4.2.1 $z/D=0.5$

The energy content and eigenvalues of the first 100 most energetic modes for the first-half data segment are displayed in figures 6(a) and 6(b) respectively. Dynamic-mode decomposition modes typically form conjugate pairs, and each pair corresponds to a specific pattern of collective oscillations, which can be physical but not necessarily recurrent because of nonlinearities, complexities and sensitive dependence on the initial and boundary conditions of turbulent swirling reacting flows. Without loss of information, only eigenvalues with a positive imaginary part (i.e. frequency) are shown in figure 6(b). All eigenvalues have a negative real part, which means globally asymptotic stability in the case of linear time-invariant systems. The mean value and the standard deviation for the real part of the eigenvalues for the robust mode are $129.6~\text{s}^{-1}$ and $12.1~\text{s}^{-1}$ respectively. Turbulent swirling reacting flows are highly nonlinear and chaotic, with at least one positive Lyapunov exponent (Wolf et al. Reference Wolf, Swift, Swinney and Vastano1985; Rosenstein, Collins & De Luca Reference Rosenstein, Collins and De Luca1993); thus, the negative real part of the eigenvalues is an artefact of Koopman decomposition. Figure 6(c) shows the identification of robust DMD modes. The number of robust modes reaches a plateau of 2 with the threshold for eigenvalues $(\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{R}=\unicode[STIX]{x1D6FF}_{I})$ above 2 Hz. The conjugate pair of robust modes oscillates at 800 Hz, corresponding to a Strouhal number ( $St$ ) of ${\sim}0.30$ , $St=fD/U_{0}$ , where $f$ , $D$ and $U_{0}$ denote the PVC frequency ( ${\sim}800~\text{Hz}$ ), the swirler exit diameter (19 mm) and the bulk velocity at the swirler exit ( $50~~\text{m}~\text{s}^{-1}$ ) respectively. The pair of robust modes has the smallest negative real part among all eigenvalues and the largest energy content among all DMD modes (approximately 11.9 % of the total turbulent kinetic energy). This conjugate pair of robust modes corresponds to the helical shear-layer mode and its induced motion, as demonstrated in the following discussion. Figure 7(a) shows the phases of the conjugate pair of robust modes which vary linearly with time. The phases are computed from the coefficients of the DMD modes. Figure 7(b) displays the Lissajous figure for the two robust modes. The Lissajous figure appears as a straight line, suggesting that the two modes are essentially identical. This is expected since DMD analyses are linear in the time domain and the two modes form a complex conjugate pair.

Figure 7.

Phases (a) and the Lissajous figure (b) for the conjugate pair of robust modes.

Figure 8.

Construction of a velocity snapshot with an increasing number of DMD modes. The planar velocity ( $V_{x}V_{y}$ ) is shown by arrows, with the axial velocity in colours. The original snapshot (a) is reconstructed with (b) the first six most energetic modes, (c) the first 10 most energetic modes, (d) the first 50 most energetic modes, (e) the first 500 most energetic modes and (f) the first 1500 most energetic modes.

Figure 9.

Robust constituents of velocity at $z/D=0.5$ during stable combustion across one PVC cycle: (a) $t=0.2~\text{m}\text{s}$ , (b) $t=0.4~\text{m}\text{s}$ , (c) $t=0.6~\text{m}\text{s}$ , (d) $t=0.8~\text{m}\text{s}$ , (e) $t=1.0~\text{m}\text{s}$ and (f) $t=1.2~\text{m}\text{s}$ . The planar velocity is shown by arrows with the axial velocity in colours.

Figure 8 shows the construction of an instantaneous velocity snapshot with an increasing number of DMD modes. The fine details of the instantaneous flow can be obtained by increasing the number of DMD modes. The dominant features of the snapshot are fully captured with the first six most energetic modes. In fact, as shown in figure 9, the essential features can be identified with the first two DMD modes. This strongly suggests that DMD – as a frequency-resolved data-reduction technique – has the potential to aid in the development of low-order modelling and adaptive robust control of spatio-temporal systems. Because of the chaotic nature of turbulent swirling combustion, DMD-based low-order models are not expected to yield quantitatively accurate long-term predictions; however, from the control point of view, short-term predictions would suffice (Yi et al. Reference Yi, Wee, Annaswamy and Ghiniem2003).

