3 Hysteresis studies: overview

The ranges of existence of different long-time flow states observed for columnar initial conditions and the Maxworthy inflow profile, with $\unicode[STIX]{x1D6FC}=100$, $\unicode[STIX]{x1D6FF}=0.2$, $Re=200$ and $S$ varying, as reported in Moise & Mathew (Reference Moise and Mathew2019), are reviewed first. These include the pre-VB ($S\leqslant 1.38$), bubble ($1.39\leqslant S\leqslant 1.57$) and conical ($1.58\leqslant S\leqslant 2$) forms. The VB forms were further categorized into different types. Three types of BVB were observed for columnar initial conditions: steady BVB ($1.39\leqslant S\leqslant 1.48$), BVB with spiral tail ($1.49\leqslant S\leqslant 1.5$) and pulsating BVB ($1.55\leqslant S\leqslant 1.57$). Importantly, the BVB with spiral tail had a one-celled bubble structure, while the pulsating type had an intermittent presence of a second cell in the bubble. The regular CVB was observed for $1.58\leqslant S\leqslant 1.8$, while the wide-open type was observed for higher $S$, although the exact boundaries between the two could not be determined precisely due to the large lateral spread of the flow at high swirls (see Pradeep Reference Pradeep2019, pp. 69–70). Table 1 summarizes these results, with $S_{c}^{l}$ and $S_{c}^{u}$ denoting the lower and upper swirl limits of the ranges.

Figure 2.

An approximate map of all flow states identified as swirl number ($S$) is varied, with focus on VB. The conical form is seen to coexist with various types of the bubble form. Some ambiguities remain in determining the lowest swirl for which CVB can be sustained, as highlighted by the dashed lines (see appendix A). Asymmetric BVB and two-celled BVB with spiral tail also coexist in a small overlapping range of swirl ($1.64\leqslant S\leqslant 1.66$).

Table 1.

Ranges of existence of flow states observed for columnar initial conditions, Maxworthy profile, as reported in Moise & Mathew (Reference Moise and Mathew2019). Here, $S_{c}^{l}$ and $S_{c}^{u}$ represent the lower and upper swirl number limits, respectively.

In the hysteresis studies, the ranges of existence of both the bubble and conical forms could be extended beyond those observed in Moise & Mathew (Reference Moise and Mathew2019), as summarized in figure 2. Hysteresis effects between different types of a given form were also studied for all cases except the two CVB types. The latter was not carried out due to numerical expense. Thus, for CVB, only the reduction of $S$ below 1.58 was considered. CVB was seen to sustain as a long-time state for $S_{t}=1.5$, the features of which are examined in § 4. The lowest $S$ at which CVB can be sustained could not be precisely determined, as discussed in appendix A.

With regard to the bubble form, the studies showed that the three BVB types mentioned above (see table 1) do not coexist with each other. It is emphasized that the absence of the hysteresis phenomenon is only between BVB types, as CVB was observed to exist for $S\geqslant 1.5$. Also, the bubble form was not found to exhibit hysteresis at its swirl threshold, implying that it does not coexist with pre-VB flow states. By contrast, Billant et al. (Reference Billant, Chomaz and Huerre1998) have observed in hysteresis studies that the bubble form sustained at lower swirls than usually observed. Possible explanations for this discrepancy include the differences in inflow profiles and the somewhat lower $Re$ used here. Indeed, Billant et al. (Reference Billant, Chomaz and Huerre1998) reported hysteresis effects to be present only above a critical Reynolds number for one type of nozzle used (see their figure 11a, p. 200). Similar observations indicating critical Reynolds-number requirement have also been made for other swirling flow profiles (Beran & Culick Reference Beran and Culick1992; Wang & Rusak Reference Wang and Rusak1997).

Increasing the swirl beyond $S=1.57$, the bubble form was found to sustain up to $S=1.78$, while it transitioned to a CVB at $S=1.8$. The pulsating type of BVB was found to persist up to $S=1.59$, beyond which the spatio-temporal features of BVB were distinct in different swirl ranges, allowing for classification into two new BVB types. These were identified as a two-celled BVB with spiral tail and an asymmetric BVB. For the two-celled BVB with spiral tail, a double ring structure was observed for the bubble, similar to that in Faler & Leibovich (Reference Faler and Leibovich1978), but the remaining features were essentially the same as that of BVB with spiral tail seen for columnar initial conditions. The asymmetric type of BVB also had a two-celled bubble structure, but it is characterized by strong unsteady non-axisymmetric flow in almost the entire bubble region. The features of these two types are discussed in further detail in §§ 5.1 and 5.2. Interestingly, these two types themselves were found to coexist with each other over a small range of $S$, as reported in § 6.

Figure 2 provides a map of all long-time flow states observed as the swirl number is changed for the Maxworthy profile, updated based on the results of the present hysteresis study. The $x$-axis does not quantify any specific variable but represents the flow states observed. This map is only approximate – for example, the lower limit of CVB’s range of existence is based on D20 (appendix A) and the wide-open CVB was simulated for $S=2$ alone (Moise & Mathew Reference Moise and Mathew2019). The dashed lines are used to emphasize the former and additionally the result that CVB has been sustained in D80 for swirls not lower than $S_{t}=1.5$. In the following sections, the hysteresis features of the bubble and conical forms are examined in further detail.

