2.1 Measurement methodology

Time-resolved PIV

It is well known that the inherent instabilities present in swirl flow lead to complex coherent structures. It is hence mandated to employ time-resolved measurements to understand the flow dynamics. In general, experimental techniques like hot-wire anemometry (HWA), laser Doppler velocimetry (LDV) are more powerful in terms of dynamic (temporal) response (usually ${\sim}20{-}50~\text{kHz}$ ). However, being point measurement techniques, their usages are limited to local measurements. Hence, planar measurements like PIV are usually preferred to resolve the large spatial scales. In this study, time-resolved PIV is employed to meticulously analyse the spatio-temporal characteristics of the swirling flow field.

Figure 2. Schematic showing the optical arrangements of PIV system (a) $y{-}r$ plane; (b) $r{-}\unicode[STIX]{x1D703}$ plane; (c) out of plane motion in an $r{-}\unicode[STIX]{x1D703}$ plane; (d) optimal displacement of particles across two frames in an $r{-}\unicode[STIX]{x1D703}$ plane.

A high repetition rate ( ${\sim}10~\text{kHz}$ ; make: Photonics Inc.) dual pulsed Nd:YLF laser with a pulse energy of $30~\text{mJ}~\text{pulse}^{-1}$ is used as an illumination source. The cylindrical beam of the laser output is converted into a thin sheet ( ${\sim}1~\text{mm}$ ) using sheet making optics. This optics comprises two telescopic lenses and one cylindrical lens. The purpose of the telescopic lens is to adjust the beam diameter thereby the light sheet thickness. The aperture angle $(\unicode[STIX]{x1D6FC})$ of the sheet is controlled by the focal length ( $f$ ) of the cylindrical lens. In this study, ‘ $f$ ’ is chosen as 10  mm to ensure proper illumination of the required field of view.

The flow has been seeded with diethyl hexyl sebacate oil (DEHS) as tracer particles. The size of the particles is chosen as ${\sim}1{-}3~\unicode[STIX]{x03BC}\text{m}$ to maintain better tracing accuracy (i.e. Stokes number, $St\ll 1$ ). The seeded flow is illuminated with the laser and the light scattered by the oil particles is recorded with a high-speed camera (make: Photonics Inc; $7000~\text{frames}~\text{s}^{-1}$ at the full resolution of $1024~\text{pixel}\times 1024~\text{pixel}$ ). The laser pulse and camera shutter are synchronized using a programmable timing unit (PTU) to yield the time frozen snapshots. Furthermore, an optical bandpass filter of 527 nm is coupled with a camera lens to avoid noisy scattered signals from the ambient. The camera is operated in double frame mode with the optimal time delay ‘ $\text{d}t$ ’ between two frames.

In this study, PIV is done in the $y{-}r$ as well as the $r{-}\unicode[STIX]{x1D703}$ plane. For $y{-}r$ plane measurements the laser sheet is aligned with the longitudinal plane of the burner (see figure 2 a). The laser sheet is flipped to horizontal orientation and aligned parallel to the burner exit (figure 2 b). This facilitates measurements at different $y/d$ locations. In both cases, the camera is positioned orthogonal to the plane of illumination. During $r{-}\unicode[STIX]{x1D703}$ measurements, the plane of illumination is projected towards the overhead mirror which is oriented at ${\sim}45^{\circ }$ with respect to the vertical stand (see figure 2 b). A Scheimpflug adapter is coupled to the camera lens to enable fine adjustment between the image and lens planes without altering the camera position. Since the camera is viewing the plane in an oblique manner, it is necessary to correct for angular distortion. This is accomplished by employing a calibration plate with predefined spatial dots. The Scheimpflug adapter is finely adjusted to make sure all the dots in the calibration plate are clearly visible. It is common practice to select ‘ $\text{d}t$ ’ (pulse separation time in PIV) in such a way that the spatial displacement of the particle across the two frames should be less than $1/4$ of the interrogation window (IW) size. To evaluate the optimal ‘ $\text{d}t$ ’, experiments are started with zero displacement (i.e. a very low value say, ${\sim}6~\unicode[STIX]{x03BC}\text{s}$ ), then $\text{d}t$ is slowly increased till ${\leqslant}1/4$ IW. In the present study, the final IW size is chosen to be $32~\text{pixel}\times 32~\text{pixel}$ for both $r{-}\unicode[STIX]{x1D703}$ and $y{-}r$ planes.

In particular, ‘ $\text{d}t$ ’ is chosen with special care for $r{-}\unicode[STIX]{x1D703}$ measurements, to ensure the detection of the same particle in the second frame since the strong axial velocity can lead to out of plane motion of the particles.

For instance, with $Re_{a}\sim 4896$ case, the maximum axial velocity $(U_{y})$ at the burner exit is found to be $1.7~\text{m}~\text{s}^{-1}$ . The maximum axial displacement of the particle for the chosen ‘ $\text{d}t$ ’ (i.e. $95~\unicode[STIX]{x03BC}\text{s}$ ) is found to be ${\sim}0.095~\text{mm}$ (figure 2 c), which is much less than the laser sheet thickness ${\sim}1~\text{mm}$ . The corresponding values for $\mathit{Re}_{a}\sim 10\,545$ , 17 456 are found to be 0.11 mm, 0.19 mm respectively (all ${<}1~\text{mm}$ ). In addition, the selected ‘ $\text{d}t$ ’ also satisfies the optimal displacement criteria (i.e. $\unicode[STIX]{x0394}r$ , $\unicode[STIX]{x0394}\unicode[STIX]{x1D703}<(1/4)IW$ ; ${\sim}5~\text{pixel}$ ; figure 2 d). The values of ‘ $\text{d}t$ ’ in the $r{-}\unicode[STIX]{x1D703}$ plane for $\mathit{Re}_{a}\sim 4896$ , 10 545, 17 546 are chosen as $95~\unicode[STIX]{x03BC}\text{s}$ , $50~\unicode[STIX]{x03BC}\text{s}$ , $25~\unicode[STIX]{x03BC}\text{s}$ respectively.

The same optimal pixel displacement criteria have been also extended to $y{-}r$ plane too (i.e. $\unicode[STIX]{x0394}r$ , $\unicode[STIX]{x0394}y<(1/4)IW$ ). In the $y{-}r$ plane, the optimal ‘ $\text{d}t$ ’ for $Re_{a}\sim 4896$ , 10 545, 17 546 are chosen as $105~\unicode[STIX]{x03BC}\text{s}$ , $77~\unicode[STIX]{x03BC}\text{s}$ , $40~\unicode[STIX]{x03BC}\text{s}$ respectively.

Furthermore, the accuracy of the flow field perceived from PIV is greatly influenced by laser sheet thickness $(\unicode[STIX]{x1D6FF}_{t})$ , particle displacement across two frames $(ds)$ and number of particles ( $N$ ) per IW. The correctness of the above-mentioned parameters is validated by employing ‘correlation statistics’ method (Sciacchitano, Wieneke & Scarano Reference Sciacchitano, Wieneke and Scarano2013; Wieneke Reference Wieneke2015). This method attempts to evaluate the uncertainty in the measured velocity values $(u_{error})$ based on the positional disparity between the correlation peaks observed across the two frames. In the present study, the uncertainty in the measured velocity values is found to be ${\sim}\pm 1.3\,\%$ ( $r{-}\unicode[STIX]{x1D703}$ ) and 1 % ( $y{-}r$ plane) of the local velocity values. More technical operational details involved in PIV like optimal seeding density, $\text{d}t$ , laser sheet thickness can found in Raffel et al. (Reference Raffel, Willert, Wereley and Kompenhans2013).

