3.1. Overview of the steady flow topologies

As discussed in the Introduction, the various physical processes involved in swirling jet mechanics tend to manifest an intricate state space underlying a wide range of distinct flow topologies. Therefore, before delving into the particular details of each of flow regime in the following sections, this section presents briefly the basic flow topologies of annular swirling jets. To this end, we consider a case where ${ {d}}=0.5$ and ${ {Re}}=100$ via numerical continuation along ${ {S}}$. This particular value of ${ {d}}$ was selected to enable direct comparisons with the results of Sheen et al. (Reference Sheen, Chen and Jeng1996), and the value of the Reynolds number was chosen as it is sufficiently low such that no unsteady solutions are identified for any ${ {S}}$, yet sufficiently high that an interesting sequence of distinct axisymmetric steady states exists. A qualitatively similar evolution with ${ {S}}$ of the steady flow topology occurs at other ${ {Re}}$ and ${ {d}}$ values. The results of this analysis are presented in the bifurcation diagrams and flow visualisations of figure 2. There, $\min u_x(x,0)$ denotes the minimum axial velocity along the central axis. This quantity is extracted from each steady state, allowing any solutions exhibiting central recirculation features to be identified easily as those satisfying $\min u_x(x,0)<0$.

Figure 2. Dynamics of the swirling annular jet at ${ {Re}}=100$ and ${ {d}}=0.5$. (a) ${ {S}}$–$\min u_x(x,0)$ bifurcation diagram. Solid curves indicate stable nodes, while the dashed curve indicates an unstable saddle. (b) Contour plot showing the evolution of $u_x(x,0)$ as a function of ${ {S}}$. Here, the black curves indicate on-axis stagnation points. (c) Flow visualisations from points indicated on (a,b) illustrating how the steady flow topology evolves under varying ${ {S}}$. The flow visualisations show coloured contours of the axial velocity and axisymmetric streamline projections over $(x,r)\in [-0.5,2.5]\times [0,1.5]$. Black curves are used to indicate stagnation streamlines.

Based on figure 2, we classify the annular jet flow at ${ {d}}=0.5$ and ${ {Re}}=100$ into three main flow regimes, which typically also occur at other ${ {Re}}$ and ${ {d}}$ values. First is the wake regime, which occurs at relatively low ${ {S}}$ values. In the wake regime, the primary flow feature is a small central recirculation vortex immediately beyond the expansion (see figure 2(c) subpanels 1 and 2) caused by the momentum deficit behind the centrebody. As the rotation rate increases in the wake regime, the extent of this recirculation zone decreases (figure 2(c) subpanel 3) until a small counter-rotating vortex forms along the centrebody wall. Eventually, this secondary vortex penetrates the main wake vortex, causing the recirculation zone to recede to a narrow toroid along the centrebody lip such that the axial flow along the centreline is non-negative (figure 2(c) subpanel 4). The mechanics of this collapse is consistent with the description by Sheen et al. (Reference Sheen, Chen and Jeng1996) reviewed in the Introduction, which they referred to as the ‘penetration’ of the ‘inverted triangle region’.

Further increase in swirl leads to the formation of a new momentum deficit downstream of the expansion due to the vortex breakdown phenomenon. The beginning of the second regime, termed the breakdown regime, is demarcated by the point where this momentum deficit is sufficient to trigger a flow reversal along the centreline (figure 2(c) subpanel 5). Once vortex breakdown occurs, the new isolated central recirculation zone enlarges rapidly with increasing swirl until it collides and merges with the lip vortex (figure 2(c) subpanel 6). The newly merged recirculation zone continues to grow in size as swirl increases (figure 2(c) subpanel 7) until eventually it reaches a crisis (figure 2(c) subpanel 8) where the recirculation bubble can no longer grow continuously with increasing ${ {S}}$. At this point, the solution manifold experiences a saddle–node bifurcation beyond which the bubble grows tremendously with decreasing ${ {S}}$ along an unstable solution branch (figure 2(c) subpanel 9). The mechanism behind this dynamics is identical to that in swirling circular jets (Douglas et al. Reference Douglas, Emerson and Lieuwen2021b). It is driven by a nonlinear exchange of dominance between the central low-pressure region associated with the vortex breakdown recirculation zone and the outer low-pressure region associated with restricted entrainment of the ambient fluid near the wall as the bubble expands (i.e. the Coandă effect). It should be noted that this dynamics is affected strongly by the particular geometry of our configuration and, in particular, the presence of the flush wall along $x=0$ (Douglas & Lesshafft Reference Douglas and Lesshafft2022).

Once another saddle–node bifurcation signals that this nonlinear transition is complete (figure 2(c) subpanel 10), the flow enters the third regime. Here, a Coandă-type attachment of the jet occurs along the outer wall, resulting in a wall jet configuration similar to that observed by Ogus et al. (Reference Ogus, Baelmans and Vanierschot2016). As ${ {S}}$ increases further from this point, the strength of the outer separation vortex intensifies, reducing the size of the separation zone and causing the flow to cling ever more closely to the outer wall (figure 2(c) subpanels 11 and 12). No additional flow states were identified within the range of rotation rates investigated (up to ${ {S}}=6$).

