This paper presents a vortex breakdown study due to the interaction between a Batchelor vortex and the crossing of oblique shock waves with Mach numbers of 3.5 and 5.0, favourable for supersonic mixing and combustion, respectively. Numerical simulations were conducted to investigate the effects of the circulation intensity and shock angle on vortex breakdown. The results indicate that a breakdown occurs at the shock angle $\beta \geqslant 45^\circ$ or the vortex circulation $q = 0.32$, and the configuration is a bubble structure with a recirculation region; most of the breakdowns possess a stagnation point. Furthermore, the structure differs from that of a normal shock wave and vortex interaction because the bubble region is subsonic and does not comprise a normal shock wave on the inside. Additionally, this vortex breakdown shows that the momentum flux on the centreline decreases once at the tip of the bubble owing to a sudden drop in velocity in the subsonic region. In addition, the enstrophy production resulting from vortex stretching and tilting is found to have a significant advantage in the interaction region. Based on these results, the threshold required for a bubble vortex breakdown was theoretically derived as an inequality. The numerical simulation results support the theoretical criterion obtained from the proposed inequality. Therefore, a streamwise vortex breakdown resulting from the interaction between the vortex and intersecting oblique-shocks should be reasonably predicted.

The purpose of this research was to provide further understanding of turbulent dynamics and heat transfer mechanisms in accelerating flows with thermophysical variations and pressure drops in micron tubes. Direct numerical simulations were conducted to investigate the turbulence to supercritical pressure ${\rm CO}_2$ in heated micron tubes with inner diameter $99.2~\mathrm {\mu }$m. In general, the turbulent heat transfer enhancement/deterioration at supercritical pressure is dominated by variations in thermophysical properties, buoyancy and thermal acceleration; however, the mechanism differs in micron tubes ($d^* < 100\ \mathrm {\mu }$m). The results showed that the pressure drop and scale effect made significant contributions to the development of turbulence flows heated at supercritical pressure in micron tubes, leading to the prominent property change and flow acceleration in the inlet fully developed turbulent flow. The deviation on temperature distribution because of pressure changes was non-negligible. The primary contribution of the acceleration was the decay of a boundary layer, which significantly suppressed the production of turbulence and decreased heat transfer. The acceleration had stabilizing effects on the ejection and sweep motions of the turbulent flow. The high-speed fluid contributed to a new disturbance scenario of the flow with a larger spanwise wavenumber superimposed on existing perturbations. The high-speed streak width in the quasilaminar region was approximately $150$–$160\nu /u_\tau$ in accelerating flow. In the micron tubes, the Reynolds stress events of quadrant Q4 contributed 60 % of the Reynolds stress, greater than those of quadrant Q2.

The formation of swirling vortex rings and their early time evolution, resulting from the controlled discharge of an incompressible, Newtonian fluid into a stationary equivalent fluid bulk, is explored for weak to moderate swirl number $S \in [0, 1]$. Two practically realisable inlet conditions are investigated with swirl simultaneously superposed onto a linear momentum discharge; the corresponding circulation based Reynolds number is 7500. The results obtained reveal that for $S > 1/2$, the addition of swirl promotes the breakdown of the leading primary vortex ring structure, giving rise to the striking feature of significant negative azimuthal vorticity generation in the region surrounding the primary vortex ring core, whose strength scales with ${S}^2$. Through a nonlinear interaction with the vortex breakdown, the radius of the primary toroidal vortex core is rapidly increased; consequently, the self-induced propagation velocity of the leading ring decreases with $S$ and vortex stretching along the circular primary vortex core increases counteracting viscous diffusion effects. The latter governs the evolution of the peak vorticity intensity and the swirl velocity magnitude in the primary ring core, the circulation growth rate of the primary ring, as well as the vorticity intensity of the trailing jet and hence its stability. This combination of effects leads to an increased dimensionless kinetic energy for the primary ring with increasing $S$ and results in an almost linearly decreasing circulation based formation number, $F$.