Further, we acknowledge that stereoscopic (three-dimensional) high-speed PIV may bring additional insights into the flow dynamics (especially in the $r{-}\unicode[STIX]{x1D703}$ plane). The three-dimensional (3-D) vector field can be confidently calculated only in the overlapped region of cameras 1 and 2 (Arroyo & Greated Reference Arroyo and Greated1991; Prasad & Adrian Reference Prasad and Adrian1993) using stereo or tomographic PIV. In the present work, the flow spread area is much higher (especially with conical breakdown) than the overlap area one can expect in stereo mode. Hence, we have not attempted stereoscopic PIV in this study due to equipment limitations. At the same time, the acquisition of all three component of velocities ( $U_{r}$ , $U_{y}$ , $U_{\unicode[STIX]{x1D703}}$ ) in simultaneous fashion improves the understanding of the problem. Since, the swirling flow is known to be highly three-dimensional flow, 3-D measurements may help to delineate the vortex breakdown mechanisms in a more insightful manner. In the present work, all three components of velocity have been acquired, but not in the same temporal space. In addition, 2-D PIV also ignores particles with strong out of plane motion leading to errors. The absence of tomographic PIV data is therefore considered a limitation in the present study. However, the same will be taken up in our future work. However, we have tried to keep the errors incurred due to 2-D PIV to a minimum (as explained previously).

Throughout the experiments, images are acquired at $1500~\text{frames}~\text{s}^{-1}$ with an acquisition time of 2 s (i.e. 3000 images are considered per each test case). For most of the $y{-}r$ measurements, the field of view (FOV) is chosen as $140~\text{mm}\times 140~\text{mm}$ with a spatial resolution of ${\sim}7.2~\text{pixels}~\text{mm}^{-1}$ . During $r{-}\unicode[STIX]{x1D703}$ measurements, the same is kept at ${\sim}280~\text{mm}\times 280~\text{mm}$ ( ${\sim}3.65~\text{pixels}~\text{mm}^{-1}$ ). It should be noted that for most of the near-field measurements ( $y/R_{a}\leqslant 0.5$ ), smaller field of view ( ${\sim}180~\text{mm}\times 180~\text{mm}$ ) is adapted to yield a high spatial resolution of ${\sim}5.68~\text{pixels}~\text{mm}^{-1}$ .

The acquired double frame PIV raw images are post-processed using Lavision 8.4 software to reconstruct the vector fields. For all the test cases, multipass interrogation windows (IW) with decreasing sizes are employed with initial and final window sizing being $48~\text{pixel}\times 48~\text{pixel}$ and $32~\text{pixel}\times 32~\text{pixel}$ respectively. In addition to this, adaptive PIV option is employed to enable auto calculation of window shape based on the gradient in the flow field.

2.2 Planar laser Mie scattering experiments (PLMS)

In addition to PIV measurements, a Nikon D7200 DSLR camera is used to perform a planar laser Mie scattering experiment. This experiment provides qualitative visualization of flow topologies at the burner exit across various operating conditions. Here, the optical arrangements are similar to the $y{-}r$ plane PIV measurements, except that the camera is operated in video grabbing mode. The flow is seeded with the same DEHS oil; the light scattered by the oil particles from the illuminated plane is recorded with the DSLR camera. It is important to note that a higher seeding density is employed to generate smoke-like flow visualization images. PLMS experiments are carried out only in the $y{-}r$ plane. Two macro lenses, 100 mm (make: Tokino) and 50 mm (make: Nikon), are used to image the near field and far field of the flow. With both lenses, videos are recorded at 30 f.p.s. (full HD 1080 p).

2.3 Flow conditions

The present study involves air flow across three different passages (A, B, C; see figure 1), hence it is necessary to define non-dimensional numbers to identify the parametric relationships among the different flow configurations. The experiment has been carried out at three sets of flow rates across annular (A) and central pipe (C) passages. At each set, the flow rate across tangential entry (B) is progressively varied to impart a different level of swirl intensity to the annular flow. The experimental test cases are globally characterized using three important non-dimensional numbers namely, the Reynolds number, momentum ratio and swirl number.

Reynolds number $(Re)$ can be defined as

(2.1) $$\begin{eqnarray}Re_{g}=\frac{\unicode[STIX]{x1D70C}_{a}U_{o}D_{eff}}{\unicode[STIX]{x1D707}_{a}},\end{eqnarray}$$

where, $\unicode[STIX]{x1D70C}_{a}$ , $\unicode[STIX]{x1D707}_{a}$ are air density ( $\text{kg}~\text{m}^{-3}$ ) and viscosity (Pa s), $D_{eff}$ is effective burner diameter (m); $U_{o}$ is bulk-averaged velocity of the fluid at burner exit ( $\text{m}~\text{s}^{-1}$ ):

(2.2) $$\begin{eqnarray}U_{o}\sim \frac{A_{c}U_{c}+A_{a}U_{a}+A_{t}U_{t}}{A_{c}+A_{a}+A_{t}}.\end{eqnarray}$$

Here, $A_{c},A_{a},A_{t}$ are cross-sectional areas available for the central, annular and swirl jet, respectively. $U_{c},U_{a},U_{t}$ are corresponding bulk average velocities computed based on the volumetric flow rate. Hence it should be noted that $Re_{g}$ is basically a bulk-averaged flow Reynolds number at the burner exit.

Further, only the annular flow (i.e. flow across A and B) Reynolds number $(Re_{a})$ is defined by neglecting the ${A_{t},U}_{t}$ terms in the expression (2.2).

In coaxial jets, it is customary to define the momentum ratio (MR) as a quick estimate to quantitatively understand the momentum transfer pathways among coaxial jets. In general, the jet whose flow rate is variable is defined in the numerator part and the jet with constant air flow rate is kept in the denominator. Since, in this study, for the given co-annular flow across A and C, the flow rate across the tangential port is progressively varied, the MR can be defined as follows,

(2.3) $$\begin{eqnarray}MR=\frac{\unicode[STIX]{x1D70C}_{a}U_{t}^{2}}{(\unicode[STIX]{x1D70C}_{a}U_{a}^{2}+\unicode[STIX]{x1D70C}_{c}U_{c}^{2})}.\end{eqnarray}$$

In single phase coaxial jets, since the density is same for all of the fluids, equation (2.3) becomes the velocity ratio

(2.4) $$\begin{eqnarray}MR=\frac{U_{t}^{2}}{(U_{a}^{2}+U_{c}^{2})}.\end{eqnarray}$$

Swirl number is quantified as a function of the axial flux of tangential momentum and axial flux of axial momentum:

(2.5) $$\begin{eqnarray}S=\frac{\displaystyle \int _{R_{h}}^{R_{o}}\unicode[STIX]{x1D70C}U_{y}U_{w}2\unicode[STIX]{x03C0}r^{2}\,\text{d}r}{\displaystyle \int _{R_{h}}^{R_{o}}\unicode[STIX]{x1D70C}U_{y}^{2}2\unicode[STIX]{x03C0}r\,\text{d}r}.\end{eqnarray}$$

However, Claypole & Syred (Reference Claypole and Syred1981) showed that the swirl number can be defined in terms of the burner geometry (2.6)

(2.6) $$\begin{eqnarray}S_{G}=\frac{\unicode[STIX]{x03C0}r_{o}r_{a}}{A_{t}}\left\{\frac{m_{t}}{m_{a}+m_{c}}\right\}^{2},\end{eqnarray}$$

where, $r_{o}$ is the radial distance between the burner central axis and the tangential port, $r_{a}$ is the inner radius of the annular pipe (A), $m_{a},m_{c},m_{t}$ are respectively the annular, central, tangential flow rates (in $\text{l}~\text{min}^{-1}$ ). In common practice, geometric swirl number $S_{G}$ is related to $S$ in terms of $S\sim 0.25S_{G}$ .