In closing this subsection, it is important to emphasise that the specific ${ {S}}$ values associated with the various transitions in steady-state flow topology are strongly dependent on the other parameters. This will be discussed further in § 3.4. It should also be noted that the interval of bistability for the circular jet ($2.05\leq { {S}}\leq 2.10$) given by Douglas et al. (Reference Douglas, Emerson and Lieuwen2021b) does not approximate the corresponding bistable interval for the annular jet even at small ${ {d}}$ (investigated here down to ${ {d}}=10^{-4}$). This is attributed to the aforementioned singularity at ${ {d}}=0$ associated with the incongruity of a no-slip surface along a one-dimensional centrebody. The effects of the centrebody diameter and Reynolds number are summarised in the regime diagrams of figure 3. Most of the observed trends can be understood quite well based on physical intuition. For example, conservation of angular momentum ensures that jets with large centrebodies are endowed with more azimuthal momentum at a given rotation rate than jets with a lower value of ${ {d}}$, which tends to shift the various transitions to lower ${ {S}}$ values at higher ${ {d}}$. Similarly, higher Reynolds numbers result in less viscous entrainment of ambient fluid into the jet, causing the swirl to dissipate more gradually as the jet evolves axially. This reduced entrainment at higher ${ {Re}}$ also stalls the transition to the wall jet regime to higher values of ${ {S}}$. Finally, note the significant sensitivity of the jet's centreline stagnation behaviour with respect to ${ {S}}$ as the flow transitions between regimes (see figure 2b). This sensitivity of the number and location of the stagnation points has significant implications on the potential flame shapes in combustor flow fields with annular swirling jets.

Figure 3. Regime diagrams for the swirling annular jet, characterising the steady solutions present (a) in the ${ {S}}$–${ {d}}$ plane at ${ {Re}}=100$, and (b) in the ${ {S}}$–${ {Re}}$ plane at ${ {d}}=0.5$. In (b), the grey curve indicates the bistable region for a circular swirling jet (Douglas et al. Reference Douglas, Emerson and Lieuwen2021b). Note that unsteady solutions are present within the range of ${ {Re}}$ shown, as discussed further below.

3.2. Symmetry breaking in the non-swirling jet

Our analysis proceeds with an investigation of the flow behaviour before the introduction of swirl. With ${ {S}}=0$, the governing system and boundary conditions are symmetric under rotation and/or reflection in both $\theta$ and $t$. Thus for sufficiently low Reynolds numbers, linearity ensures that the velocity field inherits these symmetries for any value of ${ {d}}$, resulting in steady states that are both two-dimensional axisymmetric (2-DA) and 3-D plane-symmetric (3-DPS). For such flows, disturbances with positive and negative $m$ are degenerate, leading to non-rotating interference patterns (Long & Petersen Reference Long and Petersen1992). Because of this degeneracy, the signs of $m$ and $f$ are unimportant, and we will refer to $|m|$ and $|\,f|$ throughout this subsection.

Using the branch continuation and stability analysis techniques outlined in § 2.3, the elementary 2-DA/3-DPS steady solutions that appear in the low-${ {Re}}$ limit are traced along ${ {Re}}$ and ${ {d}}$ to explore how the system bifurcates into more complex flow patterns. A stability map, a bifurcation diagram, and flow visualisations that characterise the non-swirling annular jet's incipient dynamics are presented in figure 4. In the bifurcation diagram, the evolution of the velocity fields is monitored through the maximum of the transverse velocity across the centreline, $\max |u_r(x,0)|$, since $|u_r(x,0)|>0$ for any $x$ indicates a violation of axisymmetry.

Figure 4. (a) Stability map in the ${ {Re}}$–${ {d}}$ plane and plot of critical frequency along the leading Hopf curves. (b) A $\max |u_r(x,0)|$ bifurcation diagram. (c) Flow visualisations from the indicated points illustrating the dynamics of the non-swirling annular jet. Planar visualisations show contours of axial velocity and streamline projections for the unstable 2-DA steady state in the meridional plane, and for the unstable 3-DPS steady state within and orthogonal to the plane of symmetry, as well as in an axial plane at $x=0.35$. The 3-D visualisations show yellow and black axial velocity isocontours from each branch of oscillatory Hopf bifurcations at a single instance of phase overlaid upon the symmetry plane from the 3-DPS steady state.

The results of figure 4 indicate that the steady 2-DA/3-DPS non-swirling annular jet bifurcates to non-axisymmetric, 3-DPS steady states beyond a certain critical Reynolds number that depends on ${ {d}}$. This breaking of axisymmetry is triggered by a non-oscillatory ($|\,f|=0$) $|m|=1$ Hopf bifurcation (i.e. a pitchfork bifurcation with continuous rotational symmetry) that is either subcritical or supercritical in nature, for lower and higher values of ${ {d}}$, respectively. The interface between the subcritical and supercritical bifurcations occurs where the steady 3-DPS saddle–node curve intersects the steady $|m|=1$ Hopf curve at a codimension-2 fold–Hopf bifurcation point, as indicated in figure 4(b). Hence the region between this 3-DPS saddle–node curve and the steady axisymmetry-breaking Hopf bifurcation curve for ${ {d}}\lesssim 0.84$ yields an interval of bistability between the stable 2-DA/3-DPS and 3-DPS steady states, allowing hysteresis. Visualisations of the steady 2-DA/3-DPS flow and the steady 3-DPS flow are shown in figure 4(c) subpanels 1 and 2 at the point $({ {Re}},{ {d}})=(420,0.7)$.