The linear and nonlinear dynamics of two-phase swirling flows produced by the Grabowski–Berger profile under the influence of a viscosity stratification are investigated. We perform axisymmetric nonlinear simulations and fully three-dimensional linear global stability analysis of the flow for several swirl numbers $S$ and viscosity ratios $\tilde {\mu }$. They are accompanied by nonlinear three-dimensional simulations and subsequent modal analysis using the bispectral mode decomposition, recently introduced by Schmidt (Nonlinear Dyn., vol. 102, issue 4, 2020, pp. 2479–2501). We find a pronounced destabilising effect of the viscosity stratification on both the onset of axisymmetric vortex breakdown and helical instability that is linked to the required shear stress continuity across the interface. Consequently, destabilisation is shifted to lower $S$ as compared with an equivalent flow with uniform viscosity. Further, the stability analysis reveals the simultaneous destabilisation of two global modes with wavenumbers $m=1$ and $m=2$ that have harmonic frequencies. The analysis of the nonlinear flow reveals a strong triadic resonance between these modes that governs the nonlinear dynamics and leads to a rapid departure from the linear dynamics. At larger swirl, the bifurcation of additional modes initiates an interaction cascade by means of triadic resonance which is elucidated by the bispectral analysis. It leads to the emergence of a variety of additional modes in the nonlinear flow. This study contributes to an improved understanding of the influence of viscosity stratification on the onset of vortex breakdown and the destabilisation of global modes. Further, it provides a clear picture of the dynamics of swirling flows with a codimension-two point and related triadic interaction of two global modes at harmonic frequencies and wavenumbers.

The spatial structure and time evolution of tornado-like vortices in a three-dimensional cavity are studied by topological analysis and numerical simulation. The topology theory of the unsteady vortex in the rectangular coordinate system (Zhang, Zhang & Shu, J. Fluid Mech., vol. 639, 2009, pp. 343–372) is generalized to the curvilinear coordinate system. Two functions $\lambda (q_1,t)$ and $q(q_1,t)$ are obtained to determine the topology structure of the sectional streamline pattern in the cross-section perpendicular to the vortex axis and the meridional plane, respectively. The spiral direction of the sectional streamlines in the cross-section perpendicular to the vortex axis depends on the sign of $\lambda (q_1,t)$. The types of critical points in the meridional plane depend on the sign of $q(q_1,t)$. The relation between the critical points of the streamline pattern in the meridional plane and that in the cross-section perpendicular to the vortex axis is set up. The flow in a three-dimensional rectangular cavity is numerically simulated by solving the three-dimensional Navier–Stokes equations using high-order numerical methods. The spatial structures and the time evolutions of the tornado-like vortices in the cavity are analysed with our topology theory. Both the bubble type and spiral type of vortex breakdown are observed. They have a close relationship with the vortex structure in the cross-section perpendicular to the vortex axis. The bubble-type breakdown has a conical core and the core is non-axisymmetric in the sense of topology. A criterion for the bubble type and the spiral type based on the spatial structure characteristic of the two breakdown types is provided.

This paper explores the effect of axial and radial confinement on the flow topology of laminar swirling jets. Its objective is to provide a unifying perspective toward swirling jet mechanics that connects earlier reports across a variety of confined and unconfined flow situations, and over a range of swirl ratio $S$ values. The analysis focuses separately on the influence of the jet's injection depth $L$ in a radially unconfined flow and of the chamber diameter $C$ in radially confined jets. In the former case, it shows that axial confinement influences strongly the jet's behaviour when $L$ is small, allowing bistable steady states: a central jet (CJ) solution with or without a small central recirculation zone (CRZ), and a wall jet (WJ) solution with a wide-open CRZ spreading along the reservoir's edge. Similar behaviour is identified for radially confined jets, where bistable CJ and WJ states appear over a range of moderate $C$ values, and the WJ state adopts a conical CRZ. In either case, the WJ solution appears or disappears via saddle–node bifurcations when the confinement is made sufficiently strong or weak, respectively. This dynamics is attributed to an exchange of dominance between central and outer low-pressure regions as the flow transitions from CJ to WJ, or vice versa. The findings demonstrate that the hysteresis associated widely with swirling jets is controlled not just by vortex breakdown, but also by confinement through the Coandă effect. Such confinement is found to alter significantly the state-space structure even when the walls are far from the nozzle.