In addition to the MR and the swirl number, the influence of inlet flow conditions on the final flow field structure can be delineated by a parameter named the Rossby number,

(2.7) $$\begin{eqnarray}Ro\sim |\unicode[STIX]{x0394}U_{y}|/U_{t},\end{eqnarray}$$

where,

(2.8) $$\begin{eqnarray}\unicode[STIX]{x0394}U_{y}\sim U_{c}-U_{a}.\end{eqnarray}$$

Here, $U_{c},U_{a}$ indicates the bulk average exit velocities, which are calculated based on the flow rate and exit diameter of the annular and central pipes, respectively.

Table 1. Experimental test cases. PVB – pre-vortex breakdown; CB – conical breakdown; CTRZ – central toroidal recirculation zone.

Theoretically, the Rossby number $(Ro)$ represents the ratio of axial and radial pressure gradient. In swirling flows, it is well known that flow with a dominant radial pressure gradient undergoes vortex breakdown.

The experimental test cases are listed in table 1, it should be noted that the experimental runs are not only confined to the values shown here. For instance, while flow undergoing a transition from CTRZ to conical breakdown (CB) (figure 3), experiments are carried out for small variations in tangential mass flow rate ( $\unicode[STIX]{x0394}m_{t}\sim 3{-}5~\text{l}~\text{min}^{-1}$ ) to carefully identify the flow/vortex breakdown modes. The importance of employing small $\unicode[STIX]{x0394}m_{t}$ will be elucidated in greater detail in subsequent sections.

The main aim of this study is to elucidate the effect of swirl intensity on annular flows, in particular, for a given $Re_{a}$ , how the flow topologies vary with swirl number $S_{G}$ . During this process, it is observed that the flow undergoes different forms of topological variations (like PVB, CB, CTRZ etc. figure 3).

3.1 Evolution of shear layers in the flow field

The evolution of shear layer in the spatial domain of the flow field across the different flow modes is illustrated in figure 7. It is seen that multiple shear layers exist for flows which exhibit vortex breakdown (see figure 7 b–e). The PVB flow mode (figure 7 b) exhibits two inner shear layers (ISL 1, ISL 2) and one outer shear layer (OSL 1). This is due to the combined dominance of the axial and centrifugal momentum induced by co-annular and swirling jets. During the CTRZ mode, ISL 2 completely vanishes due to the resistance offered by the recirculated flow against the axial jet. However, as the flow approaches the CB from the CTRZ mode, the ISL starts to shift towards the radial direction and its extent in the axial direction is also significantly reduced. Furthermore, with the full open cone configuration, the ISL almost vanishes and persists only in the near burner exit zone. The shear layers observed from the $r{-}\unicode[STIX]{x1D703}$ measurements across various $S_{G}$ values are delineated in figure 5. Except for the CB case, the presence of the outer shear layer for all other flow modes is clearly witnessed. The outer shear layer does exist for the CB case, but it is far away from the burner central axis.

It should be noted that, excepting swirl number, parameters such as the momentum ratio (MR) and Rossby number ( $R_{o}$ ) have been used to classify the flow into two paradigms i.e. in the given coaxial jet, whether the tangential flow is dominant ( $\boldsymbol{U}_{\unicode[STIX]{x1D73D}}>\boldsymbol{U}_{\boldsymbol{y}}$ – regime II) or the axial jets are dominant ( $\boldsymbol{U}_{\unicode[STIX]{x1D73D}}<\boldsymbol{U}_{\boldsymbol{y}}$ – regime I). Furthermore, as discussed in the above section, the flow modes are distinguished into three distinct types, PVB, CTRZ, CB as a function of  $S_{G}$ .

Hence, to avoid the confusion, instead of using all the non-dimensional numbers, thereafter only the flow modes (PVB, CTRZ, CB) and the corresponding swirl numbers are widely used in the forthcoming sections.

Figure 7. Evolution of shear layers across various $S_{G}$ values. (a) $S_{G}\sim 0$ ; (b) $S_{G}\sim 1.78$ ; (c) $S_{G}\sim 2.28$ ; (d) $S_{G}\sim 2.4$ ; (e) $S_{G}\sim 2.5$ ; (f) $S_{G}\sim 2.53$ (OSL, ISL – outer, inner shear layer).

4.1 Proper orthogonal decomposition

The essential role of the POD is to extract the most energetic structures from instantaneous flow field data. Additionally, POD enables us to quantify characteristic frequencies of the dominant modes from the temporal spectrum. In this study, we have implemented the ‘method of snapshots’ based POD algorithm (Sirovich Reference Sirovich1987; Berkooz, Holmes & Lumley Reference Berkooz, Holmes and Lumley1993) to decompose the flow field. The steps involved in the implementation of POD for given flow field data is shown in figure 8. Mathematically, this algorithm decomposes the instantaneous flow field data $(U^{\prime },V^{\prime })$ into corresponding spatial $\emptyset _{n}(x)$ and temporal $(j)$ modes. Further details on POD implementation for PIV data can be found in Rajamanickam & Basu (Reference Rajamanickam and Basu2017b ).

Figure 8. Steps involved in computing POD from the instantaneous flow field. Adapted from Rajamanickam & Basu (Reference Rajamanickam and Basu2017b ).

4.1.1 Temporal dynamics and spatial eigenmodes in the $y{-}r$ and $r{-}\unicode[STIX]{x1D703}$ planes

In general, the snapshot algorithm yields a spatial eigenmode number that equals the number of instantaneous data sets. However, the resultant eigenmodes are numbered in descending order of turbulent kinetic energy (TKE) magnitude $(\unicode[STIX]{x1D706}_{n})$ . For brevity, POD is carried out for those flow conditions which exhibit the vortex breakdown phenomenon (i.e. only for PVB, CTRZ, CB; $S_{G}\geqslant 1.76$ ). Additionally, POD analysis has been extended to the $r{-}\unicode[STIX]{x1D703}$ plane to meticulously understand the dynamics of vortex breakdown across different flow modes.

Figure 9. Illustration of spatial eigenmodes perceived from POD analysis in $y{-}r$ plane at $Re_{a}\sim 4896$ . (a) $S_{G}\sim 1.78-\text{PVB}$ ; (b) $S_{G}\sim 2.28-\text{CTRZ}$ ; (c) $S_{G}\sim 2.53-\text{CB}$ ; (d) distribution of % TKE across various POD spatial modes.

POD analysis has been carried out at many locations ( $r{-}\unicode[STIX]{x1D703}$ planes), although the results pertaining to only three axial locations are explicitly considered. These three positions are considered such that key locations such as downstream and upstream of the VCC (see figure 10 a) are covered. It is well known that frequency signatures perceived from fast Fourier transform (FFT) implemented over temporal modes yield more quantitative information on nature of the flow. Hence, for brevity, spatial eigenmodes are shown only for near-field locations ( $y/R_{a}\sim 0.3$ ) and temporal modes are shown for all the defined locations.