4 Bistability of bubble and conical forms of vortex breakdown

Figure 3.

Comparison of time-averaged features of bistable BVB (blue) and CVB (black) at $S=1.5$ based on azimuthally averaged variables. (a) Projected streamlines on the meridional plane, showing the stark contrast in flow features of the bistable VB forms. (b) A close-up view of the same, showing differences in the vicinity of the stagnation point. (c) Centreline velocity variation: BVB (——) and CVB (- - - -). Only part of the lateral extent is shown.

For $S_{i}=1.6$ (CVB), $S_{t}=1.5$ and $\unicode[STIX]{x1D6FF}_{S}=-10^{-3}$, the conical form was sustained at the target swirl value (attained at $t=100$) in D80 till the end of the simulation ($t=5400$). For $S=1.5$ and columnar initial conditions, a BVB with spiral tail was observed (see table 1). Thus, for the Maxworthy inflow profile, with $S=1.5$, $\unicode[STIX]{x1D6FF}=0.2$, $\unicode[STIX]{x1D6FC}=100$ and $Re=200$, the bubble and conical forms of VB are conclusively established to be bistable by means of hysteresis studies. To compare the bistable forms at $S=1.5$, the mean velocity fields were computed for BVB in D40 and the present case of CVB in D80. For the latter, the time averaging was carried out for a duration of 2200. Assuming that the mean flow is axisymmetric for both cases, the flow field was further azimuthally averaged. For convenience, only data from the $y=0$ and $z=0$ Cartesian planes were used for this purpose. This approach has been adopted for all other cases in this study where the time-averaged axisymmetric features are considered, for better statistical convergence.

The spatio-temporal features of CVB at $S=1.5$ closely resembled that observed for $S=1.6$. Since the latter has been reported in detail previously (see Moise & Mathew Reference Moise and Mathew2019, §§ 4.2.1, 4.2.2 and movies 3, 4), only a brief description is provided here for $S=1.5$. Projected streamlines on the meridional plane for the bistable forms at $S=1.5$ are shown in figure 3(a). This highlights, somewhat strikingly, the stark contrast between the two forms of VB, especially in regard to the associated length scales. It is emphasized that only part of the lateral extent is shown. Conspicuous differences between the two forms can be observed, including the relatively enormous recirculation zone and the strong radial divergence downstream of the stagnation point for CVB. Nevertheless, interesting similarities are also present. Both forms have an associated columnar approach flow, a stagnation point, a recirculation zone, a mean-flow envelope around this zone which expands radially outwards upstream before curving inwards, and an almost columnar structure downstream of the zone. It is also noted that the overall dimensions of the recirculation zone of CVB reduced with the reduction in $S$.

Figures 3(b) and 3(c) show close-up views of projected streamlines in the vicinity of the stagnation point and the variation in mean centreline velocities, respectively. These panels further show that the position of the stagnation point is different, and that the flow is affected even upstream of this point, with the axial velocity gradients being different. The mean centreline velocity is seen to be more negative for the case of CVB. Differences in length scales between the two forms were observed to be somewhat smaller in the experiments of Billant et al. (Reference Billant, Chomaz and Huerre1998). This might be due to the presence of weak disturbances at the inflow, which might cause the observed distortions and rollup of the conical sheet, leading to a reduction in the average radial size of the recirculation zone of CVB. This possibility is corroborated by studies on the Maxworthy profile at higher $Re$ (not reported here; see Pradeep (Reference Pradeep2019)), where it was observed that the onset of turbulence can lead to strong reductions in the maximum radial size of the CVB’s recirculation zone. Another possible explanation for these differences is the smaller cross-sectional width of the tank used in the experiments.

5 Hysteresis features of bubble form of vortex breakdown

Features of the two new types of BVB that were exclusively observed in the hysteresis studies are discussed next. Unless otherwise mentioned, these studies used $S_{i}=1.5$ and $\unicode[STIX]{x1D6FF}_{S}=10^{-3}$.

5.1 Two-celled BVB with a spiral tail

Figure 4.

Hysteresis effects for BVB for $S_{i}=1.5$ and $S_{t}=1.6$. (a) Streaklines and (b) pathlines (latter case, $r_{0}=0$ alone) showing a spatial structure similar to the BVB with spiral tail.

A reference case of $S=1.6$ is considered here in detail. Features of this BVB type at higher $S$ in its range of existence were qualitatively similar, except for one difference – the stagnation point at the bubble’s nose was lost above a particular $S$. Surprisingly, this feature did not affect the flow’s overall spatial structure. Details of this stagnation point loss and its implications are provided in appendix B.

For the reference case, the increase in $S(t)$ from $S_{i}=1.5$ to $S_{t}=1.6$ was accompanied by a gradual increase in bubble size, upstream movement of the stagnation point and radial widening of the downstream end of the bubble with an intensification of its fore–aft asymmetry. Figure 4 shows features of the long-time flow state based on tracer visualizations starting from $t=7600$. It is evident that a BVB with a dominant $m=+1$ helical structure in its wake is present. The bubble’s structure is seen to be mostly unaffected by the spiral, a characteristic of the BVB type that is referred to as ‘BVB with spiral tail’ (cf. Moise & Mathew Reference Moise and Mathew2019, figure 5). The sustenance of the bubble form additionally confirms bistability with CVB at $S=1.6$ (see table 1). The flow dynamics was time-periodic with no pulsating behaviour observed. This can be inferred from figure 5, where both time series and power spectral density (PSD) of velocity components at different positions in the domain are shown (cf. Moise & Mathew Reference Moise and Mathew2019, figures 6a and 7). Here, $f$ represents the non-dimensional frequency (based on jet radius and centreline velocity; see § 2.1), while $f_{0}$ has been used to denote the frequency associated with the maximum PSD. The flow in the upstream regions of the bubble remained mostly steady. As will be shown below, the flow state closely resembled a BVB with a spiral tail, except for two interesting differences in flow structure – a second cell in the bubble and an additional recirculation zone in its wake.