This paper presents bifurcation analyses characterising the nonlinear dynamics of fully developed laminar annular jets with respect to the centrebody diameter ${ {d}}$, Reynolds number ${ {Re}}$, and swirl ratio ${ {S}}$. Similar flows appear in numerous applications and feature a vibrant range of topological and dynamical characteristics associated with phenomena including shear layer separation and vortex breakdown. Our results begin by describing the non-monotonic evolution of the axisymmetric jet's steady topology under varying ${ {S}}$. In accord with earlier reports, the jet progresses through a sequence of wake, breakdown and wall jet regimes in a qualitatively similar manner across a wide span of ${ {d}}$ and ${ {Re}}$ values. In the wake regime, the non-swirling jet bifurcates to a plane-symmetric, but not axisymmetric, steady flow pattern beyond a ${ {d}}$-dependent critical ${ {Re}}$ value. With further increase in ${ {Re}}$, the steady non-swirling jet destabilises subsequently via multiple distinct Hopf bifurcations. Introducing ${ {S}}>0$ to the jet also induces unsteadiness by twisting the singly azimuthally periodic ($|m|=1$) asymmetric wake structure and causing it to precess periodically in time about the central axis. Intermediate swirl stabilises this unsteady dynamics and restores the jet's axisymmetry. This stabilising effect is then reversed in the breakdown regime at higher ${ {S}}$, where a variety of different $|m|=1$ and $|m|=2$ instabilities bifurcate from the steady flow as ${ {S}}$ is increased. Several instances of hysteresis and subcritical behaviour are reported and discussed, including one that manifests precessing vortex core oscillations.

Theoretical predictions and numerical simulations are used to determine the transition to bubble and conical vortex breakdown in low-Mach-number laminar axisymmetric variable-density swirling jets. A critical value of the swirl number $S$ for the onset of the bubble ($S^*_B$) and the cone ($S^*_C$) is determined as the jet-to-ambient density ratio $\varLambda$ is varied, with the temperature dependence of the gas density and viscosity appropriate to that of air. The criterion of failure of the slender quasi-cylindrical approximation predicts $S^*_B$ that decreases with increasing values of $\varLambda$ for a jet in solid-body rotation emerging sharply into a quiescent atmosphere. In addition, a new criterion for the onset of conical breakdown is derived from divergence of the initial value of the radial spreading rate of the jet occurring at $S^*_C$, found to be independent of $\varLambda$, in an asymptotic analysis for small distances from the inlet plane. To maintain stable flow in the unsteady numerical simulations, an effective Reynolds number $Re_{eff}$, defined employing the geometric mean of the viscosity in the jet and ambient atmosphere, is fixed at $Re_{eff}=200$ for all $\varLambda$. Similar to the theoretical predictions, numerical calculations of $S^*_B$ decrease monotonically as $\varLambda$ is increased. The critical swirl numbers for the cone, $S^*_C$, are found to depend strongly on viscous effects; for $\varLambda =1/5$, the low jet Reynolds number (51) at $Re_{eff}=200$ delays the transition to the cone, while for $\varLambda =5$ at $Re_{eff}=200$, the large increase in kinematic viscosity in the external fluid produces a similar trend, significantly increasing $S^*_C$.

Shock-induced vortex breakdown, which occurs on the delta wings at transonic speed, causes a sudden and significant change in the aerodynamic coefficients at a moderate angle-of-attack. Wind-tunnel tests show a sudden jump in the aerodynamic coefficients such as lift force, pitching moment and centre of pressure which affect the longitudinal stability and controllability of the vehicle. A pneumatic jet operated at sonic condition blown spanwise and along the vortex core over a 60° swept delta-wing-body configuration is found to be effective in postponing this phenomenon by energising the vortical structure, pushing the vortex breakdown location downstream. The study reports that a modest level of spanwise blowing enhances the lift by about 6 to 9% and lift-to-drag ratio by about 4 to 9%, depending on the free-stream transonic Mach number, and extends the usable angle-of-attack range by 2°. The blowing is found to reduce the magnitude of unsteady pressure fluctuations by 8% to 20% in the aft portion of the wing, depending upon the method of blowing. Detailed investigations carried out on the location of blowing reveal that the blowing close to the apex of the wing maximises the benefits.