As mentioned in the Introduction, the loss of axisymmetry in non-swirling annular jet flows is a known phenomenon (Del Taglia et al. Reference Del Taglia, Blum, Gass, Ventikos and Poulikakos2004, Reference Del Taglia, Moser and Blum2009; Vanierschot et al. Reference Vanierschot, Van Dyck, Sas and Van den Bulck2014). Nonetheless, the detailed dynamics revealed by our bifurcation analysis leads to new insights. First, the identified subcritical bifurcation behaviour in this flow for ${ {d}}<0.84$ is a new result. This dynamics indicates a strong sensitivity of the flow to initial conditions and parameter fluctuations in the bistable region. Second, in comparison to the observations of Del Taglia et al. (Reference Del Taglia, Moser and Blum2009), the neutral curve associated with the loss of axisymmetry in our results is located over a nearly identical range of Reynolds numbers but across a wider span of centrebody diameters. We expect that this discrepancy may be attributed primarily to the lack of an inlet pipe in their configuration, which prevented their jet from developing any non-axisymmetry upstream of the dump plane.

At ${ {Re}}$ values immediately beyond the axisymmetry-breaking bifurcation, the 3-DPS steady solution manifold is linearly stable. However, the steady 3-DPS solution manifold loses its stability at higher ${ {Re}}$ values through oscillatory Hopf bifurcations as the degree of non-axisymmetry grows. As shown in figures 4(a,b), three neutral curves exist on this steady manifold that each give rise to instability modes with distinct frequencies and spatial structures. Two of these bifurcations are associated with 3-DPS structures, corresponding to unsteady Bénard–von Kármán-type vortex shedding behaviours. The main distinction between these 3-DPS modes is their frequency. The slower oscillation has Strouhal numbers $0.1\lesssim |\,f|\lesssim 0.2$, while the faster mode reaches $0.4\lesssim |\,f|\lesssim 0.5$. The slower mode is visualised in figure 4(c) subpanel 3 and supplementary movie 1 (available at https://doi.org/10.1017/jfm.2022.453) at the point $({ {Re}},{ {d}})=(344,0.7)$, and the faster mode is shown in figure 4(c) subpanel 4 and supplementary movie 2 at $({ {Re}},{ {d}})=(483,0.6)$. The remaining instability is associated with 3-D plane-asymmetric (3-DPA) streamwise streak structures that correspond to a gentle side-to-side ‘wobbling’ of the wake asymmetry. This mode is nearly static, with Strouhal number $|\,f|\sim 0.001$. It is visualised in figure 4(c) subpanel 5 and supplementary movie 3 at $({ {Re}},{ {d}})=(475,0.87)$. Note that each of these bifurcations corresponds to a primary instability of the steady 3-DPS flow for some part of the parameter space.

The instabilities described in the previous paragraph provide new insight into several earlier results.

First, the structure and frequency of the slower 3-DPS instability characterised in our results match closely with the $|\,f|=0.187$ vortex shedding mode found in the laminar ${ {d}}=0.65$ jet studied by Ogus et al. (Reference Ogus, Baelmans and Vanierschot2016). Our results indicate that this particular instability is not universally dominant in non-swirling annular jets, and appears only as the primary instability over a range of centrebody diameters from $0.65\lesssim { {d}}\lesssim 0.85$ for ${ {Re}}\leq 500$. Below ${ {d}}\sim 0.65$, the primary instability is the higher-frequency 3-DPS vortex shedding mode, while for ${ {d}}\geq 0.85$, the primary instability is the 3-DPA wobble mode. It should be emphasised here, however, that these neutral curves overlap at high ${ {Re}}$ values and are not mutually exclusive. It is likely that combinations of these basic instability modes and others will promote more complex interactions and time-aperiodic temporal dynamics as ${ {Re}}$ increases. As an example of this, we consider the strong similarities between the instability modes identified in our laminar study and the coherent turbulent structures observed by Vanierschot et al. (Reference Vanierschot, Percin and van Oudheusden2021). Using a proper orthogonal decomposition of their time-resolved experimental data, those authors identified three dominant coherent oscillations that mirror closely the space–time behaviour of each instability mode identified in our study. This suggests that the dominant coherent dynamics is driven by the same physical mechanisms across a broad range of Reynolds numbers, and motivates further studies in the transitional regime.