Figure 5.

Temporal features of two-celled BVB with spiral tail for $S_{t}=1.6$. (a) Time series of fluctuating component of $u_{x}$ for long-time state, showing absence of pulsation. (b) PSD of fluctuating component of $u_{y}$ (fundamental frequency, $f_{0}=0.094$) at different positions ($x,y,z$): ($1,0,0$) (blue — ⋅ —), ($5,0,0$) (red - - - -) and ($5,0,0.5$) (black ——).

5.1.1 Instantaneous flow features

Figure 6.

Comparison of three-dimensional streamlines at an arbitrary instant for (a) $S=1.5$ (columnar initial conditions) and (b) $S_{t}=1.6$, showing the presence of a second toroidal structure (green) along with the original (red) in the recirculation region for the latter. Isosurface of $u_{x}=0$ additionally shown to highlight the recirculation zone.

Three-dimensional streamlines at an arbitrary instant for the cases of BVB at $S=1.5$ (columnar initial conditions) and $S=1.6$ (present hysteresis study) are compared in figure 6. The isosurface of $u_{x}=0$, which acts to delineate the region of reverse flow, is also shown. The toroidal structure that is common to both cases is highlighted by red streamlines. For $S=1.6$, a second such structure of smaller radius (green curves) can be seen. Thus, the bubble’s structure is seen to contain a second toroid nested within another. This structure was present at all times (not shown). As noted previously, this has been observed in experiments (Faler & Leibovich Reference Faler and Leibovich1978; Billant et al. Reference Billant, Chomaz and Huerre1998) and is generally identified as a two-celled bubble. These toroidal structures should not be confused with conventional vortex rings. In this study, this type is referred to as a two-celled BVB with spiral tail, to differentiate from the one-celled BVB with spiral tail observed at lower $S$. For convenience, the bubble is considered one-celled unless explicitly stated otherwise. Interestingly, figure 6(b) additionally reveals the presence of an isosurface of $u_{x}=0$ centred around $x\approx 6$ in the core region for $S=1.6$, indicating a second recirculation zone in the wake of the bubble.

5.1.2 Time-averaged flow features

Figure 7.

Time-averaged features of two-celled BVB with spiral tail at $S_{t}=1.6$. (a) Projected streamlines based on time-averaged velocity. (b) Contours of r.m.s. of fluctuating axial velocity component, $(\overline{u_{x}^{\prime 2}})^{0.5}$ (levels shown, 10 equal steps in the range [$0,0.06$]) showing two-celled structure of time-averaged BVB (second cell highlighted by green curves, $x\approx 3$), presence of a second breakdown region ($6\leqslant x\leqslant 10$) and absence of strong fluctuations in the bubble region.

Figure 7 shows time-averaged flow features for the two-celled BVB with spiral tail for $S_{t}=1.6$. The presence of a second set of closed streamlines in the bubble at $x\approx 3$ (green curves) confirms that the two-celled structure of the bubble persists in the time-averaged fields. Interestingly, an additional streamwise-elongated recirculation zone can be identified in the wake, centred at $x\approx 8$, containing a stagnation point at its upstream end. Thus, the flow can be identified to have two regions of VB. Note that this is related not to the cell structure of the bubble, but to the number of such individual recirculation zones present. The presence of multiple breakdown regions has been commonly observed in simulations (Ruith et al. Reference Ruith, Chen, Meiburg and Maxworthy2003) and experiments (Escudier Reference Escudier1984). Explanations for their existence have been proposed using arguments based on azimuthal vorticity (Brown & Lopez Reference Brown and Lopez1990). This flow type can also be referred to as a BVB with an SVB downstream, but this nomenclature is not adopted in this study. Figure 7(b) shows contours of the root-mean-square (r.m.s.) of the fluctuating component of axial velocity, i.e. $(\overline{u_{x}^{\prime 2}})^{0.5}$. As expected, the fluctuation intensities are strong in the bubble’s wake and almost negligible in the bubble region ($x\leqslant 4$). Other components also had a similar spatial distribution. It will be shown in § 6 that the base flow for this type of BVB is strongly similar to the mean flow. Thus, these results indicate that the generation of the second toroidal structure is not to be completely ‘attributed to the action of the fluctuations’, as inferred in Faler & Leibovich (Reference Faler and Leibovich1978), but originates through axisymmetric mechanisms. Nevertheless, for the asymmetric BVB, helical instabilities do strongly modify the bubble’s structure, as will be shown in § 5.2.3 (see figure 13).