Wind-tunnel experiments have been conducted on cylindrical models with canted fins. The fins introduced a swirling motion into the wake downstream of a blunt-based afterbody aligned with a Mach 2 flow. Measurements of the velocity field downstream of the models and the pressure distribution at the model base show evidence of two wake flow patterns distinctively differing from the classical supersonic wake, depending on the degree of rotation introduced. For a fin-cant angle of 16$^\circ$, a rotating wake flow with a central, downstream-directed vortex tube and a concentric, counter-rotating, toric vortex pair forms. A higher fin-cant angle of 32$^\circ$, in turn, results in a swirling flow surrounding a region of low-momentum flow at the axis. Near the central axis of the flow field an upstream flow establishes, extending from the far wake up to the model base. Numerical simulations have been performed to explain the fluid-dynamic processes and the origins of the experimentally observed structural changes of the rotating wakes. The results of the large-scale-turbulence-resolving simulations agree qualitatively well with the measured flow fields. The numerical results show that the centrifugal forces decrease the base pressure and cause the experimentally observed structural changes in the wake.

To mimic the unsteady vortex–wall interaction of animal propulsion in a canonical test case, a vorticity-annihilating boundary layer was examined through the spin-down of a vortex from solid-body rotation. A cylindrical, water-filled tank was rapidly stopped, and the decay of the vortex from solid-body rotation was observed by means of planar and stereo particle image velocimetry. High Reynolds-number ($Re$) measurements were achieved by combining a large-scale facility (diameter, $D=13\ \textrm {m}$) with a novel approach to reduce end-wall effects. The influence of the boundary-layer formation at the tank's bottom wall was minimised by introducing a saturated salt-water layer. The experimental efforts have allowed us to assess the $Re$ dependency of the laminar–turbulent transition of the vorticity-annihilating side-wall boundary layer at scales similar to large cetaceans. The scaling of the transition mechanism and its onset time were found to agree with predictions from linear stability analysis. Furthermore, the growth rate of the curved turbulent boundary layer was also in good agreement with an empirical scaling formulated in the literature for much smaller $Re$. Eventually, the scaling of vorticity annihilation was addressed. The earlier onset of transition at high $Re$ compensates for the reduced effects of viscosity, leading to similar vorticity annihilation rates during the early stages of the spin-down for a wide $Re$ range.

This paper characterises the steady and time-periodic behaviour of swirling jets using numerical bifurcation analysis. Its objective is to elucidate the dynamics of fully developed, unconfined, laminar swirling jets under variations in the Reynolds number $\textit {Re}$ and swirl ratio $S$. Within the $(0,0)\leq (\textit {Re},S)\leq (300,3)$ range, the steady, axisymmetric flow exhibits several distinct patterns ranging from a quasi-columnar jet along the central axis at low $S$ to a radial jet attached to the containing wall at high $S$ with various forms of vortex breakdown in between. A cusp bifurcation appears in the steady solution manifold which triggers bistable behaviour due to a competition between inner and outer low pressure regions associated with vortex breakdown and entrainment of the ambient fluid, respectively. Instability of the steady flow is linked to eigenmodes which are singly ($|m|=1$) or doubly ($|m|=2$) azimuthally periodic, although additional instabilities with other azimuthal wavenumbers occur at $(\textit {Re},S)$-values beyond the leading neutral curves. The various branches of limit cycle solutions stemming from these neutral curves are associated with both super- and sub-critical Hopf bifurcations. The resulting unsteady flow fields exhibit a wide array of rotating, three-dimensional flow structures, and comparisons between the time-averaged and steady flow patterns highlight the role of these unsteady nonlinear interactions on the overall behaviour of swirling jets. Similarities and differences between this laterally unconfined jet and broader classes of swirling flows, including confined swirling jets and unconfined vortex models, are also discussed.

In this paper, we address the question of whether total helicity is conserved through viscous reconnection events in topologically complex vortex flows. To answer this question, we performed direct numerical simulations (DNS) focused on two complex vortex flow problems: (1) a trefoil knot and (2) a two-ring link, both simulated for various vortex core radii. The DNS framework relies on a block-structured adaptive mesh refinement (AMR) technique. A third simulation of a colliding pair of unlinked vortex rings, which exhibit no total helicity change, is also performed to serve as a reference case. The results show that a well-defined total helicity jump occurs during the unknotting/unlinking events of cases (1) and (2), which arises from the annihilation of the local helicity density content in the reconnection regions. Changes in total helicity become steeper as thinner core radii are considered for both cases (1) and (2). Finally, an analytical derivation based on the reconnection of two infinitesimal anti-parallel vortex filaments is provided that quantitatively links helicity annihilation and viscous circulation transfer processes, which unveils the fundamental hydrodynamic mechanisms responsible for production/destruction of total helicity during reconnection events.

