4.1 Supercritical and subcritical Hopf bifurcations

Figure 3 shows the two different types of Hopf bifurcation seen in our experiments: supercritical (a,b) and subcritical (c,d). These are distinguished by their bifurcation diagrams (a,c), which are produced by increasing and decreasing $Re$ , and hence $D/\unicode[STIX]{x1D703}_{0}$ , while keeping $S$ and $L/D$ fixed. The vertical axis is the amplitude of the velocity fluctuation normalised by the time-averaged velocity, $|A|\equiv |u^{\prime }|/\overline{u}$ . Also shown are the amplitude evolution and its time derivative (b,d), which are measured in transient experiments performed at the onset of global instability (figure 3 a,c, moving forward from the Hopf point along the blue arrow, due to an impulsive increase in flow rate) and are used to verify the super/subcritical nature of the bifurcation via the Landau model (as per § 3). We examine the two types of bifurcation in turn.

Figure 3. (a,c) Bifurcation diagram and (b,d) derivative of the amplitude evolution for a low-density jet at $S=0.14$ produced by two nozzle extensions: (a,b)  $L/D=1$ , for which the Hopf bifurcation is supercritical, and (c,d)  $L/D=4$ , for which it is subcritical. In (b,d), the dashed grey line is a local linear fit showing the slope and extent of linearity at small $|A|^{2}$ , which is used for comparison with the Landau model (§ 3). In the insets of (b,d), the red line is a time trace of the normalised velocity fluctuation, the blue line is its instantaneous amplitude and the vertical dashed line shows when the amplitude first rises above the noise floor, indicating the start of the growth phase for plotting $\text{d}\log |A|/\text{d}t$ versus  $|A|^{2}$ .

Figure 4. Universal scaling of the onset of global instability in a low-density jet: global-instability maps at (a)  $S=0.14$ and (b–d)  $S=0.14$ –0.60, with hollow markers denoting subcritical bifurcations and solid markers denoting supercritical bifurcations. The error bars are smaller than the markers themselves, so are not shown separately. Also shown are least-squares fits to the Hopf points at constant $S$ , which are hyperbolic in (a,b) (dashed–dotted lines) but linear in (c,d) (solid lines). Panels (b–d) share a common legend. In (d), our proposed scaling is represented by a solid purple line, which is extended beyond the actual range of our data with dotted lines to facilitate comparisons with previous experiments and numerical global stability analyses. The schlieren images show the two different states of the jet. The globally stable state is characterised by a steady potential core, whereas the globally unstable state is characterised by axisymmetric oscillations in the wavemaker region, followed by three-dimensional side jets and then turbulence in the far field.

1. (i) Supercritical bifurcation (figure 3 a,b). At this flow condition ( $S=0.14$ , $L/D=1$ ), the bifurcation diagram shows $|A|$ increasing and decreasing along the same path, with the forward bifurcation occurring at the same Hopf point as the backward bifurcation, indicating the absence of hysteresis. The increase/decrease in $|A|$ scales with the square root of the control parameters. This is the classical behaviour near a supercritical Hopf bifurcation and has been seen before in other open shear flows (§ 1.2). Crucially, it can be modelled with a cubic Landau equation (§ 3). The insets of figure 3(b) show the jet bifurcating from a fixed point to a limit cycle, with $A$ growing and saturating with time. In the plot of $\text{d}\log |A|/\text{d}t$ versus $|A|^{2}$ , the negative slope ( $-l=-49.0~\text{s}^{-1}$ ) and linear behaviour near the vertical axis (i.e. the linear fit is satisfied up to $|A|^{2}\approx 2\times 10^{-3}$ ) confirm via the Landau model that this Hopf bifurcation is supercritical (§ 3).

2. (ii) Subcritical bifurcation (figure 3 c,d). At a different flow condition ( $S=0.14$ , $L/D=4$ ), the bifurcation diagram shows $|A|$ increasing and decreasing along different paths, with the forward bifurcation (Hopf point) occurring at higher $D/\unicode[STIX]{x1D703}_{0}$ and $Re$ values than the backward bifurcation (saddle-node point), indicating the presence of a hysteretic bistable region (shaded in grey). This is the classical behaviour near a subcritical Hopf bifurcation, which has been seen in wake flows (Sheard et al. Reference Sheard, Thompson and Hourigan2004) but has yet to be established in jet flows (§ 1.2). In figure 3(d), the positive slope ( $-l=638.9~\text{s}^{-1}$ ) and nonlinear behaviour near the vertical axis (i.e. deviation from the linear fit occurs at small $|A|^{2}$ ) confirm via the Landau model that this Hopf bifurcation is subcritical (§ 3).