Second, it is interesting to speculate to possible connections between the elongated streak structures associated with the self-excited 3-DPA instability in annular jets and the streak structures that are known to be temporally stable but strongly spatially amplified in round jets (Nogueira et al. Reference Nogueira, Cavalieri, Jordan and Jaunet2019; Pickering et al. Reference Pickering, Rigas, Nogueira, Cavalieri, Schmidt and Colonius2020; Wang et al. Reference Wang, Lesshafft, Cavalieri and Jordan2021). It seems plausible that the recirculation zone in the centrebody wake could provide an avenue for intrinsic feedback within the lift-up mechanism that could destabilise such structures. Nevertheless, more work would certainly be necessary to evaluate this hypothesis.

Finally, it should be remarked that all of the bifurcations described in this subsection are influenced strongly by the physics of the centrebody wake, and have no analogues in circular jets (Douglas et al. Reference Douglas, Emerson and Lieuwen2021b). They do, however, possess clear similarities to the bifurcations of axisymmetric wake flows (Tomboulides & Orszag Reference Tomboulides and Orszag2000; Fabre, Auguste & Magnaudet Reference Fabre, Auguste and Magnaudet2008; Bohorquez et al. Reference Bohorquez, Sanmiguel-Rojas, Sevilla, Jiménez-González and Martínez-Bazán2011; Rigas, Esclapez & Magri Reference Rigas, Esclapez and Magri2016). The onset of unsteadiness in both cases involves an initial steady break of axisymmetry followed by subsequent instabilities of a steady 3-D solution manifold. Hence oscillatory linear instability arises from a 3-D state rather than an axisymmetric one. To illustrate this point better, we have performed stability calculations on the axisymmetric manifold beyond the leading neutral curve associated with the steady loss of axisymmetry. For the unstable steady axisymmetric flow, the minimum Reynolds number associated with an oscillatory instability is ${ {Re}}=574$, where an $|m|=1$ Hopf bifurcation associated with vortex shedding at $|\,f|=0.325$ occurs at ${ {d}}=0.68$. Nonetheless, this instability is of limited physical significance as neither its critical parameter values nor its frequency are representative of the actual instabilities stemming from the 3-DPS state. This has important implications towards modelling and controlling non-swirling annular jet flows, as the instability characteristics of the axisymmetric steady state do not reflect those of the true 3-D flow.

3.3. Effect of low rotation rates on wake asymmetry

Though the system and boundary conditions remain 2-DA with ${ {S}}\neq 0$ at low ${ {Re}}$, introducing any finite amount of rotation to the pipe immediately destroys the system's 3-DPS property. Hence for ${ {S}}\neq 0$, disturbances with positive and negative $m$ are no longer degenerate, leading to 3-D structures that generally rotate in time. (Note that in systems not exhibiting such exact temporal–azimuthal symmetries as ours, it is possible for robust non-axisymmetric steady or time-averaged solutions to arise at non-zero swirl; Pacheco, Lopez & Marques Reference Pacheco, Lopez and Marques2011.) In this work, as in Douglas et al. (Reference Douglas, Emerson and Lieuwen2021b), we have restricted our analysis to ${ {S}}\geq 0$ and $m\leq 0$ without loss of generality such that the sign of $f$ determines whether a non-axisymmetric disturbance co-rotates ( $f>0$) or counter-rotates ($\,f<0$) along $\theta$. Nevertheless, all results will be reported in terms of $|m|$ to avoid confusion with other studies, which sometimes use the sign of $m$ to distinguish among disturbances with different rotation directions.

The stability map of figure 5(a) illustrates the effect of weak swirl for a range of ${ {d}}$ and ${ {S}}\leq 0.3$ with the Reynolds number fixed at ${ {Re}}=400$. It also shows that the subcritical and bistable behaviour described above for the non-swirling jet at certain ${ {d}}$ values persists for non-zero ${ {S}}$. Taking the case of ${ {d}}=0.8$ as a representative example, figure 5(b) demonstrates how the steady 3-DPS solution from the non-swirling case inherits $f>0$ for any ${ {S}}>0$, manifesting a co-rotating $|m|=1$ limit cycle manifold. Flow visualisations provided in figure 5(c) subpanel 1 and supplementary movie 4 further reveal how the spatial structure of these limit cycle oscillations follows from a swirl-induced twisting of the steady 3-DPS structure from the non-swirling case. The end result is a co-rotating $|m|=1$ spiral structure that winds about the central axis in the direction opposing the pipe's rotation as the jet proceeds axially. Nonetheless, the amplitude of this limit cycle decays with increasing ${ {S}}$ as the structure of the wake recirculation region changes. The upward trend of $\min u_x(x,0)$ in figure 5(b) indicates that the reversed axial flow in the centrebody wake region diminishes with increasing swirl. As this wake recirculation weakens, the amplitude of the $|m|=1$ limit cycle decays until eventually the flow becomes unable to sustain the wake asymmetry. At this point, depicted in figure 5(c) subpanel 2, the system bifurcates to a stable 2-DA steady state. With further increase in ${ {S}}$, the secondary ‘inverted triangle zone’ vortex apparent in figure 5(c) subpanel 2 ‘penetrates’ through the primary wake vortex to yield the topology shown in figure 5(c) subpanel 3. This evolution of the time-mean and steady flow fields with increasing ${ {S}}$ occurs in a manner consistent with the description in § 3.1 and earlier reports (Sheen et al. Reference Sheen, Chen and Jeng1996; Ogus et al. Reference Ogus, Baelmans and Vanierschot2016).