Figure 10. (a) Schematic illustration of spatial locations where POD analysis is carried out in an $r{-}\unicode[STIX]{x1D703}$ plane; (b–d) spatial eigenmodes of PVB, CTRZ, CB in near-field locations at $Re_{a}\sim 4896$ ( $y/R_{a}\sim 0.3$ ).

Dynamics of pre-vortex breakdown (PVB)

The POD spatial eigenmodes pertaining to PVB, CTRZ, CB of the $y{-}r$ plane are respectively shown in figure 9. Here vorticity maps are used to delineate the energetic modes. Since the cumulative energy of the first few modes contribute 90 % of the total energy, only those spatial modes are shown. In the PVB case, the most energetic mode (figure 9 a, M1) clearly shows the dominance of axial shear induced by KH vortices of the co-annular jets at ISL 2. PVB case exhibits strong counter-rotating eddies in the near field of the burner (see figure 3  $S_{G}\sim 1.76$ ), leading to vortex shedding in the outer shear layer (OSL 1). This phenomenon is not intensified in mode 1, whereas, the same is clearly visible in modes 2–4 (figure 9 a). More quantitative information on this observation can be perceived from the frequency signatures derived from the POD temporal modes. In general, it is known that shedding eddies in the flow lead to frequency harmonics. From figure 11(a), it is evident that modes 2–4 show the existence of harmonics in the frequency signatures. On the other hand, mode 1 shows one strong peak around ${\sim}20~\text{Hz}$ followed by higher-order frequencies with very small magnitude (this is due to the absence of shedding in mode 1; figure 9 a M1). Furthermore, the instantaneous flow field suggests that the activation of shedding mode in the PVB case may be due to unsteadiness induced by the interaction of the swirling and co-annular jets in the near field. This can be interpreted in a quantitative manner from the POD analysis implemented over the $r{-}\unicode[STIX]{x1D703}$ plane.

Figure 11. Schematic illustration of frequency signatures perceived from POD temporal modes in $y{-}r$ plane across various $S_{G}$ values. ( $Re_{a}\sim 4896$ .)

As mentioned earlier, the activation of shedding modes in the PVB case is more pronounced in the near field (figure 12 a; $y/R_{a}\sim 0.3$ , 1.5 and figure 10 b), due to the strong interaction of the two jets (i.e. co-annular and swirl jet). Particularly, this phenomenon induces intense oscillations in the outer shear layer (see figure 10 b, movie 3). In the far downstream position, after the vortex core centre (VCC), the deficiency in the momentum of two streams leads to a reduction in intensity of these oscillations. Frequency signatures derived from FFT (figure 12 a; $y/R_{a}\sim 3$ ) are also in line with this argument.

Figure 12. Illustration of frequency signatures observed across various axial locations in the $r{-}\unicode[STIX]{x1D703}$ plane ( $Re_{a}\sim 4896$ ).

Dynamics of central toroidal recirculation zone (CTRZ)

For the CTRZ case, the dominant mode is identified as centrifugally unstable (see M1 in figure 9 b). This is due to the strong influence of the azimuthal shear induced by high swirl $(\unicode[STIX]{x1D70C}U_{\unicode[STIX]{x1D703}}^{2}>\unicode[STIX]{x1D70C}U_{y}^{2})$ . Rayleigh’s inviscid criterion and the theoretical work done by Coles (Reference Coles1965) state that the shear layer becomes unstable either due to a circulation intensity decrease in the radial direction or due to a stratification in angular momentum.

(4.1a,b ) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D6E4}}{\text{d}r}<0;\quad \unicode[STIX]{x1D714}_{z}\unicode[STIX]{x1D6FA}<0.\end{eqnarray}$$

Hence, the dominance of centrifugal effects in the CTRZ mode is evident from the negative vorticity zone in the central region (figure 9 b, M1). The second mode is anticipated as the axial shear mode, which may arise due to the interaction between the inner and outer shear layers (OSL, ISL) induced by vortex bubble breakdown (VBB). Unlike the PVB case, the shedding mode in CTRZ carries a much lower energy (figure 9 b, M4). As a result, harmonics are not observed with the shedding mode. All the modes exhibit frequency peaks concentrated around a very narrow band of ${\sim}10{-}20~\text{Hz}$ (figure 11 b). In general, the helical shear mode exhibits higher energy content for flows with weak swirl (i.e. prior to VBB). However, CTRZ exhibits strong vortex breakdown; hence the helical mode appears as the third mode (figure 9 b, M3).

Contrary to observations made for PVB in the $r{-}\unicode[STIX]{x1D703}$ plane, with the CTRZ flow mode, for all $y{-}R_{a}$ planes, activation of the shedding mode is not perceived. The frequency signatures of CTRZ pertaining to the VCC plane show dual peaks in mode 1 (figure 12 b, $y/R_{a}\sim 1.5$ ). This may due to mild oscillations induced by the precessing vortex core (PVC). It is well known that PVC in strong swirling jets (the ones which exhibit VBB) induces oscillations in the flow (Syred Reference Syred2006; Oberleithner et al. Reference Oberleithner, Stöhr, Im, Arndt and Steinberg2015). However, in a global sense, azimuthal and axial shear modes appeared to be the most energetic flow modes in the CTRZ conditions (see figures 9 b and10 c).

Dynamics of conical breakdown (CB)

The spatial eigenmodes pertaining to the conical breakdown case is shown in figure 9(c). Contrary to the PVB and CTRZ conditions, the CB case shows highly energetic shedding structures (see figure 9 c, M1). Here, almost all of the four modes show the dominance of shedding vortices. The dominance of shedding structures in CB results in the appearance of frequency harmonics across all of the modes (figure 11 c,d). In order to understand the underlying dynamics of CB, POD is implemented over two small intervals of $\unicode[STIX]{x0394}S_{G}$ ( ${\sim}0.05$ ). To achieve this, flow rates across the tangential ports are varied in small steps of ${\sim}4~1~\text{min}^{-1}$ . The corresponding frequency signature is shown in figure 11(c,d) (for eigenmodes figure 11 d shows a similar pattern and hence is not shown here). It is seen that in both cases, frequency peaks shifted towards the higher side ${\sim}50{-}100~\text{Hz}$ and exhibit harmonics. Further, it is very interesting to note that for the flow conditions which recover back to CTRZ from CB, the frequency values also shifted to the lower side and harmonics are not evident (figure 11 e).

For the CB case, since the breakdown occurs very close to the burner exit with rapid radially outward flow expansion (not in the axial direction), POD in the $r{-}\unicode[STIX]{x1D703}$ plane is computed only in the near-field plane close to $y/R_{a}\sim 0.3$ . The frequency signature at this location delineates the strong dominance of shedding across all modes. In the $y{-}r$ plane POD, it is mentioned that the sudden shift in frequencies towards the higher side may be the reason why the flow undergoes conical breakdown. The POD analysis carried out in an $r{-}\unicode[STIX]{x1D703}$ plane also shows similar behaviour.

The higher-order frequencies exhibited by the shedding vortices may be the energy source responsible for the conical breakdown. This argument will be further proved by DMD analysis in subsequent sections.

Figure 13. Sequence of PIV raw images illustrating the intense bubble bursting phenomenon in the conditions close to CB.

