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Onset of global instability in low-density jets

Published online by Cambridge University Press:  04 September 2017

Yuanhang Zhu ,
Vikrant Gupta Open the ORCID record for Vikrant Gupta [Opens in a new window]  and
Larry K. B. Li Open the ORCID record for Larry K. B. Li [Opens in a new window]
Yuanhang Zhu
Affiliation:
Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
Vikrant Gupta
Affiliation:
Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen, China
Larry K. B. Li*
Affiliation:
Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
*
†Email address for correspondence: larryli@ust.hk
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Abstract

In low-density axisymmetric jets, the onset of global instability is known to depend on three control parameters, namely the jet-to-ambient density ratio $S$, the initial momentum thickness $\unicode[STIX]{x1D703}_{0}$ and the Reynolds number $Re$. For sufficiently low values of $S$ and $\unicode[STIX]{x1D703}_{0}$, these jets bifurcate from a steady state (a fixed point) to a self-excited oscillatory state (a limit cycle) when $Re$ increases above a critical value corresponding to the Hopf point, $Re_{H}$. In the literature, this Hopf bifurcation is often regarded as supercritical. In this experimental study, however, we find that under some conditions, there exists a hysteretic bistable region at $Re_{SN}<Re<Re_{H}$, where $Re_{SN}$ denotes a saddle-node point. This shows that, contrary to expectations, the Hopf bifurcation can also be subcritical, which we explore by evaluating the coefficients of a truncated Landau model. The existence of subcritical bifurcations implies the potential for triggering and the need for weakly nonlinear analyses to be performed to at least fifth order if one is to be able to predict saturation and bistability. We conclude by proposing a universal scaling for $Re_{H}$ in terms of $S$ and $\unicode[STIX]{x1D703}_{0}$. This scaling, which is insensitive to the super/subcritical nature of the bifurcations, can be used to predict the onset of self-excited oscillations, providing further evidence to support Hallberg & Strykowski’s concept (J. Fluid Mech., vol. 569, 2006, pp. 493–507) of universal global modes in low-density jets.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Wakes/Jets: Jets Instability: Nonlinear instability
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 828 , 10 October 2017 , R1
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Copyright
© 2017 Cambridge University Press 

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