Figure 5. Characterisation of the weakly swirling jet at ${ {Re}}=400$. (a) Stability map in the ${ {S}}$–${ {d}}$ plane. (b) Results for the case ${ {d}}=0.8$ with varying ${ {S}}$, showing a bifurcation diagram for $\min u_x(x,0)$, a contour plot for $u_x(x,0)$ of the steady and periodic solutions, and a bifurcation diagram for $f$. (c) Flow visualisations at ${ {d}}=0.8$ and three points in ${ {S}}$ showing meridional axial velocity contours and streamline projections for $(x,r)\in [-0.5,2.5]\times [0,1]$ of the 2-DA states derived from the mean component of the $|m|=1$ limit cycle and the steady 2-DA solutions. A 3-D visualisation of the instantaneous flow on the $|m|=1$ limit cycle at ${ {S}}=0.1$ is also shown, using a meridional slice plane and yellow and black isocontours of $\acute {u}_x=\pm 0.1$ for $(x,r,\theta )\in [-1,9]\times [0,2.5]\times [0,2{\rm \pi} ]$.

It is interesting to note that the behaviour of the co-rotating, counter-winding $|m|=1$ limit cycle described above and in figure 5 bears a striking resemblance to the observations of Vanierschot et al. (Reference Vanierschot, Van Dyck, Sas and Van den Bulck2014) in turbulent jets. This similarity suggests that the breaking of planar symmetry caused by non-zero swirl and the subsequent weakening of the wake recirculation zone with increasing swirl are both significant dynamical effects that control the jet's coherent structures even at much higher Reynolds numbers. Hence it is plausible that low swirl could be used to manipulate the dominant frequencies present in annular jet applications.

Here, we should point out that the techniques used in our study characterise explicitly only steady and time-periodic solutions. Nonetheless, our results can be extended straightforwardly to more complex dynamics and bifurcations using the continuous parameter dependence and symmetry of (2.3). For example, the Hopf bifurcations of the 3-DPS steady state associated with 3-DPS vortex shedding at ${ {S}}=0$ (indicated in figure 5 and detailed in figure 4) must correspond to Neimark–Sacker bifurcations of the co-rotating $|m|=1$ limit cycle state for ${ {S}}>0$ due to the destruction of the 3-DPS symmetry under finite ${ {S}}$. Such dynamics indicate the existence of quasi-periodic solution manifolds (i.e. limit tori) within the weakly swirling annular jet state space composed of ‘fast’ vortex shedding and ‘slow’ wake precession motions, consistent with the observations of Ogus et al. (Reference Ogus, Baelmans and Vanierschot2016). Furthermore, at $({ {Re}},{ {d}})$-values where at least two modes are unstable for ${ {S}}=0$, the slow precession that appears for ${ {S}}>0$ may manifest a strange attractor with chaotic dynamics via a Ruelle–Takens–Newhouse scenario (Eckmann Reference Eckmann1981).

It will be shown in the next subsection that sufficiently strong rotation promotes flow instability. Yet, based on the above results, weaker rotation levels may be viewed to have a stabilising effect from the perspective of the annular jet's intrinsic dynamics. It is worth mentioning, however, that a non-modal perspective towards flow instability may lead to a very different conclusion. Indeed, we did find ‘arc branches’ of slightly damped eigenvalues (Garnaud et al. Reference Garnaud, Lesshafft, Schmid and Huerre2013; Coenen et al. Reference Coenen, Lesshafft, Garnaud and Sevilla2017) throughout the wake regime, and weakly swirling jets are well known to amplify transiently very small extrinsic perturbations through pseudo-resonance in a manner that is not captured directly by the normal modes approach used in this study (Schmid Reference Schmid2007). For example, Montagnani & Auteri (Reference Montagnani and Auteri2019) considered weakly swirling co-axial jets (which do not have a centrebody wake) with underdeveloped velocity profiles at Reynolds numbers similar to those studied here, and showed that the optimal degree of non-modal amplification grows monotonically with swirl under the conditions that they investigated. They also showed that tremendous algebraic growth of small transient perturbations can give rise to nonlinear interactions that are sufficient to sustain oscillations and/or turbulence. Nonetheless, as their transition was triggered by extrinsic forcing, such dynamics may not be accessible to the purely self-excited system considered in our investigation. The least stable subcritical eigenvalue branches in our study suggest a preference of the steady flow towards non-modal amplification of counter-rotating $|m|=1$ and $|m|=2$ disturbances, with lower ${ {S}}$ conditions favouring $|m|=1$ growth. However, non-modal analysis would be necessary to confirm this.