Furthermore, the Mie scattered raw images acquired during time-resolved measurements suggest intense bubble bursting in the central core region (figure 13 a–c). The bursting intensifies with an increase in swirl number, until the flow undergoes conical breakdown. At the transition state $(S_{G})_{c1}$ , although the bubble starts to burst, the flow is still attached to the rotating core (figure 13 a,b; see movie 4). However, at the conical breakdown state, the increased centrifugal momentum causes the flow to stay predominantly at the periphery (figure 13 c,f). The flow conditions prior/closer to CB are identified as precursor events. We can see severe intermittency in bubble bursts across the temporal space (figure 13 g). On account of the intermittency associated with bubble burst, these precursor events exhibit highly unstable behaviour with shear layer oscillations observed across longitudinal ( $y{-}r$ ) and azimuthal ( $r{-}\unicode[STIX]{x1D703}$ ) planes.

The above-mentioned observations are clearly delineated in movie 4 ( $r{-}\unicode[STIX]{x1D703}$ plane) and movie 5 ( $y{-}r$ plane).

4.1.2 Influence of sampling frequency

The frequency signatures perceived from the POD temporal modes corresponding to CB (figures 11 d; 12) exhibit a wide band of frequency oscillations in the range of ${\sim}100{-}600~\text{Hz}$ . To precisely probe the existence of any further frequencies beyond this range, experiments have been conducted at higher sampling rates (i.e. ${\sim}3.5~\text{kHz}$ ; 5 kHz). For instance, the comparison made between the frequency signatures observed across three sampling rates (1.5 kHz, 3.5 kHz and 5 kHz) for $Re_{a}\sim 17\,456$ is graphically depicted in figure 14.

Figure 14. Comparison between frequency signatures observed at CB case for $Re_{a}\sim 17\,456$ . (a) 1.5 kHz; (b) 3.5 kHz; (c) 5 kHz. (d) Time-averaged flow field acquired as various sampling rates.

It is clear that in all the cases, the frequency signatures follow a similar trend. Interestingly, as seen from figure 14(c), even at higher sampling rates (i.e. 5 kHz), there are no significant higher-order frequencies. It should be noted that 5 kHz is the maximum attainable sampling frequency used in this study (equipment limitations). Since, the observed maximum frequencies are in the range of ${\leqslant}550~\text{Hz}$ , the dynamics extracted from the flow field using a sampling rate of 1.5–5 kHz is reasonable. However, minor shifts in peak frequencies (figure 14 without any loss of major physical insight) can also be partially attributed to the limitation in sampling rate. In addition, a lower sampling rate may cause undetected high frequency signatures (although this is unlikely in the current set of experiments) in swirling flows. Furthermore, it should be noted that we have observed a slight variation ( ${\sim}10{-}15~\text{Hz}$ ) in the dominant frequency even when the experiments are repeated for the CB case at the same sampling rate. However, these variations are very minimal/negligible for the CTRZ and PVB cases. Hence, the slight difference in frequency (noticed in figure 14 a–c) is attributed as an effect of unsteadiness associated with the CB mode itself. However this slight difference could also be partially due to the limited sampling rate. In particular, CB is always known to induce severe oscillations in the outer shear layer, which further induce multiple shedding vortices in the shear layer. This has been reported by Billant et al. (Reference Billant, Chomaz and Huerre1998) through smoke flow visualization experiments. Billant et al. (Reference Billant, Chomaz and Huerre1998) also showed the intermittency in the emptying and filling process of fluid inside the recirculation zone. In addition, their hysteresis analysis delineated the evolution to variations in the temporal dynamics of conical breakdown.

Owing to the low bulk velocity of the flow ( $U_{o}\sim 1.23~\text{m}~\text{s}^{-1}$ , $2.497~\text{m}~\text{s}^{-1}$ , $4.01~\text{m}~\text{s}^{-1}$ respectively at $Re_{a}\sim 4896$ , 10 545, 17 546), we have found that temporally ${\sim}2~\text{s}$ of data are needed to achieve a proper temporal average of the flow field. This is particularity significant with the low $Re_{a}$ case. For instance, the time-averaged streamline plot of $Re_{a}\sim 4896$ sampled at 3.5 kHz (see figure 14 c) shows an erroneous temporal average mainly due to the reduction in the sampling time. The lowering of the sampling time occurs due to the limitation of the on board storage of the camera at 3.5 kHz sampling. Therefore, acquisition of images at 1.5 kHz becomes more of an optimum choice for the present study.

4.2 Identification of robust modes using dynamic mode decomposition

In the previous section, POD has been employed to elucidate the energetic structures present in each vortex breakdown mode. In general, POD is viewed as a modal reduction technique which quantifies the dominant structures that govern the flow field (Taira et al. Reference Taira, Brunton, Dawson, Rowley, Colonius, McKeon, Schmidt, Gordeyev, Theofilis and Ukeiley2017), or in simple terms, it is basically an ‘energy ranking method’. Several researchers have quoted that although POD precisely resolves the dominant spatial structures, it may be difficult to predict the contribution of each frequency in the flow field. For instance, M2 in figure 12(c), shows multiple frequency values. Likewise, other modes also show several frequency values. Hence, from POD analysis, it is too cumbersome to predict the contribution of a particular frequency value towards the dynamics/stability of the flow. Recently however, another modal analysis technique named dynamic mode decomposition (DMD) has received greater attention in the understanding of the dynamics of a given flow field. This technique was originally developed by Schmid (Reference Schmid2010) and later improved in several ways by many researchers (Williams, Kevrekidis & Rowley Reference Williams, Kevrekidis and Rowley2015; Proctor, Brunton & Kutz Reference Proctor, Brunton and Kutz2016). The essential role of DMD is to evaluate the contribution of each frequency found in the flow field. In essence, it attempts to predict whether a particular frequency forces the flow to become stable or unstable. More details on the strengths and weaknesses of modal analysis tools like POD, DMD, single value decomposition (SVD) etc. in interpreting fluid flows are elaborated with illustrative examples in Taira et al. (Reference Taira, Brunton, Dawson, Rowley, Colonius, McKeon, Schmidt, Gordeyev, Theofilis and Ukeiley2017).

4.2.1 Implementation of DMD

In the first step, the acquired set of instantaneous snapshots (i.e. velocity fields) needs to be converted into a matrix form. In particular, the velocity data set pertaining to each instant of time should be represented in a sequential way (i.e. over ‘ $n$ ’ time instants) in the form of a column matrix:

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D653}_{1}^{N}=\{v_{1},v_{2},v_{3},\ldots \ldots ,v_{N}\}.\end{eqnarray}$$

The matrix obtained using (4.2) from instantaneous data sets can be further treated as follows

(4.3) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D653}_{1}^{N-1}=\{v_{1},v_{2},v_{3},\ldots \ldots ,v_{N-1}\}, & \displaystyle\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D653}_{2}^{N}=\{v_{2},v_{3},v_{4},\ldots \ldots ,v_{N}\}. & \displaystyle\end{eqnarray}$$

Now, the constant linear mapping or dynamical propagator $Y$ needs to be defined to represent the velocity fields at each time instant

(4.5) $$\begin{eqnarray}\displaystyle & v_{k+1}=Yv_{k}, & \displaystyle\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D653}_{2}^{N}=Y\unicode[STIX]{x1D653}_{1}^{N-1}. & \displaystyle\end{eqnarray}$$