5.2 Asymmetric BVB

At $S_{t}=1.67$, the envelope of the bubble was found to precess in the same sense as the swirling flow, leading to a non-axisymmetry persisting in the bubble region for the long-time flow state. This appears to be the asymmetric BVB reported in Billant et al. (Reference Billant, Chomaz and Huerre1998), although the asymmetry observed here is not as pronounced. Additionally, Billant et al. (Reference Billant, Chomaz and Huerre1998) reported this behaviour only for high Reynolds numbers in the parametric range that was studied. A representative swirl of $S_{t}=1.7$ is chosen for further examination of this BVB type. Incidentally, this swirl number, along with that of $S=1.5$ (with other parameters the same as that of this study), were the only two cases examined previously for the Maxworthy profile in Ruith et al. (Reference Ruith, Chen and Meiburg2004). The authors observed a two-celled bubble structure for $S=1.7$, but the dynamical aspects were not reported.

5.2.1 Spatio-temporal features

Figure 8.

Contours on the $z=0$ plane of vorticity magnitude (levels shown, seven equal steps in the range [$0,4.5$]) for case $S_{t}=1.7$, at two instants, capturing features of the asymmetric BVB. The bubble’s structure is seen to be non-axisymmetric.

Figure 8 shows features of asymmetric BVB at two arbitrary instants using contours of vorticity magnitude for case $S_{t}=1.7$. The top–bottom asymmetry of the envelope in the $z=0$ plane is evident. Oscillations of the bubble envelope in a non-axisymmetric fashion is best inferred from the animation of the visualizations provided as supplementary movie 1 (available at https://doi.org/10.1017/jfm.2020.105). For swirling jets, due to the presence of the shear layer, the choice of vorticity magnitude can be misleading in identifying vortical structures. Therefore, the $\unicode[STIX]{x1D706}_{2}$-criterion proposed in Jeong & Hussain (Reference Jeong and Hussain1995) was additionally used for visualizations. A three-dimensional isosurface of $\unicode[STIX]{x1D706}_{2}=-0.1$ is shown in figure 9, where a two-armed vortical structure in the bubble’s wake is evident.

Figure 9.

Isosurface $\unicode[STIX]{x1D706}_{2}=-0.1$ at $t=4500$ for asymmetric BVB, $S_{t}=1.7$.

Figure 10 shows time series and PSDs of different velocity components as a function of frequency $f$, at various positions. The positions were chosen to study the temporal behaviour in the bubble ($x=2$), its near wake ($x=5$) and further downstream ($x=8$). Figures 10(a), 10(b) and 10(c) show features on the jet axis, while figure 10(d) is based on a probe positioned at a coordinate $(8,0,0.5)$. The time series in figure 10(a) shows a strong periodic component after the initial transients (approximately $t\geqslant 1000$), although non-coherent/stochastic features are also present. This is further highlighted in figure 10(b), where a shorter time interval ($4000\leqslant t\leqslant 4500$) is shown for clarity. These aspects are also reflected in figure 10(c), with a sharp peak in the PSD seen for $f_{0}=0.049$, but with the PSD spread out over a range of other frequencies also. To clearly highlight the former feature, a linear scale is used in the figure for plotting the PSDs. The energy content at a lower frequency can also be observed off-axis, as shown in figure 10(d). The origin of the non-coherent features of the time series remain unclear, although it might be associated with weak intermittent streamwise motions. This temporal behaviour can be inferred from tracer visualizations, as discussed next.

Figure 10.

Temporal features of asymmetric BVB at $S_{t}=1.7$ shown using time series of (a) $u_{x}$ and (b) $u_{y}$ on the axis and PSDs based on fluctuating component of $u_{y}$ (c) on the axis and (d) at $y=0$ and $z=0.5$. Streamwise positions: $x=2$ (red - - - -), $x=5$ (blue — ⋅ —) and $x=8$ (black ——). A dominant frequency of $f_{0}=0.049$ was observed.

5.2.2 Visualization using tracer particles

Figure 11 shows visualization of flow features using tracer particles. The time $t$, for convenience, was measured starting from the instant when the tracers were released. In the wake of the bubble, an abrupt radial expansion can be observed ($x\approx 13$), where an almost axisymmetric toroidal structure can be seen. This is probably associated with the development of the second arm in the wake (figure 9), although the two-arm structure is not discernible from this visualization. The streamwise position of this structure was found to change intermittently in time, as can be inferred from the two times shown. The flow downstream of this structure, as visualized by the tracers, was strongly distorted with no clear helix visible.

Figure 11.

Instantaneous positions of tracers released from $r_{0}=0$ and $r_{0}=0.1$ at different times for $S_{t}=1.7$. Meridional (a,b) and three-dimensional perspective (c,d) views are shown. A somewhat abrupt radial expansion in the bubble’s wake ($x\approx 15$) can be observed.

Figure 12.

Paths taken by tracer particles released at the inflow plane at radial positions $r_{0}=0$ for asymmetric BVB, $S_{t}=1.7$ are plotted. Only the bubble and its immediate wake is shown. The plot reveals that the particles do not enter into the bubble.