3.4. Influence of centrebody size on vortex breakdown and precession

As the rate of rotation increases beyond the relatively small values considered above, eventually vortex breakdown and flow instability begin to play prominent roles in the jet's dynamics. The basic mechanics of this process for the steady solutions has already been discussed in § 3.1. However, at higher Reynolds numbers, this steady dynamics is complicated by the emergence of unsteady behaviours that will be the main focus of this subsection. The present discussion will highlight the effects of varying ${ {d}}$ and ${ {S}}$ at a fixed Reynolds number ${ {Re}}=400$. A stability map and flow regime diagram that summarise the topology (shading) and stability limits (lines) of the 2-DA steady flow at ${ {Re}}=400$ are shown in figure 6.

Figure 6. Summary of the dynamics of the steady annular swirling jet at ${ {Re}}=400$. (a) Stability map and flow regime diagram. (b) Plot of critical frequency along the primary $|m|=1$ and $|m|=2$ Hopf curves shown in (a). In (a), the bistable region associated with the transition between breakdown and wall jet solutions is demarcated by the black curve. The neutral curves corresponding to Hopf bifurcations associated with $|m|=2$ or $|m|=1$ structures are indicated by the solid red and blue curves. The regions where a 2-DA solution is linearly unstable are shaded grey, while the green regions indicate stable 2-DA solutions. The shading of these green regions denotes the number of stagnation points present along the centreline, which is used to classify the steady flow topology into six categories.

We begin with a brief discussion of the 2-DA steady flow topologies, and their evolution with varying ${ {S}}$ and ${ {d}}$. Figure 6(a) shows that the evolution of the flow topology with ${ {S}}$ described by Sheen et al. (Reference Sheen, Chen and Jeng1996) and detailed in figure 2 is typical of other centrebody sizes and higher Reynolds numbers. As the rotation rate and/or the centrebody diameter change, the flow progresses through this sequence of states in a manner that can be distinguished precisely by the number of stagnation points identified on the flow centreline. As noted earlier, unlike the circular jet, the presence of the centrebody in the annular jet ensures that at least one stagnation point is always present on the centreline. Nonetheless, the presence of the centrebody wake (regime A, e.g. figure 2(c) subpanel 1), the formation of the ‘inverted triangle region’ (regime B, e.g. figure 5(c) subpanel 2), the ‘penetration’ of the inverted triangle region (regime C, e.g. figure 5(c) subpanel 3), and the formation and attachment of the vortex breakdown recirculation zone (regimes D and E, e.g. figure 2(c) subpanels 5 and 7, respectively) all introduce or eliminate stagnation points, as reflected in figure 6(a). With the exception of the bistability between the merged vortex breakdown state and the wall jet state (regime F, e.g. figure 2(c) subpanels 7 and 11, respectively), the steady 2-DA solution manifold for the swirling annular jet is single-valued, and there is no parameter overlap among these distinct steady states. Furthermore, aside from the change between regimes E and F, the transitions among these steady flow patterns are not associated with bifurcations. It will be shown below that this single-valuedness does not carry over into the unsteady regime, and that some of the steady states do experience multi-valuedness with unsteady solutions.

As was mentioned in § 2.1, the presence of the centrebody introduces a no-slip condition that distinguishes the annular jet from a circular jet even at very small values of ${ {d}}$. Such configurations are realised practically in applications where the centrebody tapers gradually to a point (as in Ko & Chan Reference Ko and Chan1979, for example). Nonetheless, when the centrebody is sufficiently small, viscous effects dissipate quickly the central momentum deficit, and the centrebody wake plays very little role in the jet's instability behaviour. As a result, we find that the jet's unsteady dynamics is dominated by instabilities of the outer shear layer that behave very similarly to the case of a circular jet for a range of ${ {d}}$ dependent on the Reynolds number and the nature of the instability. Thus for annular jets at ${ {Re}}=400$ where the centrebody diameter lies within the range $0<{ {d}}\lesssim 0.4$, the primary instability occurs in the wake regime via a slowly co-rotating $|m|=2$ mode, as shown in figure 6. The resulting periodic orbit emerges via a subcritical Hopf bifurcation to a limit cycle associated with a slowly co-rotating $|m|=2$ spiral structure that also appears in the quasi-columnar regime of circular swirling jets (Billant et al. Reference Billant, Chomaz and Huerre1998; Douglas et al. Reference Douglas, Emerson and Lieuwen2021b) and is driven by azimuthal shear along the outer shear layer (Gallaire & Chomaz Reference Gallaire and Chomaz2003). Apart from this leading $|m|=2$ instability, the steady jet also experiences instability towards a counter-rotating $|m|=1$ mode, co-rotating $|m|=3$ modes, and other $|m|=2$ modes at ${ {S}}$ values beyond the range shown here, in a manner consistent with the description in our earlier paper. Generally, these instabilities exhibit large spatial amplification over an extended axial span, and structural sensitivity analysis shows that they are not strongly influenced by the flow structure in the interior of the jet. Thus for small ${ {d}}$, the annular swirling jet becomes unsteady and non-axisymmetric at ${ {S}}$ values before any central momentum deficit forms, and therefore does not experience vortex breakdown via a steady or axisymmetric process. Instead, the transition to vortex breakdown likely involves several competing solution manifolds (which need not be steady or periodic), with the selection of any particular attractor being dependent on the specific initial conditions. Because of the similarity of the unsteady dynamics between the small ${ {d}}$ annular jet in the wake regime and the circular jet in the quasi-columnar regime, we will move on to situations where the effect of the centrebody-driven inner shear layer is more prominent. The reader is referred to Douglas et al. (Reference Douglas, Emerson and Lieuwen2021b) for further details of these outer shear layer dynamics, though we must again emphasise the fundamental difference between a small-${ {d}}$ annular jet and a circular jet.