The single value decomposition of (4.2) yields,

(4.7) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D653}_{1}^{N-1}=\unicode[STIX]{x1D650}\unicode[STIX]{x1D64E}\unicode[STIX]{x1D651}^{J}, & \displaystyle\end{eqnarray}$$

(4.8) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D653}_{2}^{N}=\unicode[STIX]{x1D650}\unicode[STIX]{x1D64E}\unicode[STIX]{x1D651}^{J}\unicode[STIX]{x1D653}_{1}^{N-1}, & \displaystyle\end{eqnarray}$$

(4.9) $$\begin{eqnarray}\displaystyle & Y=\unicode[STIX]{x1D650}^{J}\unicode[STIX]{x1D653}_{2}^{N}\unicode[STIX]{x1D651}\bar{\unicode[STIX]{x1D64E}}. & \displaystyle\end{eqnarray}$$

Here, $\unicode[STIX]{x1D650}$ , $\unicode[STIX]{x1D651}$ are orthonormal matrices and $\unicode[STIX]{x1D64E},\bar{\unicode[STIX]{x1D64E}}$ represent the diagonal and pseudo-inverse of the diagonal matrix respectively.

Finally, the diagonal matrix with eigen values $(\unicode[STIX]{x1D707})$ is obtained. Further, these values are converted to Floquet values as follows

(4.10) $$\begin{eqnarray}\unicode[STIX]{x1D714}=\frac{\ln (\unicode[STIX]{x1D707})}{\unicode[STIX]{x0394}t},\end{eqnarray}$$

where, $\unicode[STIX]{x0394}t$ is the time separation between successive frames/snapshots.

The parameter $\unicode[STIX]{x1D714}$ is represented as a combination of real and imaginary parts $(\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{r}+\unicode[STIX]{x1D714}_{i})$ , from which frequency can be extracted as follows

(4.11) $$\begin{eqnarray}f=\frac{\unicode[STIX]{x1D714}_{i}}{2\unicode[STIX]{x03C0}}.\end{eqnarray}$$

The value and sign of $\unicode[STIX]{x1D714}$ is used as a criterion to analyse the dynamics of the flow. For instance, positive and negative values of $\unicode[STIX]{x1D714}$ denote unstable and stable modes respectively. In general practice, modes (modes $1,2,3,\ldots ,n$ ) in the DMD are interpreted based on the magnitude of $\unicode[STIX]{x1D714}$ in descending order. Recently, ranking the DMD modes based on their amplitude coefficients (derived using the least squares method) has become popular. The mathematical procedure involved with sorting the DMD modes based on modal amplitude is illustrated with details in Rowley et al. (Reference Rowley, Mezić, Bagheri, Schlatter and Henningson2009) and Mohan, Gaitonde & Visbal (Reference Mohan, Gaitonde and Visbal2015). DMD usually assigns the highest growth rates to the flow with small-scale eddies originating from wakes, even though these eddies possess less energy. In applications where the objective is to find the dynamic nature of the frequency associated with energetic coherent structures, DMD modes ranked based on amplitude are highly preferred. To illustrate this phenomenon, comparison is made between the growth rate $(\unicode[STIX]{x1D714}_{r})$ and amplitude $(\unicode[STIX]{x1D709})$ based DMD modes (figure 15) for the CTRZ case at $Re_{a}=4896$ .

Figure 15. (a) Illustration of comparison between dynamic modes extracted using growth rate versus amplitude based algorithms: (i) DMD growth rate versus frequency; (ii) DMD growth rate (only unstable) versus frequency; (iii) DMD amplitude versus frequency. (b) Spatial modes corresponding to the frequencies marked with i, j, k in (a).

The DMD computed based on the growth rate algorithm shows a dominant value around ${\sim}400~\text{Hz}$ (marked as ‘ $j$ ’ in figure 15 a (ii)), whereas the amplitude versus frequency plot shows dominance around ${\sim}30~\text{Hz}$ . The spatial modes corresponding to the frequencies marked with $i$ , $j$ , $k$ in figure 15(a) (ii), (iii) respectively are shown in figure 15(b). The spatial modes clearly delineate that the higher-order frequencies shown by the growth rate algorithm are contributed by the outer shear layer shedding in the flow. On the contrary, the frequency values ranked based on the amplitude $(\unicode[STIX]{x1D709})$ show that the shear mode is dominant which is identified as a most energetic mode in the POD analysis. This clearly delineates the disclaimer associated with $(\unicode[STIX]{x1D714}_{r})$ and $(\unicode[STIX]{x1D709})$ in identifying the dominant modes in the given flow field.

Since, in this study, the main objective is to identify the fluid dynamic cause behind various transition and flow modes (which are believed to be mostly governed by dominant coherent structures in the flow field), we have computed DMD modes based on modal amplitudes $\unicode[STIX]{x1D709}$ .

The obtained DMD modes across various swirl numbers for $Re_{a}=4896$ are graphically illustrated in figure 16. It should be noted that the only first fifty modes (frequency) sorted based on amplitude $(\unicode[STIX]{x1D709})$ are shown. For brevity, the frequencies pertaining to the $y{-}r$ plane are shown in figure 16. However, the comparison between frequencies corresponding to $\unicode[STIX]{x1D709}_{max}$ observed across the $y{-}r$ and $r{-}\unicode[STIX]{x1D703}$ planes is graphically depicted in figure 18(a).

Figure 16. DMD amplitude ranked frequency signatures across various $S_{G}$ values in the $y{-}r$ plane ( $Re_{a}\sim 4896$ ).

4.2.2 Insights into the vortex breakdown dynamics using combined POD and DMD analysis

In general, DMD modes are compared with POD modes by extracting the spatial structure pertaining to each frequency obtained in the DMD spectrum. This is to corroborate whether the energy content of the spatial structure pertaining to the dominant frequency in the DMD plot is physically relevant or not. For instance, the spatial structure (shown in figure 17 a; $S_{G}\sim 2.28$ ) corresponding to $f_{1}$ of $S_{G}\sim 2.28$ (figure 16) is similar to M1 in figure 9(b), can be interpreted as showing that the corresponding flow condition is dominated by the KH shear instability.

The frequency signature plotted against amplitude $(\unicode[STIX]{x1D709})$ delineates the rapid shift of $f_{1}$ towards higher values (300–50 Hz) when the flow approaches conical breakdown (see $S_{G}\sim 2.40,2.53$ in figure 16). However, for the other cases, it is clear that the dominant frequency $f_{1}$ concentrates around ${<}100~\text{Hz}$ (marked as zone A). Particularly for pre- and post-CTRZ cases, $f_{1}$ is observed around ${\sim}30~\text{Hz}$ . Though, all the cases show some frequencies in the range of 300–500 Hz, excepting the CB case, the same is not observed within the first 20 DMD modes.

As mentioned earlier, it is necessary to evaluate the spatial structure pertaining to $f_{1}$ to yield quantitative information on the dynamics governing each flow state (i.e. PVB, CTRZ, CB). The reconstructed spatial structures of $f_{1}$ resemble the most dominant (M1) spatial eigenmodes perceived from POD analysis (see M1 in figures 9 a–c and 10 a,b). The spatial structures of the higher-order frequencies observed with CTRZ, PVB predominantly show shedding which is identified as low energetic structures (% TKE ${\sim}$ 15 %) in POD reconstruction. Hence, the contribution of higher-order frequencies with these flow conditions (i.e. CTRZ, PVB) is insignificant.

