Zhao
Runjie Song
Affiliation:
State Key laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China
Ming Dong*
Affiliation:
State Key laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, PR China Sino-Russian Mathematics Center, Peking University, Beijing 100871, PR China
Lei Zhao*
Affiliation:
Department of Mechanics, Tianjin University, Tianjin, 300072, PR China
*
†Email addresses for correspondence: dongming@imech.ac.cn, lei_zhao@tju.edu.cn
†Email addresses for correspondence: dongming@imech.ac.cn, lei_zhao@tju.edu.cn
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Abstract

In this paper, we present a systematic study of the nonlinear evolution of the travelling Mack modes in a Mach 3 supersonic boundary layer over a rotating cone with a $7^{\circ }$ half-apex angle using the nonlinear parabolic stability equation (NPSE). To quantify the effect of cone rotation, six cases with different rotation rates are considered, and from the same streamwise position, a pair of oblique Mack modes with the same frequency but opposite circumferential wavenumbers are introduced as the initial perturbations for NPSE calculations. As the angular rotation rate $\varOmega$ increases such that $\bar \varOmega$ (defined as the ratio of the rotation speed of the cone to the streamwise velocity at the boundary-layer edge) varies from 0 to $O(1)$, three distinguished nonlinear regimes appear, namely the oblique-mode breakdown, the generalised fundamental resonance and the centrifugal-instability-induced transition. For each regime, the mechanisms for the amplifications of the streak mode and the harmonic travelling waves are explained in detail, and the dominant role of the streak mode in triggering the breakdown of the laminar flow is particularly highlighted. Additionally, from the linear stability theory, the dominant travelling mode undergoes the greatest amplification for a moderate $\varOmega$, which, according to the $e^N$ transition-prediction method, indicates premature transition to turbulence. However, this is in contrast to the NPSE results, in which a delay of the transition onset is observed for a moderate $\varOmega$. Such a disagreement is attributed to the different nonlinear regimes appearing for different rotation rates. Therefore, the traditional transition-prediction method based on the linear instability should be carefully employed if multiple nonlinear regimes may appear.

JFM classification

Boundary Layers: Boundary layer stability Compressible Flows: Supersonic flow Instability: Transition to turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 971 , 25 September 2023 , A4
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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