At intermediate ${ {d}}$ values, figure 6 indicates that a neutral curve associated with an $|m|=1$ instability crosses over the critical $|m|=2$ curve at ${ {d}}\sim 0.45$ to become the primary instability mode. The diagram reveals that just prior to criticality, the steady flow exists in the wake regime with a steady flow topology analogous to that of figure 5(c) subpanel 3. Thus the emergence of the $|m|=1$ instability appears prior to the emergence of the vortex breakdown recirculation bubble for $0.45\lesssim { {d}}\lesssim 0.6$. This indicates that the annular swirling jet at ${ {Re}}=400$ follows an unsteady non-axisymmetric route to vortex breakdown for this range of centrebody diameters, and similar results were found for lower Reynolds numbers. An example of the resulting dynamics at ${ {d}}=0.5$ is considered in the bifurcation diagrams and flow visualisations of figure 7 and in supplementary movie 5. Among the bifurcation diagrams, figure 7 shows the amplitude of the limit cycle solutions as measured using the unsteady energy norm, $\|\acute {\boldsymbol {u}}\|=\|\boldsymbol {u}-\bar {\boldsymbol {u}}\|$. This spatially integrated quantity includes all contributions from the oscillation's fundamental and harmonic tones over all space.

Figure 7. (a,c,d) Bifurcation diagrams illustrating the dynamics of the vortex breakdown process at $({ {Re}},{ {d}})=(400,0.5)$, and (b) flow visualisations of the $|m|=1$ PVC solution at the indicated point (${ {S}}=0.88$). The bifurcation diagrams show the minimum velocity along the centreline, the limit cycle amplitude, and the oscillation frequency against ${ {S}}$. Visualisations consist of a comparison of the axisymmetric steady and mean flow fields in the meridional plane, as well as a snapshot of the instantaneous flow from the $|m|=1$ limit cycle via planar and 3-D representations. Planar graphics show contours of the axial velocity and with projected streamlines, while the 3-D graphic includes a yellow/black isocontour representation of the radial velocity fluctuations at $\acute {u}_r=\pm 0.15$. The black dotted line shows the intersection of the axial and meridional planes.

For the case ${ {Re}}=400$ and ${ {d}}=0.5$, figure 7 shows that the base flow becomes unstable with respect to several distinct $|m|=1$ and $|m|=2$ disturbances as ${ {S}}$ increases. These dynamics are led by the emergence of an $|m|=1$ limit cycle, which bifurcates from the steady 2-DA flow before the steady 2-DA flow begins to express a central vortex breakdown recirculation zone. As this limit cycle's amplitude increases, nonlinear interactions between its mean and fluctuating components eventually begin to aid the deficit of momentum along the centreline more than a corresponding increase in ${ {S}}$, leading to subcritical behaviour. The resulting oscillation consists of a co-rotating, counter-winding $|m|=1$ PVC structure that exhibits a significant interval of multi-valuedness between itself and the steady 2-DA solution. This therefore represents a marked departure from an interpretation common to many existing studies of swirling flows. Such studies have typically adopted a linear framework for analysing flow oscillations and have focused mostly on simple swirling flow models like the Grabowski–Berger vortex. In such works, the formation of the vortex breakdown recirculation zone via an axisymmetric, steady or quasi-steady process is typically regarded as a prerequisite for unsteady vortex breakdown (Escudier, Bornstein & Maxworthy Reference Escudier, Bornstein and Maxworthy1982; Ruith et al. Reference Ruith, Chen, Meiburg and Maxworthy2003; Herrada & Fernandez-Feria Reference Herrada and Fernandez-Feria2006; Meliga et al. Reference Meliga, Gallaire and Chomaz2012). Instead, our results demonstrate that the nonlinear Reynolds stresses induced by finite-amplitude oscillations can yield a time-averaged breakdown zone and instantaneous PVC at parameter values where vortex breakdown does not occur for the steady flow. This indicates an alternative subcritical route to unsteady vortex breakdown that exists in addition to the supercritical route associated with a simultaneous emergence of the central recirculation zone and PVC oscillations (Manoharan et al. Reference Manoharan, Frederick, Clees, O'Connor and Hemchandra2020).