To the best of our knowledge, this is the first definitive evidence of a subcritical Hopf bifurcation occurring in a low-density jet that is from experiments supported by a theoretical model. In fact, as figure 4(b) will show, most of the bifurcations (32 out of 45) in our experiments are subcritical. This is unexpected, as the default assumption in previous studies on low-density jets has been that their bifurcations are supercritical (§ 1.2). This discrepancy could be due to two reasons: (i) the small width of the bistable region – typically we find $(Re_{H}-Re_{SN})/Re_{H}\approx 2\,\%$ –5 %, where the subscripts $H$ and $SN$ denote the Hopf and saddle-node points respectively – and (ii) our deliberate use of extremely small increments in the control parameters – typically we use $\unicode[STIX]{x0394}Re/Re_{H}\approx 0.2\,\%$ –0.5 %, which approaches the uncertainty limits of most mass flow controllers, including ours (§ 2). It is thus plausible that bistable regions existed in previous studies but were overlooked due to the use of insufficiently small control increments. We note that despite the small width of the bistable region, it is still important to know whether the Hopf bifurcation is supercritical or subcritical, not least because the two require different modelling treatments (§ 3).

In addition to subcritical bifurcations, supercritical bifurcations can also be found, albeit mainly for the shortest and longest nozzle extensions (figure 4). For the longest extension ( $L/D=32$ ), we attribute this to the greater length available for spatial amplification of background noise by convective instabilities within the nozzle. This could be compounded by the higher spatial growth rates expected at the higher critical $Re_{H}$ arising from the longer $L/D$ and lower $D/\unicode[STIX]{x1D703}_{0}$ (Hallberg et al. Reference Hallberg, Srinivasan, Gorse and Strykowski2007). Studies on the Duffing–van der Pol oscillator (Zakharova et al. Reference Zakharova, Vadivasova, Anishchenko, Koseska and Kurths2010) and thermoacoustic systems (Gopalakrishnan & Sujith Reference Gopalakrishnan and Sujith2015) have shown that such noise amplification can induce triggering, reducing the effective width of the bistable region, making subcritical bifurcations appear like supercritical bifurcations. Although not shown here, our base-flow measurements at $x/D\approx 0$ support this hypothesis by showing that the normalised velocity fluctuation in the shear layers increases with $L/D$ . For the shortest extension ( $L/D=1$ ), which is the nozzle configuration most similar to those of previous studies in which only supercritical bifurcations were found, we believe that the bifurcations are genuinely supercritical. We speculate that this could be due to the particularly thin shear layers and the way in which they modify the shape and alignment of the velocity profile with respect to the density profile. This has been shown to influence the local – and therefore global – instability properties in a non-trivial manner (Raynal et al. Reference Raynal, Harion, Favre-Marinet and Binder1996; Coenen et al. Reference Coenen, Sevilla and Sánchez2008; Lesshafft & Marquet Reference Lesshafft and Marquet2010; Srinivasan et al. Reference Srinivasan, Hallberg and Strykowski2010; Coenen & Sevilla Reference Coenen and Sevilla2012).

4.2 Universal scaling of the critical Reynolds number

Figure 4(a) shows the states and bifurcation points of the jet in the $D/\unicode[STIX]{x1D703}_{0}$ – $Re$ domain at $S=0.14$ . The global-instability boundary, defined by the Hopf-point trajectory or $Re_{H}$ (black dashed–dotted line), scales hyperbolically with $D/\unicode[STIX]{x1D703}_{0}$ , with regions to its upper right being globally unstable (stable limit cycle) and regions to its lower left being globally stable (stable fixed point). As the bistable region is small ( ${\approx}2\,\%$ –5 % of $Re_{H}$ ), the Hopf and saddle-node points appear to coincide but actually do not (see inset).