Figure 12 shows pathlines associated with tracers released from $r_{0}=0$ alone. Surprisingly, these particles do not enter into the bubble region, which is in stark contrast to the other BVB types reported before (see figures 4 and 23 for swirl numbers 1.6 and 1.65, and Moise & Mathew (Reference Moise and Mathew2019, figure 5) for $S=1.5$). It must be clarified that those particles released from positions not on the centreline ($r_{0}\neq 0$) do enter into the bubble for the asymmetric BVB, as can be inferred from the visualizations shown in figure 11. These observations suggest that if the dye visualization of the flow was carried out using only a thin dye filament injected at the inflow axis, the bubble’s presence would not be apparent. Indeed, based on the investigations of Brücker (Reference Brücker1993) and precession of the bubble for the asymmetric BVB, Billant et al. (Reference Billant, Chomaz and Huerre1998) predicted that ‘if a single filament of dye were used, the present asymmetric bubble would appear as a spiral breakdown’. This aspect is examined in appendix C.

5.2.3 Time-averaged features

Figure 13.

Time-averaged features for $S_{t}=1.7$. Projected streamlines on meridional plane based on azimuthally averaged (a) mean velocity field and (b) base-flow solutions obtained using SFD. (c) Contours of r.m.s. of the fluctuating component of $u_{x}$ on meridional plane (levels shown, 10 equal steps in the range [$0,0.2$]) showing unsteadiness in the upstream region.

Figure 13(a) shows the time-averaged features of the asymmetric BVB for $S_{t}=1.7$ using projected streamlines on a meridional plane based on the azimuthally averaged mean velocity field. Interestingly, a stagnation point was present at the nose of the bubble. Figure 13(b) shows projected streamlines based on the steady axisymmetric base flow for the same $S$ obtained by means of SFD starting from the long-time solutions as initial conditions. Significant differences exist between the two, suggesting strong mean-flow modifications due to the unsteady helical structure. For the mean field, the bubble is relatively enlarged, while in the wake region, a smaller recirculation zone ($9<x<11$) is seen to replace the elongated recirculation zone of the base flow ($5<x<12$). The two-celled bubble structure observed in the time-averaged flow should be compared with that reported in tube-and-vane experiments (see Faler & Leibovich Reference Faler and Leibovich1978, figure 10). While the inflow configurations for the two cases differ significantly, most of the qualitative features of the two-celled structure can be seen to be similar for the two cases. Thus, while the two-celled structure might be generated due to axisymmetric mechanisms, helical instabilities appear to play an important role in modifying it.

The contours of the r.m.s. of the fluctuating part of $u_{x}$ for $S_{t}=1.7$ are shown in figure 13(c). Strong unsteadiness is observed in the wake region, which might account for the modifications to the base flow. Additionally, the intensity is somewhat stronger in the bubble region in comparison to the two-celled BVB with spiral tail (figure 7b). Other components of the fluctuating part of the velocity were found to have significant intensities further upstream in the bubble region. These aspects will also be confirmed using results from stability analysis (see figure 19).

6 Hysteresis between types of the bubble form of vortex breakdown

Starting from the asymmetric type of BVB observed at $S_{i}=1.67$, when swirl was reduced gradually with $\unicode[STIX]{x1D6FF}_{S}=-10^{-4}$, this type was found to sustain as a long-time state for $S_{t}=1.65$ and 1.64 (simulation times of $t\approx 6300$ and 7300, respectively), revealing a small overlapping range of existence ($1.64\leqslant S\leqslant 1.65$) with the two-celled BVB with spiral tail. The features of these bistable types are discussed next for a reference swirl of $S=1.65$. It is noted that the two-celled BVB with spiral tail did not sustain when increasing swirl starting from $S_{i}=1.65$ to $S_{t}=1.67$, for $\unicode[STIX]{x1D6FF}_{S}=10^{-4}$.

6.1 Bistable BVB types at $S=1.65$

Figure 14.

Contours of axial velocity on $z=0$ plane at an arbitrary instant showing features of long-time bistable BVB types at $S=1.65$: (a) two-celled BVB with spiral tail and (b) asymmetric BVB.

Figure 15.

Isosurface $\unicode[STIX]{x1D706}_{2}=-0.1$ at an arbitrary instant for long-time bistable BVB types at $S=1.65$: (a) two-celled BVB with spiral tail and (b) asymmetric BVB.

The features of the bistable two-celled with spiral tail and asymmetric types of BVB for $S_{t}=1.65$ are compared in figure 14 using axial velocity contours on the $z=0$ plane at arbitrary instants of $t=5310$ and $t=5490$, respectively. The instantaneous vortical structures based on $\unicode[STIX]{x1D706}_{2}$ isosurfaces are additionally compared in figure 15. These figures highlight the differences between the two, with a single helical structure observed in the wake of the two-celled case, while a double helix can be seen for the asymmetric BVB. Axisymmetry of the bubble for the former case can also be inferred.

Figure 16.

Comparison of time-averaged characteristics of bistable BVB types at $S=1.65$ shown using (a) projected streamlines on meridional plane based on mean velocity and (b) streamwise variation of centreline velocity (inset showing variation in vicinity of stagnation point) and radial variation of $u_{x}$ at $x=3$: two-celled BVB with spiral tail (black ——), asymmetric BVB (blue — ⋅ —) and base flow (red - - - -).

The time-averaged features of the two are compared in figure 16. The steady base flow attained using SFD is additionally provided for comparison in figures 16(b) and 16(c). As seen from these, the time-averaged flow for two-celled BVB with spiral tail closely resembles the base flow, and thus, for clarity, has not been included in figure 16(a). The projected streamlines show that, for the asymmetric BVB, the bubble is relatively enlarged, while the downstream recirculation zone in its wake is shortened along $x$, as is typical for this BVB type (see figure 13a). The absence of a stagnation point for the base flow and the two-celled BVB with spiral tail can also be inferred from figure 16(b) (see inset).

