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Linear, nonlinear and transitional regimes of second-mode instability

Published online by Cambridge University Press:  28 October 2020

S. Unnikrishnan Open the ORCID record for S. Unnikrishnan [Opens in a new window]  and
Datta V. Gaitonde Open the ORCID record for Datta V. Gaitonde [Opens in a new window]
S. Unnikrishnan*
Affiliation:
Mechanical Engineering, Florida State University, Tallahassee, FL32310, USA
Datta V. Gaitonde
Affiliation:
Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH43210, USA
*
†Email address for correspondence: usasidharannair@fsu.edu
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Abstract

The evolution of the potent second-mode instability in hypersonic boundary layers (HBLs) is examined holistically, by tracking its linear and nonlinear evolution, followed by its role in initiating transition and eventual breakdown of the HBL into a fully turbulent state. Linear stability theory is utilized to first identify the features of the second-mode wave after $FS$-synchronization. These are then employed in separate linearly and nonlinearly forced two-dimensional (2-D) and three-dimensional (3-D) direct numerical simulations (DNS). The nonlinear 2-D DNS shows saturation of the fundamental frequency, and the resulting superharmonics induce tightly braided ‘rope-like’ patterns near the generalized inflection point (GIP). The instability exhibits a second region of growth constituted by the fundamental frequency downstream of the primary envelope, which is absent in the linear scenario. Subsequent fully 3-D DNS identify this region as crucial in amplifying oblique instabilities riding on the 2-D second-mode ‘rollers’. This results in lambda vortices below the GIP, which are detached from the rollers in the inner boundary layer. Streamwise vortex-stretching results in a localized peak in length scales inside the HBL, eventually forming hairpin vortices. Spectral analyses track the transformation of harmonic peaks into a turbulent spectrum. The appearance of oblique modes at the fundamental frequency suggests that fundamental resonance is the most dominant mechanism of transition. The bispectrum reveals coupled nonlinear interactions between the fundamental and its superharmonics leading to spectral broadening, as well as traces of subharmonic resonance. The global forms of the fundamental and subharmonic modes show that the former disintegrate at the location of spanwise breakdown, beyond which the latter amplify. Statistical analyses of the near-wall flow field indicate an increase in large-scale ‘splatting’ motions immediately following transition, resulting in extreme skin-friction events, which equilibrate as turbulence sets in. Fundamental resonance results in complete breakdown of streamwise streaks in the lower log-layer, ultimately resulting in a fully turbulent HBL.

JFM classification

Compressible Flows: Compressible boundary layers Instability: Transition to turbulence Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 905 , 25 December 2020 , A25
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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