Figure 17. Robust dynamic modes observed across various vortex breakdown modes. (a) $y{-}r$ plane; (b) $r{-}\unicode[STIX]{x1D703}$ plane  at  $y/R_{a}\sim 0.3$ ( $Re_{a}\sim 4896$ ).

Furthermore, the energy ratio $(\unicode[STIX]{x1D713})$ between the shedding and shear modes is computed to quantitatively represent the contribution of spatial structures in the given flow conditions. The parameter $\unicode[STIX]{x1D713}$ can be defined as follows,

(4.12) $$\begin{eqnarray}\unicode[STIX]{x1D713}=\frac{(\%~\text{TKE})_{shedding}}{(\%~\text{TKE})_{shear}}.\end{eqnarray}$$

The energy ratio $(\unicode[STIX]{x1D713})$ calculated across various $S_{G}$ values are graphically illustrated in figure 18(b). Additionally, only peak frequencies (i.e. $f_{1}$ ) obtained from the DMD plot are also shown in tandem (figure 18 a). The values clearly demarcate, that with the swirl number increasing from 1.5 to 3, the frequencies show a sudden rise in a very narrow band $(S_{G}\sim 2.45{-}2.55)$ . Correspondingly, the energy contribution of the shedding modes also shows a sharp peak in this zone. Hence, it is interpreted that the occurrence of the sudden rise in the higher-order frequencies together with the dominant nature of shedding mode causes the flow to undergo conical breakdown. Further, figure 18(a,b) is constructed in the form of a regime map from which states of various flow modes under the influence of $S_{G}$ can be identified.

Figure 18. (a) DMD-based amplitude ranked frequencies across various $S_{G}$ values; (b) energy-ranked VB modes derived from POD analysis ( $Re_{a}\sim 4896$ ).

The observations made with POD and DMD analysis can be summarized as follows

(4.13) $$\begin{eqnarray}\displaystyle & f_{1}\leqslant 100;\unicode[STIX]{x1D713}<1\rightarrow PVB,CTRZ, & \displaystyle\end{eqnarray}$$

(4.14) $$\begin{eqnarray}\displaystyle & f_{1}\geqslant 300;\unicode[STIX]{x1D713}\gg 1\rightarrow CB. & \displaystyle\end{eqnarray}$$

4.3 Conical breakdown mechanism

The understanding related to the occurrence of conical breakdown in swirling flows still remains hypothetical since the first report made by Billant et al. (Reference Billant, Chomaz and Huerre1998) and later by Santhosh et al. (Reference Santhosh, Miglani and Basu2013) although extensive works (Benjamin Reference Benjamin1962; Sarpkaya Reference Sarpkaya1971b ; Hall Reference Hall1972; Leibovich Reference Leibovich1978) have been reported on the vortex breakdown phenomenon in swirling flows, occurrence of CB is not understood clearly. Billant et al. (Reference Billant, Chomaz and Huerre1998) showed that flow undergoes CB if the pressure at the stagnation point and the ambient are identical. This essentially shows the absence of steady motion between the fluid in the stagnation zone and the ambient fluid. In general, when the flow is moving towards a higher swirl number, first helical vortices appear in the flow at moderate $S_{G}$ followed by bubble breakdown at higher $S_{G}$ . The transfer of energy from the azimuthal waves to the helical waves is viewed as a source for vortex bubble breakdown at higher $S_{G}$ values.

In the present study, the combined DMD and POD analyses carried out over the time-resolved PIV data acquired at various $y{-}r$ and $r{-}\unicode[STIX]{x1D703}$ planes reveal the dominance of shedding modes at the onset of conical breakdown. In general, although shedding is observed at the outer shear layer (OSL) across all flow conditions, its intensity may be reduced due to the presence of strong shear induced by two counter-rotating eddies in the near field (see figure 3). This can be viewed as a holder for OSL shedding, whereas in the CB case, the flow is not constrained by these two counter-rotating eddies (figure 3; $S_{G}\sim 2.53$ ). The flow starts to shed from the burner exit itself. The vanishing nature of the counter-rotating eddies are clearly visible in the flow case close to CB (figure 3; $S_{G}\sim 2.4,2.67$ ). Furthermore, this can also be interpreted from the DMD spectrum shown in figure 16, for instance, higher-order frequencies (say ${>}300~\text{Hz}$ ) do appear for PVB, CTRZ but the strong shear (axial $+$ azimuthal) induced by counter-rotating vortices prevents the flow from undergoing conical breakdown.

The present and previous investigations clearly delineate that temporally, the CB state undergoes oscillations attributed to strong shear layer shedding. Previous studies (Chanaud Reference Chanaud1965; Spall et al. Reference Spall, Gatski and Grosch1987) have proved that swirl flow undergoes vortex breakdown if and only if the magnitude of $R_{O}$ is less than 1. Although CB always appears in the azimuthal momentum dominant condition (i.e. $R_{O}<1$ ), it happens over a very narrow band and quickly recovers back to CTRZ even when the flow rate is continuously increased, favouring azimuthal momentum. However, the issues which remain as hypotheses relate to why conical breakdown happens in the very narrow band and why it recovers back to the classical CTRZ mode.

The occurrence of conical breakdown in a narrow band can be hypothesized as follows.

In general, swirl flows exhibit instability waves in the azimuthal as well as axial direction. The classical CTRZ structure in swirl flow is an evolution resulting from helical waves occurring at low swirl number. Stability analysis performed with swirling jets clearly delineates the evolution of various forms of shear layer instability mechanisms as a function of $S_{G}$ (Afanasyev & Peltier Reference Afanasyev and Peltier1998; Gallaire & Chomaz Reference Gallaire and Chomaz2003a ). The results presented by Gallaire & Chomaz (Reference Gallaire and Chomaz2003a ) reveal the activation of centrifugal instabilities in addition to KH (axial $+$ azimuthal) instabilities even with the presence of axial plug flow.

Likewise, in the present study, an increase in $S_{G}$ from the CTRZ mode may predominantly pump energy into azimuthal waves rather than into axial KH waves $(\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}>\unicode[STIX]{x1D706}_{axial})$ . The highly energetic unstable waves induced in the azimuthal plane may cause the flow to fan out completely (see figure 20 a). At the instant of CB, the raw images clearly suggest that the axial jets are completely engulfed by the coaxial swirling jet. Moreover, the raw images pertaining to pre- and post-CB also suggests that the instability waves induced due to shear inside the jet (position where the swirl jet interacts with the axial jets) are not intensified (see figure 19 b). However, during CB, the waves are clearly visible at the inner and outer surface of the jet (figure 19 a). This may be due to the dominance of centrifugal instabilities as pointed out by Gallaire & Chomaz (Reference Gallaire and Chomaz2003a ) in their stability analysis. However, the resultant flow structure during the dominance of centrifugal instabilities is not shown in Billant et al. (Reference Billant, Chomaz and Huerre1998).

Figure 19. Schematic illustration of shear waves formed at inner and outer surfaces of the jets.

Figure 20. Hypothetical illustration of azimuthal and axial instability waves. (a) Conical breakdown; (b) CTRZ; (c) spatial evolution of vortices in swirling flow field.