At higher ${ {d}}$ values along the boundary of regions C and D in figure 6(a), the vortex breakdown recirculation zone forms in a steady manner as an isolated axisymmetric bubble downstream of the centrebody. It is only after the rotation increases well beyond the point where the internal stagnation point forms initially that self-excited flow instabilities become relevant to the overall dynamics. The primary unsteady dynamics subsequent to vortex breakdown includes co-rotating, counter-winding structures with $|m|=1$ or $|m|=2$ spatial patterns depending on the specific ${ {d}}$ and ${ {Re}}$ values. Much like the intermediate-${ {d}}$ case considered just above, the limit cycle solutions that emerge from these Hopf bifurcations may arise in either a subcritical or supercritical manner depending on the specific flow circumstances at the considered parameter values. Finally, it is interesting to note from figure 6(b) that the frequencies associated with the unsteady $|m|=1$ and $|m|=2$ solutions in this regime tend to decrease with increasing ${ {d}}$ along the critical curves, with relatively little effect from the variations in ${ {S}}$.

It is interesting to compare the results described above with existing works in the annular swirling jet literature that have considered the influence of the centrebody diameter. For example, in the turbulent regime, Mukherjee et al. (Reference Mukherjee, Muthichur, More, Gupta and Hemchandra2021) and Gupta et al. (Reference Gupta, Shanbhogue, Shimura, Ghoniem and Hemchandra2021) have both suggested that PVC oscillation amplitudes may be diminished by increasing the centrebody diameter in order to merge the bubble and wake recirculation zones. This is akin to shifting the operating regime from a D- to E-type solution in figure 7. As pointed out by a referee, however, our results suggest the opposite effect, as increasing ${ {d}}$ near ${ {S}}=0.8$ is associated with the destabilisation of an $|m|=1$ structure near the D/E transition. Even so, there are several plausible explanations for this potential discrepancy between our laminar results and the turbulent results cited above. These include effects related to configurational differences such as the swirl generation mechanism (rotating walls versus static vanes), jet profile (parabolic versus top-hat), or nozzle injection angle (perpendicular versus angled inward). It is also possible that higher Reynolds numbers and turbulence are sufficient to explain these different trends without such configurational influences. Further work will be necessary to evaluate systematically these various sensitivities and relate definitively these laminar and turbulent results.

3.5. Wall interactions and attachment

This subsection focuses specifically on the wall jet regime, which occurs at high levels of rotation. As shown in figure 3, the transition to a wall jet solution occurs via a sequence of two saddle–node bifurcations of the steady 2-DA flow whereby the vortex breakdown recirculation zone expands dramatically due to the realisation of a favourable pressure gradient along the dump plane wall related to the Coandă effect. A detailed account of the flow topology and flow physics in the wall jet regime, and its hysteretic relationship with quasi-columnar-type solutions, has been discussed in other works (Ogus et al. Reference Ogus, Baelmans and Vanierschot2016; Douglas et al. Reference Douglas, Emerson and Lieuwen2021b; Douglas & Lesshafft Reference Douglas and Lesshafft2022). The main purpose of this subsection is to note briefly the extensive interval of bistability that forms in the annular swirling jet at moderate Reynolds numbers, and to mention the implications of this physics in practical situations.

As suggested above, compared to a circular swirling jet, the annular swirling jet exhibits a very large interval of bistability in ${ {S}}$. For ${ {Re}}\geq 100$, Douglas et al. (Reference Douglas, Emerson and Lieuwen2021b) showed that typically, fully developed jets without a centrebody and with flush injection exhibit a hysteresis interval of width $\Delta { {S}}\sim O(0.1)$. However, for annular swirling jets, figure 3 shows that $\Delta { {S}}\sim O(1)$ for ${ {d}}\lesssim 0.5$ at ${ {Re}}=100$, and indicates that this range grows dramatically as the Reynolds number increases. For instance, at ${ {Re}}=250$, $\Delta { {S}}\sim O(10)$ for the same range of ${ {d}}$, and this hysteresis interval continues to widen for higher ${ {Re}}$. This stark broadening of the hysteresis region indicates that a transition to or from the wall jet regime is unlikely to occur spontaneously due to fluctuations in the incoming flow rate and swirl levels pushing a given state beyond the saddle–node bifurcation at ${ {S}}_b$. Instead, such a transition would likely require external excitation or special initial conditions that cause the flow to ‘jump’ spontaneously between the breakdown and wall jet solution manifolds. Some experimental evidence for this scenario has been found previously in the forced turbulent study of O'Connor & Lieuwen (Reference O'Connor and Lieuwen2012), where a switch from a breakdown state to a wall jet state was reported under sufficiently strong external acoustic forcing.

The possibility of such an extreme shift in the bulk flow topology at high swirl conditions due to noise or other factors is a significant concern for engineering systems featuring swirling annular jets. The flat dump plane wall considered in our study offers a simple example of the possible dynamics associated with wall attachment at high swirl. However, more complex nozzle geometries and confinement situations could lead to hysteresis at more moderate swirl conditions (Douglas & Lesshafft Reference Douglas and Lesshafft2022). For example, Ogus et al. (Reference Ogus, Baelmans and Vanierschot2016) have provided numerical and experimental evidence of wall jet–central jet transitions in their recessed and tapered nozzle design, which seems to have shifted the interval of hysteresis to swirl levels lower than those seen in our study. Similar findings were also reported in the turbulent regime by Vanierschot & Van den Bulck (Reference Vanierschot and Van den Bulck2007a,Reference Vanierschot and Van den Bulckb).