7 Hysteresis diagrams

Figure 17.

(a) An approximate hysteresis curve based on the maximum radial extent achieved by the streamline with $\unicode[STIX]{x1D70E}_{0}=0.2$ ($\unicode[STIX]{x1D70E}_{max}=\max \{\unicode[STIX]{x1D70E}(x)\}$), highlighting the differences between VB forms: BVB (○) and CVB (▿). Domain dimensions are insufficient for cases where $S\geqslant 1.8$ (CVB) and hysteresis features of asymmetric BVB not considered. (b) Hysteresis plot based on the dominant frequency observed in flow simulations for BVB types: two-celled with spiral tail (♢) and asymmetric (▵).

A hysteresis diagram is attempted for the coexisting bubble and conical forms based on the maximum radial expansion of an arbitrarily chosen projected streamline computed from the time-averaged mean velocity field. The radial position of the required streamline at the inflow was chosen here as $\unicode[STIX]{x1D70E}(x=0)=\unicode[STIX]{x1D70E}_{0}=0.2$. A comparison based on the maximum radius, $\unicode[STIX]{x1D70E}_{max}=\max \{\unicode[STIX]{x1D70E}(x)\}$, obtained over all $x$, is plotted in figure 17(a) as a function of $S$. It must be emphasized that this is only approximate for $S\geqslant 1.8$ (CVB), since the solutions from the simulations were found to be severely affected by the domain dimensions used (D80). Additionally, for $S=2$ alone, the instantaneous velocity field at $t=5225$ (a time later than that shown in Moise & Mathew (Reference Moise and Mathew2019, figure 14a)) was used instead of the mean velocity field, for convenience, as the flow was found to have expanded almost to the lateral boundary and continued to evolve very slowly. The bistable asymmetric BVB has not been shown in the figure for clarity. The stark difference between the maximum radial extents achieved by the two forms is highlighted by this plot. By examining this streamline for different $S$, it was also observed that, for CVB, the cone’s angle and the recirculation zone’s streamwise extent increase with increasing $S$ (not shown).

The hysteresis features of the two-celled with spiral tail and asymmetric types of BVB are compared in figure 17(b) using a different parameter – the dominant frequency observed in the flow simulations. Differences in the associated range of the dominant frequencies for each BVB type further indicate that the two are associated with different helical modes. This aspect is examined in the following section by carrying out a global linear stability analysis.

8 Global linear stability analysis

The helical structures seen for the two-celled BVB with spiral tail and the asymmetric BVB are examined using the tool of global linear stability analysis. Firstly, results obtained using the base flow are discussed followed by those based on the mean flow. As noted in § 2.2, unstable modes were observed only for $m=+1$ for both cases, features of which are discussed below. It is noted that coherent features were not observed for the CVB and thus stability analysis for these states was not attempted.

8.1 Stability of base flow

The eigenvalue spectrum computed using the base flow in the swirl range $1.49\leqslant S\leqslant 1.65$ showed a single unstable mode with the same features as those observed for $S=1.5$. Note that for $1.6\leqslant S\leqslant 1.65$ the two-celled BVB with spiral tail exists (see figure 2), while a BVB with spiral tail (one celled) is observed for $1.49\leqslant S\leqslant 1.5$ and a pulsating type for the intermediate range (table 1). This implies that the instability features remain mostly unaffected due to the development of the second toroidal structure in the bubble. This can be anticipated, since this helical structure has negligible energy in the bubble region. This eigenmode will henceforth be referred to as the ‘wake mode’ and is associated with the spiral tail of BVB observed in flow simulations. Features of this mode have been reported in Moise & Mathew (Reference Moise and Mathew2019) for $S=1.5$ and are not shown here for other $S$.

Figure 18.

(a) Eigenvalue spectrum from stability analysis (base flow) in the vicinity of the unstable ‘wake’ mode (red symbols) showing the destabilization of a second ‘bubble’ mode (blue symbols) for swirl numbers $S=1.65$ (○) and $S=1.67$ ($+$, arrows added to highlight trends with increasing $S$). (b) Comparison of global angular frequency $\unicode[STIX]{x1D714}_{g,r}$ of unstable modes for various $S$ from stability analysis of base and mean flow with the fundamental frequency seen in flow simulations: two-celled BVB (○), asymmetric BVB (▫), (base) wake mode (▿), (base) bubble mode (♢), (mean) wake mode ($\times$) and (mean) bubble mode ($+$).

For $S\geqslant 1.67$, a second mode was observed to become unstable. This is shown in figure 18(a), where the eigenvalue spectra in the vicinity of the unstable global modes are plotted for the two cases of $S=1.65$ and $S=1.67$. For all higher $S$ where the BVB was observed, both these modes remained unstable. In contrast to the wake mode, the second unstable eigenmode showed high energy content in the bubble region. This can be inferred from figure 19(a) (discussion in following section), where contours based on $\hat{u} _{x}$ (equation (2.3)) are shown for the case of $S=1.7$. Since the asymmetric BVB is seen for $S\geqslant 1.67$ in hysteresis studies when starting from $S_{i}=1.5$ (see § B.3), it is likely to be the cause of the bubble envelope’s non-axisymmetric oscillation, which is a characteristic of this BVB type. Hence, this mode will be referred to as the ‘bubble mode’ and is associated with the asymmetric type of BVB.