Misaligned wind turbine rotors redirect the wake, and manipulate the wake shape by introducing a counter-rotating vortex pair. This mechanism is of great interest for improving wind farm power output through static or dynamic misalignment. In this study, cross-plane stereo-particle image velocimetry measurements are used to characterize the wake evolution for tilt misalignment and verify differences with yaw misalignment. Blockage from the ground, shear in the velocity profile, turbulence levels, hub-vortices and tip-vortices are found to strongly affect wake evolution for a tilted wind turbine resulting in a non-symmetric behaviour for upwards deflecting or downwards deflecting tilt. The downwards deflection of a negatively tilted wind turbine is found to result in the most benefits for wake recovery and power availability downstream through increased wake-curling, faster wake-recovery, and downdraft of high-momentum flow.

A method is developed to estimate the properties of a global hydrodynamic instability in turbulent flows from measurement data of the limit-cycle oscillations. For this purpose, the flow dynamics is separated into deterministic contributions representing the global mode and a stochastic contribution representing the intrinsic turbulent forcing. Stochastic models are developed that account for the interaction between the two and allow the determination of the dynamic properties of the flow from stationary data. The deterministic contributions are modelled by an amplitude equation, which describes the oscillatory dynamics of the instability, and in a second approach by a mean-field model, which additionally captures the interaction between the instability and the mean-flow corrections. The stochastic contributions are considered as coloured noise forcing, representing the spectral characteristics of the stochastic turbulent perturbations. The methodology is applied to a turbulent swirling jet with a dominant global mode. Particle image velocimetry measurements are conducted to ensure that the mode is the most dominant coherent structure, and further pressure measurements provide long time series for the model calibration. The supercritical Hopf bifurcation is identified from the linear growth rate of the global mode, and the excellent agreement between measured and estimated statistics suggest that the model captures the relevant dynamics. This work demonstrates that the sole observation of limit-cycle oscillations is not sufficient to determine the stability of turbulent flows, since the stochastic perturbations obscure the actual bifurcation point. However, the proposed separation of deterministic and stochastic contributions in the dynamical model allows the identification of the flow state from stationary measurements.

Vortex breakdown (VB) in unconfined swirling jets occurs as either a bubble form of vortex breakdown (BVB) or a conical form of vortex breakdown (CVB). This computational study examines flow features of these forms for a Reynolds number at which VB is accompanied by a transition to turbulence ($Re = 1000$, based on inflow jet radius and centreline velocity). Large eddy simulations were performed with the inflow condition as the Maxworthy profile which models a laminar, axisymmetric swirling jet, and the effect of varying inflow swirl strength was investigated. BVB was observed at lower swirl strengths than those at which CVB occurs. With increasing swirl, the regular and wide-open types of CVB occur. Spiral coherent structures that develop in the flow were examined using spectral proper orthogonal decomposition. Further, by means of hysteresis studies, it is established that the turbulent BVB and regular CVB are bistable forms. Similarly, it is shown that the two types of CVB are also bistable. The difference in recirculation zone (RZ) sizes between the turbulent BVB and regular CVB is greatly reduced when compared to the laminar counterparts. This is postulated as a reason for misidentification of CVB (RZ approximately conical in shape) as BVB (spheroidal RZ) in some previous studies. The present study highlights the distinct features of turbulent BVB and CVB, which can potentially be used towards improving designs of swirl-stabilized combustors.

A swirling wake flow submitted to an adverse pressure gradient is studied by bifurcation analysis, modal analysis and direct numerical simulations. In contrast to experiments in diverging tubes, the adverse pressure gradient is imposed by the presence of a downstream axisymmetric obstacle centred on the vortex axis. Different adverse pressure gradients are investigated by modifying the obstacle radius, which results in the deceleration of the vortex axial velocity. Hence, vortex breakdown occurs for a sufficiently large pressure gradient. We observe a spiral vortex breakdown type without any recirculation bubble, which contrasts with classical spiral vortex breakdown developing in the bubble wake. A weakly nonlinear analysis is performed to characterize this self-sustained instability. The resulting Landau equation reveals the sub-critical character of this Hopf bifurcation, highlighting a sub-critical vortex breakdown. In addition, the stabilization mechanism of this spiral vortex breakdown caused by an off-centre displacement of the downstream axisymmetric obstacle is investigated by direct numerical simulations. Nonlinear dynamics, such as a quasi-periodic state, is observed as a consequence of nonlinear interactions between the spiral vortex breakdown and the misalignment of the obstacle before the stabilization occurs.