The hyperbolic scaling identified in figure 4(a) applies to all eight density ratios of this study ( $S=0.14$ –0.45), as can be seen in the global-instability map of figure 4(b). To collapse the data, we exploit the hyperbolic scaling by changing the vertical axis to the inverse of the Reynolds number at the Hopf point ( $Re_{H}^{-1}$ ), producing a linear fit for each value of $S$ , figure 4(c). Recognising that the slopes of these linear fits increase with decreasing $S$ , we tried including various forms of $S$ in the functional dependence – including those used by Hallberg & Strykowski (Reference Hallberg and Strykowski2006) to scale the global frequency – but found that the optimal data collapse occurred with $\unicode[STIX]{x0394}S=(S^{-1}-1)$    $=(\unicode[STIX]{x1D70C}_{\infty }-\unicode[STIX]{x1D70C}_{j})/\unicode[STIX]{x1D70C}_{j}$ , which is simply the non-dimensional density difference. Using this difference, we show in figure 4(d) the collapse of all 45 Hopf points onto a single line. A least-squares regression gives a linear relationship of the form $Re_{H}^{-1}=a+b(D/\unicode[STIX]{x1D703}_{0})(S^{-1}-1)$ , where $a=-3.785\times 10^{-4}$ and $b=1.019\times 10^{-5}$ , with a correlation coefficient of 0.986. Both supercritical (solid markers) and subcritical (hollow markers) Hopf points follow the same scaling, indicating its insensitivity to the super/subcritical nature of the bifurcation.

For comparison with previous experiments, we include in figure 4(d) the Hopf points of Kyle & Sreenivasan (Reference Kyle and Sreenivasan1993), Hallberg & Strykowski (Reference Hallberg and Strykowski2006) and Hallberg et al. (Reference Hallberg, Srinivasan, Gorse and Strykowski2007). These all follow our proposed scaling fairly well. We further validate our data in the frequency domain against the global-frequency scaling of Hallberg & Strykowski (Reference Hallberg and Strykowski2006); the results (not shown in the interests of brevity) are also in good agreement.

For comparison with numerical global stability analyses, we include in figure 4(d) the linear neutral points of Coenen et al. (Reference Coenen, Lesshafft, Garnaud and Sevilla2017, figure 12). These are offset towards higher $Re$ when compared with our proposed scaling and the experimental data mentioned above. Coenen et al. (Reference Coenen, Lesshafft, Garnaud and Sevilla2017) attempted to account for this offset by including buoyancy and variable viscosity, but found both effects to be insufficient. It is believed that this offset between experimental and numerical data could be due to two factors: (i) subtle differences in the base flow just downstream of the nozzle outlet, which sensitivity analyses (e.g. Coenen et al. Reference Coenen, Lesshafft, Garnaud and Sevilla2017, figure 7d) predict could have a strong influence on the growth rate of global modes, and (ii) noise amplification by convective instabilities in the experiments, which induces triggering in the bistable region, reducing the Reynolds number at which the Hopf point is found through measurements. This underestimation of $Re_{H}$ cannot occur in the linear global stability framework of Coenen et al. (Reference Coenen, Lesshafft, Garnaud and Sevilla2017).

These two factors may also explain some of the scattering seen in the experimental data around our proposed scaling (figure 4 d). Although our scaling contains all three of the primary control parameters ( $S$ , $D/\unicode[STIX]{x1D703}_{0}$ , $Re$ ), none of them can explicitly account for variations in noise amplification or in the shape and alignment of the velocity and density profiles. These are determined by the subtle geometrical details of the nozzle, such as the contour of the contraction section, the quality of the flow conditioning and the shape of the nozzle lip. The latter has a particularly strong influence on the local entrainment pattern at the jet base, which can affect the degree of coflow and therefore the instability properties (Hallberg et al. Reference Hallberg, Srinivasan, Gorse and Strykowski2007). As in most experiments, our nozzle lip is carefully machined to a sharp knife edge, but the rest of the nozzle is known to vary from facility to facility. A possible way to account for such variation is to add a new parameter to our scaling to quantify the alignment between the velocity and density profiles (e.g. the baroclinic torque integral of Nichols, Schmid & Riley Reference Nichols, Schmid and Riley2007, equation (4.12)). However, this would require detailed measurements of those profiles to be taken, which is being considered but is beyond the scope of this study.