Figure 18(b) compares the unstable global angular frequencies, $\unicode[STIX]{x1D714}_{g,r}$, predicted using base flow of the wake and bubble modes for different $S$ with those obtained in the flow simulations. Only the frequencies for those modes whose associated BVB types were observed in flow simulations are shown for clarity. It should be clarified that $\unicode[STIX]{x1D714}$, which is mainly used in this section, is related to $f$ used in previous sections by a factor of $2\unicode[STIX]{x03C0}$. There are considerable differences between the frequencies predicted from stability analysis of base flow and simulations at all $S$ and especially for the asymmetric BVB. This should be contrasted with the close match observed for a lower swirl number of $S=1.5$ close to the bifurcation point associated with the wake mode (see Moise & Mathew Reference Moise and Mathew2019, table 2).

From these results, it is seen that, although many qualitative features seen in the simulations can be understood from the stability analysis using the base flow, the quantitative predictions of the global frequencies are inaccurate. Also, the wake mode remains unstable for $1.49\leqslant S\leqslant 1.78$, although the spiral tail was observed in simulations only up to $S=1.65$. Additionally, the wake mode remains the dominantly unstable mode for $1.67\leqslant S<1.69$. To further investigate these aspects, a stability analysis of the mean flow was carried out.

8.2 Stability of mean flow

For flows in the wake of a circular cylinder, where a supercritical Hopf bifurcation is observed when Reynolds number is increased above a critical value, Barkley (Reference Barkley2006) showed that linear global stability analysis of the mean flow is better at predicting saturated flow features as compared to analysis of the base flow. Sipp & Lebedev (Reference Sipp and Lebedev2007) showed for this flow that a global weakly nonlinear expansion about the critical Reynolds number leads to a Stuart–Landau amplitude equation and elucidated the physical significance of its coefficients. While a weakly nonlinear analysis for the two unstable modes is beyond the scope of this study, a stability analysis of the mean flow was easily carried out, the results of which are presented below.

For all cases studied, only a single marginally unstable mode was observed. The eigenvalue spectrum for $S=1.7$ is provided and discussed in appendix D. The predicted global angular frequencies, $\unicode[STIX]{x1D714}_{g,r}$ are compared with those for the cases of base-flow stability analysis and numerical simulations in figure 18(b). It is seen that the stability analysis of the mean flow gives more precise predictions of the global frequency than that using the base flow. Thus, nonlinear effects are clearly seen to play an important role in determining the mode frequency associated with the sustained states observed in simulations. The frequencies are approximately constant for each BVB type in their respective swirl range of existence. The Strouhal number based on the jet diameter and centreline velocity ($St=\unicode[STIX]{x1D714}/\unicode[STIX]{x03C0}=2f$) for these modes associated with the two-celled with spiral tail and asymmetric types are approximately 0.2 and 0.1, respectively.

The unstable eigenmode associated with the two-celled BVB with spiral tail had an energy distribution almost the same as that seen in the simulations (see figure 7) and is not discussed here for conciseness. The unstable eigenmode associated with the asymmetric BVB is examined further for a reference case of $S=1.7$. Features of this unstable global eigenmode are shown in figure 19. It is clearly seen that this mode has non-negligible energy content in the bubble region ($1\leqslant x\leqslant 4$, see figure 8). An additional difference between this and the mode associated with the spiral tail type is that the former has strong energy in the shear layer regions of the flow ($1\leqslant r\leqslant 2$) in addition to the core region ($r\leqslant 1$), especially due to the axial component $\hat{u} _{x}$ (figure 19a).

Figure 19.

Contours on meridional plane based on the real part of the axial component of eigenmodes computed from (a) base-flow and (b) mean-flow stability analysis, compared with (c) the corresponding component of the most energetic SPOD mode at dominant frequency.

Comparisons of $\hat{u} _{x}(x,r)$ computed from stability analysis of the base- and mean-flow fields with the mode extracted from flow simulations are also presented in figure 19. As noted in § 2.3, to extract the features of the coherent structure from the flow simulations, an SPOD was carried out. Examining $\unicode[STIX]{x1D706}(f)$ (2.4) showed a clear dominant peak associated with a frequency of $0.047$ (see appendix E, figure 26), which closely matches with that observed in the simulations, $f_{0}=0.049$ (figure 8). Further discussions on the SPOD results are provided in appendix E. Other velocity components showed similar qualitative features for the three cases and are not shown for brevity.

From figure 19, it can be seen that many qualitative features of the SPOD mode match with those of both eigenmodes, including the finite energy content in the bubble region. One important difference between the eigenmode computed using the base flow from the rest is its relatively smaller radial extent. Scrutiny of figures 19(b) and 19(c) shows that the respective modes are enlarged in comparison. This further confirms that the analysis of the mean flow better predicts the mode features, implying that a strong mean-flow modification arises due to the unstable mode. Only minor differences exist between the SPOD and mean-flow eigenmode except in the region where the two-arm structure is present ($x\approx 10$). The exact origin of the development of the two-arm structure is unclear, although the results show that it is not due to any $|m|=2$ global mode but probably arises as a harmonic of the unstable $m=1$ mode (see appendix D and E and figures 25b and 26b).