Vortex breakdown (VB) in swirling jets can be classified as either a bubble (BVB) or a conical (CVB) form based on the shape of its recirculation zone. The present study investigates the hysteresis features of these forms in laminar swirling jets using direct numerical simulations. It is established here that BVB and CVB are bistable forms in a large swirl range and for a Reynolds number of 200 (based on jet radius and centreline velocity). Considerable differences were observed in the length scales associated with the two, with the approximate recirculation zone diameters of the BVB and CVB being 1 and 15 jet diameters, respectively. Additionally, two types of BVB were observed, identified as a two-celled BVB with spiral tail and an asymmetric BVB. The former is characterized by an almost steady bubble with a two-celled structure. By contrast, the entire bubble envelope oscillated in a non-axisymmetric fashion for the latter. These two types of BVB themselves were found to coexist in a small swirl range. A global linear stability analysis was used to show that two different unstable single helical modes are associated with these two types. In comparison to using the base flow, a stability analysis performed on the mean flow was found to predict the coherent features of asymmetric BVB observed in the simulations more precisely. This study highlights the rich variety of VB flow states that coexist in various ranges of swirl strengths and the significance of hysteresis effects in laminar swirling jets.

When the cross-section is reduced, a vortex displays spiralling and elasticity, bursting when the external velocity drops. How are these properties affected near a free surface? To answer, a visualization experiment in water is considered where an oscillating obstruction is pitched orthogonally to a rolling oscillation expending minimum energy. We show that short aerated vortices in adverse pressure gradient and shear remain stable during stretching, bursting into bubbles when relaxed after maximum stretching. A vertical aerated double helix (DH) root vortex of contra-rotating vortex tubes is produced near a Rossby number of 0.20. The vortex approaches an inclination of $45^{\circ }$ to the vertical, where stretching intensifies the vorticity to the maximum extent. Vortex inclinations up to $45^{\circ }$ are stable and unstable thereafter. Vortex bursting commences only when the inclination crosses $45^{\circ }$ by the slightest amount. When relaxed, oscillations are produced, breaking the vortex into arrays of bubbles, sometimes precisely at $45^{\circ }$ inclination. The bubble diameter, modelled by equating the effects of the centrifugal force (CF) on the vortex core pressure, to surface tension scales with the inverse of the square of the rotational velocity. When a high CF is withdrawn, the bursting DH aerated vortex cone is crushed due to surface tension. A root vortex and a coiled-up trailing edge vortex together form a compact Hill’s vortex when the highest CF is relieved, bursting into bubbles spectacularly scattered in the unstable sector. The DH vortex breakup is the connection point of vortices in proximity. At high CF, the DH has stacks of Taylor air tubes at $45^{\circ }$ to the external flow bursting when relaxed. Kelvin waves abound when bursting.

Bubble-type vortex breakdown in axial vortices is investigated numerically through a model problem of flow inside a cylinder with a top rotating lid, referred to as ‘Vogel–Escuider flow’. The parameters of the flow are Reynolds number ($Re$), based on the rotation speed of the top plate, and aspect ratio ($\unicode[STIX]{x1D6E4}$), which is the ratio of height to radius of the cylinder, depending on which the flow exhibits steady or unsteady breakdown bubble topologies. The flow is analysed for Reynolds number up to 5000 for $\unicode[STIX]{x1D6E4}=2.5$ using helicity density. In the absence of vortex breakdown, the helicity density does not change the sign in the bulk, while in the event of a breakdown, it changes the sign from positive to negative in the vicinity of the breakdown bubble. The three-dimensional flow is further represented as the sum of a two-dimensional velocity field in the $rz$ plane and an out-of-plane velocity vector based on the respective energies, referred to as two-dimensional three-component flow. Here $r$ is the radial coordinate, $z$ is the axial coordinate and $\unicode[STIX]{x1D703}$ is the azimuthal coordinate. Helicity density of the flow is then decomposed into planar helicity $(h_{r,z})$ and out-of-plane helicity $(h_{\unicode[STIX]{x1D703}})$. We show that a correlation exists between planar helicity and the vortex breakdown bubble. We also show that the topology of the breakdown bubbles is described by the planar helicity. Using only this planar helicity, the entire breakdown bubble is reconstructed for axisymmetric as well as non-axisymmetric flows.