Onset of global instability in low-density jets

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.

Coherence resonance in low-density jets

“…We attribute this jump to this particular case being near the supercritical-subcritical border, where the Hopf and SN points are so close together as to be indistinguishable within experimental uncertainty (Lee et al 2019). This interpretation of supercritical-like behaviour can also explain similar jumps in |A| observed in the low-density jet experiments of Hallberg & Strykowski (2006) and Zhu et al (2017). In § 4, we will explore the implications of this jump on the modelling of CR.…”

“…As a purely nonlinear phenomenon, CR is active only in systems with sufficient nonlinearity. In our previous study (Zhu et al 2017), we established the role of nonlinearity in low-density jets by demonstrating that they can undergo both supercritical and subcritical Hopf bifurcations. Here we go further to show that CR exists in such jets, producing noise-induced dynamics that can be used to detect and distinguish between the two types of bifurcation, even before the stability boundaries are reached.…”

“…In figure 1(c), the forward and backward paths do not overlap, owing to the presence of a hysteretic bistable regime between the Hopf and SN points (grey region). These features are characteristic of a subcritical Hopf bifurcation and require at least quintic nonlinearity to be accurately modelled (Zhu et al 2017).…”

“…The transition to global instability in low-density jets was originally thought to occur solely via a supercritical Hopf bifurcation (Monkewitz et al 1990;Raghu & Monkewitz 1991;Kyle & Sreenivasan 1993). Recently, however, Zhu, Gupta & Li (2017) have shown that this transition can also occur via a subcritical Hopf bifurcation, with a hysteric bistable regime between the Hopf and saddle-node (SN) points. Specifically, through transient hot-wire measurements, these researchers were able to evaluate the coefficients of a truncated Landau equation, establishing that cubic nonlinearity is stabilising in supercritical bifurcations but is destabilising in subcritical bifurcations, which require quintic or higher-order terms for saturation.…”

Stability of Low-Reynolds-Number Separated Flow Around an Airfoil Near a Wavy Ground

“…Thus, at higher values of Reynolds number, the KH mode is expected to cause the airfoil wake to transition from a spatial amplifier of extrinsic perturbations to a self-excited oscillator with an intrinsic natural frequency [33]. Such a transition, which is often referred to as a Hopf bifurcation [34], has been reported before not only in airfoil wakes [35], but also in cylinder wakes [36], crossflowing jets [37], and low-density jets [38][39][40]. To examine the evolution of the optimal perturbations, we performed a transient growth analysis by estimating the energy gain Gτ over a time horizon τ.…”