9 Summary and conclusions

The simple addition of swirl to a jet introduces interesting variations in the observed flow structure and dynamics. Multiple long-time flow states, each with unique distinguishing features, have been previously identified in different ranges of swirls for the swirling jet at a low Reynolds number in Moise & Mathew (Reference Moise and Mathew2019). The main objective of the present study is to investigate the coexistence of these flow states for the same parametric values by means of hysteresis studies, and is motivated by indications of the same in previous experiments (Billant et al. Reference Billant, Chomaz and Huerre1998). A similar methodology to that adopted in Moise & Mathew (Reference Moise and Mathew2019) was used here, with three-dimensional numerical simulations carried out with the Maxworthy profile chosen as inflow boundary condition (Ruith et al. Reference Ruith, Chen and Meiburg2004).

A consolidated summary of all flow states observed for the Maxworthy inflow profile, based on the results from Moise & Mathew (Reference Moise and Mathew2019) and the present study, is provided next. The three major flow forms observed were the pre-VB states, BVB and CVB. Five types of BVB, including the steady, one-celled with spiral tail, pulsating, two-celled with spiral tail and asymmetric BVB, were also observed. The one-celled and two-celled types differed with respect to the number of toroidal structures present in the bubble, while the pulsating BVB was seen to be an intermediate state with an intermittent development of a second cell. A spiral structure in the bubble’s wake was observed for these three unsteady BVB types. This did not affect the upstream regions of the bubble. By contrast, the asymmetric BVB was characterized by strong oscillations of the entire bubble envelope and a two-arm structure in the bubble’s wake. Two types of CVB, including the regular and wide-open type, were identified, although only for few swirl numbers. The CVB was seen to coexist with BVB for a large swirl range, while the two-celled BVB with spiral tail was also observed to coexist with the asymmetric BVB for a small range in which the CVB could also be sustained.

These results have been obtained for the Maxworthy inflow profile with only the swirl number varied. Variations with changes in other parameters of this profile and, more generally, the effects of inflow conditions on these flow states require further investigation. Preliminary studies on the effect of shear layer thickness, $\unicode[STIX]{x1D6FF}$, showed no significant qualitative changes in the features of the VB forms and were not pursued further. Also, many of the qualitative features (including hysteresis effects) of the various flow states observed here, including CVB, two-celled BVB with spiral tail and asymmetric BVB, closely match those reported in Billant et al. (Reference Billant, Chomaz and Huerre1998), although the inflow conditions for the latter differ from this study due to the use of a converging nozzle (see Moise & Mathew Reference Moise and Mathew2019, appendix B). Similarly, BVB and CVB have been observed for other inflow profiles, including coaxial and annular swirling jets, while hysteresis effects have also been reported. Together, these results indicate that the bubble and conical forms of VB are robust features of swirling jets and that their spatio-temporal behaviours reported here might be preserved across various families of swirling jets. Nevertheless, a systematic study on the sensitivity of the observed coexisting VB forms to inflow conditions is required. It would additionally provide a better understanding of the mechanisms that sustain these states. Another interesting avenue for exploration would be to examine the effect of axial and azimuthal inflow excitations, which have been shown to give rise to fascinating behaviours in non-swirling jets, including bifurcation and blooming of jets (Reynolds et al. Reference Reynolds, Parekh, Juvet and Lee2003).

A major confounding factor that strongly influenced the breakdown forms in experiments was the presence of mild temperature variations and associated buoyancy forces (Billant et al. Reference Billant, Chomaz and Huerre1998). In the constant-density simulations used in this study, it was possible to eliminate these effects and establish the bistability of the bubble and conical forms. It would be interesting to consider the effect of buoyancy on these VB forms in future studies. The study also shows that the BVB and CVB can coexist even in the absence of Coanda or confinement effects. An important conclusion from the above results is that initial conditions play a crucial role in determining the sustained long-time flow states in laminar swirling jets. Whether bistability will be preserved for turbulent swirling jets at higher Reynolds numbers requires further scrutiny. Experimental and simulation results at high Reynolds numbers for various other similar inflow conditions resembling swirling jets suggest that this might be the case (Jiang & Shen Reference Jiang and Shen1994; Vanierschot & Van den Bulck Reference Vanierschot and Van den Bulck2007a,Reference Vanierschot and Van den Bulckb; Falese et al. Reference Falese, Gicquel and Poinsot2014; Rajamanickam & Basu Reference Rajamanickam and Basu2018).

Another significant aspect of the present study is the identification and examination of the two-celled BVB with spiral tail and asymmetric BVB. Features of these two experimentally observed BVB types, which have not been simulated previously, have been examined here in detail. For the latter type, an SPOD was implemented to extract the coherent features of the flow accurately. A global linear stability analysis was carried out to examine the origins of the helical structures observed in the simulations. Two characteristically different single helical ($m=1$) modes were found to be associated with the two types of BVB. It has been shown that the stability analysis of the mean flow is more accurate than analysis of the base flow in predicting the flow features observed in the simulations. This is especially the case for the asymmetric BVB, for which it was shown that strong mean-flow modifications are present. A weakly nonlinear analysis, similar to that performed in Meliga, Gallaire & Chomaz (Reference Meliga, Gallaire and Chomaz2012) for the Grabowski profile, might shed light on the mode selection behaviour and the bistability features observed between the two types.