The force induced by axial and azimuthal waves can be written as follows

(4.15) $$\begin{eqnarray}\displaystyle & F_{KH_{axial}}\sim \unicode[STIX]{x1D70C}U_{o}^{2}/\unicode[STIX]{x1D706}_{axial}, & \displaystyle\end{eqnarray}$$

(4.16) $$\begin{eqnarray}\displaystyle & F_{KH_{\unicode[STIX]{x1D703}}}\sim \unicode[STIX]{x1D70C}U_{o}^{2}/\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}. & \displaystyle\end{eqnarray}$$

Beyond CB, a perturbation caused to the flow in terms of increased $S_{G}$ might result in the damping of azimuthal waves ( $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ ; $F_{KH_{axial}}>F_{KH_{\unicode[STIX]{x1D703}}}$ ). The occurrence of this event in the small $S_{G}$ space clearly delineates CB as a highly unstable phenomenon. It should be noted however that the above discussed argument is purely a hypothesis. To precisely probe the mechanism of conical breakdown (particularly to answer why it occurs in a very narrow band), it is necessary to acquire the three-dimensional flow field data in a simultaneous fashion to check for the evolution of vortices in the spatial domain. In swirling flows, the momentum will be spread in the azimuthal as well as axial direction (figure 20 c). As a result, azimuthal and axial shear waves will be induced in the flow field. In addition, Garcia-Villalba, Frohlich & Rodi (Reference Garcia-Villalba, Frohlich and Rodi2005) and Komarek & Polifke (Reference Komarek and Polifke2010) have identified the ‘indirect mechanism’ by which axial vorticity is excited, which ultimately results in fluctuations in the waves. The influence of these waves as flow transits to CB from the CTRZ state can be identified only if these wavenumbers are identified in a quantitative fashion. However, these demand either three-dimensional flow field data or a rigorous three-dimensional stability analysis (Oberleithner et al. Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011), which are not within the scope of this manuscript.

It should be noted that the present study is not aimed at a quantitative understanding of why conical breakdown occurs in a narrow band. Rather, we are more focused on how the flow dynamics evolves as the flow approaches and passes the conical breakdown states. Even with the limitations of time-resolved two-dimensional PIV measurements, we have precisely delineated the precursor events such as the activation of multiband oscillations when the flow approaches the CB state. This information provides more insights into dynamical features of CB rather than the mechanism which governs its formation. This feature has not documented in the literature to date.

Furthermore, while progressively inducing swirl in the axial jets, the flow exhibits strong helical waves followed by vortex breakdown at high swirl number. At the instant of CB, helical waves might vanish completely due to centrifugal effects and may result in the flow detaching from the vortex core. However, to validate this point, three-dimensional flow field data are required to visualize the evolution of helical waves while $S_{G}$ is progressively varied.

Moreover, CB is also observed with a similar type of test rig which discharges swirl flow through a vane swirler. Here also CB appears within a very narrow band when the flow across the swirler is progressively varied (see movie 6). This observation indicates that the occurrence of CB in coaxial swirling flow may be a universal phenomenon. Furthermore, it should be very interesting to note that in both the experiments, CB is witnessed only with the presence of co-annular axial jets. We have not observed any CB in the absence of axial jets, even when the swirl number is progressively varied in very small steps. This clearly indicates that the switching between various instability modes across coaxial jets may cause the flow to undergo conical breakdown.

In both cases, the cone undergoes a series of temporal oscillations in the azimuthal as well as axial direction, which ultimately leads to the dominance of the shedding mode. Previous studies attempted to distinguish CB from other forms of vortex breakdown based on the theoretical arguments deduced from pressure gradients in the flow (Billant et al. Reference Billant, Chomaz and Huerre1998; Santhosh et al. Reference Santhosh, Miglani and Basu2014). However, the time-resolved PIV measurements carried out in this study yield new insights into the dynamics of conical breakdown.

Figure 21. Hysteresis phenomenon associated with conical breakdown pathways. (a) $Re_{a}\sim 4896$ ; (b) $Re_{a}\sim 10\,545$ ; (c) $Re_{a}\sim 17\,456$ .

4.4 Hysteresis phenomenon in conical breakdown pathways

In the global observation section, it was stated that though the time-averaged flow field is similar across all $Re_{a}$ , the vortex breakdown pathways exhibit contradictory behaviour. Particular emphasis is placed on the repeatability associated with the conical breakdown state from CTRZ. To illustrate this phenomenon, experiments have been meticulously carried out in a forward (F) and backward (R) manner, i.e. $S_{G}$ values increased from 1.5 to 3 or decreased from 3 to 1.5 (this can also be interpreted as pre- and post-CTRZ).

Figure 22. Frequency spectrum observed for CTRZ case across various $Re_{a}$ values.

The observations are transformed into a three-dimensional map as illustrated in figure 21. Two parameters namely % TKE (from POD) and frequency pertaining to $\unicode[STIX]{x1D709}_{max}$ of DMD are considered across various $S_{G}$ . These experiments are carried out with very close variation in $\unicode[STIX]{x0394}S_{G}$ values ( ${\sim}\pm 0.05$ ).

From POD and DMD analyses, it is clear that the activation of the higher-order frequencies contributed by shedding eddies in the flow leads to the CB state. Hence, in figure 21, % TKE is considered for the shedding mode. The values of other modes are assigned as zero. For the $Re_{a}\sim 4896$ case, during the forward set of experiments, conical breakdown starts to appear at $S_{G}\sim 2.4$ and is observed until $S_{G}\sim 2.6$ (figure 21 a F1). Progressive decrease in $S_{G}$ from 3 to 1.5 follows the same path (figure 21 b R1) in terms of flow topology. Whereas, for $Re_{a}\sim 17\,456$ , the flow path to the CB state during the forward direction is completely different from the backward direction. In the forward condition, the flow undergoes CB around $S_{G}\sim 2.4{-}2.6$ and transits back to the CTRZ state, whereas during the backward case, there is no evidence of conical breakdown (figure 21 b R1, see blue circles). It should be noted that the activation of higher-order frequencies, which is believed to be a cause for CB, also does not appear with a decreasing value of $S_{G}$ . However, CB is observed (shown as pink dashed lines in figure 21 b) only for a sudden and large decrease in $S_{G}$ (i.e. $\unicode[STIX]{x0394}S_{G}\sim -0.5$ ) for $Re_{a}\sim 10\,545$ . The hysteresis analysis performed by Billant et al. (Reference Billant, Chomaz and Huerre1998) reports that CB appears in forward as well as backward cases. This may be due to the low Reynolds number, as is the case for $Re_{a}\sim 4896$ . Their analysis also pointed out the upward and downward shift of stagnation points during forward and backward conditions respectively. It should be noted that upshift of the stagnation point during flow conditions near to CB was already elucidated in the global observation section (see figure 3).

Figure 23. Illustration of key governing mechanisms observed across various vortex breakdown (VB) modes.

The hysteresis observed at high $Re_{a}$ may be due to the insufficient residence time $(\unicode[STIX]{x1D70F}_{r})$ associated with the flow.

For instance, if the flow is provided with a higher residence time at $Re_{a}\sim 17\,456$ as compared to $Re_{a}\sim 4896$ , then CB is also possible in the backward path. Furthermore, it should be noted that, although the CTRZ mode exhibits a dominant frequency around ${\sim}24~\text{Hz}$ across all the $Re_{a}$ values, some secondary frequency peaks are observed at $Re_{a}\sim 17\,456$ (see figure 22). These may be attributed to the small-scale shedding eddies in the flow field, which are intensified due to the higher  $Re_{a}$ .