Time traces of robust modes, i.e. robust constituents, are constructed using (3.7). Robust constituents have very high signal-to-noise ratios since they correspond to collective oscillations at distinct frequencies. Figure 9 displays robust constituents of velocity at $z/D=0.5$ . Two symmetric tadpole regions with almost equal magnitude but opposite signs in axial velocity, i.e. skew symmetric, appear along the inner swirling shear layer. The two skew-symmetric kidney-shaped regions of peak axial velocity correspond to the intersection of the helical PVC with the $R{-}\unicode[STIX]{x1D703}$ plane at $z/D=0.5$ . Two skew-symmetric long-tailed regions with much smaller amplitude of axial velocity extend more than one-half of a perimeter at larger radius because of the radially expanding helical shear-layer disturbance. The two skew-symmetric tadpole regions rotate clockwise, i.e. in the same direction as the mean tangential flow. The rotational frequency $V_{t}/2\unicode[STIX]{x03C0}r$ varies both radially and axially. It is nearly a constant from the jet centreline to the inner swirling shear layer, roughly flattens off across the swirling jet and decreases sharply radially outwards across the outer swirling shear layer. The rotational frequency decreases sharply downstream of the air swirler, around $5800~\text{rad}~\text{s}^{-1}$ (i.e. 923 Hz) at $z/D=0.5$ , around $2800~\text{rad}~\text{s}^{-1}$ (i.e. 446 Hz) at $z/D=1.0$ and around $950~\text{rad}~\text{s}^{-1}$ (i.e. 151 Hz) at $z/D=1.5$ . The PVC frequency of ${\sim}800~\text{Hz}$ falls between the VBD bubble rotating frequency at $z/D=0.5$ and that at $z/D=1.0$ . A linear stability analysis by Rukes et al. (Reference Rukes, Sieber, Paschereit and Oberleithner2015) shows that the mode shape of the helical instability is not affected by the vortex core size, but the frequency is inversely related to the core size. The helical disturbance, as a large coherent structure in strong swirling flows, is driven by (or gains energy from) the mean tangential flow, in much the same way as large coherent structures in separating shear flows, mixing layers, jets and wakes (Ho & Huerre Reference Ho and Huerre1984; Huerre & Monkewitz Reference Huerre and Monkewitz1985, Reference Huerre and Monkewitz1990; Chomaz et al. Reference Chomaz, Huerre and Redekopp1991; Wee et al. Reference Wee, Yi, Annaswamy and Ghoniem2004). This is also supported by recent 3D stability and energy analysis of swirling flows by Wang et al. (Reference Wang, Rusak, Gong and Liu2016). It should be noted that POD analysis of a swirling reacting flow shows a dominant flow structure similar to figure 9 (Stöhr et al. Reference Stöhr, Sadanandan and Meier2011).

Figure 10.

Azimuthal decomposition of axial velocity at $r/D=0.25$ and $z/D=0.5$ during stable combustion. Here, $V_{0}$ refers to the average velocity at the exit of the air swirler at standard atmospheric conditions, $50~\text{m}~\text{s}^{-1}$ .

Azimuthal decomposition – essentially a Fourier series expansion with a periodicity of $2\unicode[STIX]{x03C0}$ along the circumferential direction – has also been used to decompose swirling flow motions (O’Connor & Lieuwen Reference O’Connor and Lieuwen2011). Figure 10 shows the normalized amplitude of azimuthal mode numbers ( $\hat{V}_{z,m}$ ), i.e. the magnitude of the complex Fourier mode coefficient, for axial velocity at $r/D=0.25$ at the axial plane $z/D=0.5$ . It should be noted that $\hat{V}_{z,m}$ is defined as

where $\hat{V}_{z}(r,\unicode[STIX]{x1D703},z,\unicode[STIX]{x1D714})$ refers to the spectrum of axial velocity. The figure shows that the energy is concentrated in three of these modes. The $m=-1$ mode has the largest amplitude among all azimuthal modes. Its amplitude is approximately two times the amplitude of the $m=-2$ azimuthal mode and much larger than the $m=0$ mode, i.e. the axisymmetric mode. Both the $m=-1$ mode and the $m=-2$ mode peak around 800 Hz, suggesting that the two azimuthal modes actually belong to a single structure, i.e. the helical shear-layer mode shown in figure 9.

Figure 11.

Robust constituents of axial vorticity (shown in colours) and planar velocity (shown by arrows) for stable combustion across one PVC cycle at $z/D=0.5$ : (a) $t=0.2~\text{m}\text{s}$ , (b) $t=0.4~\text{m}\text{s}$ , (c) $t=0.6~\text{m}\text{s}$ , (d) $t=0.8~\text{m}\text{s}$ , (e) $t=1.0~\text{m}\text{s}$ and (f) $t=1.2~\text{m}\text{s}$ .

Figure 12.

Robust constituents of OH-PLIF (shown in colours) and planar velocity (shown with arrows) at $z/D=0.5$ during stable combustion across one disturbance mode cycle: (a) $t=0.2~\text{m}\text{s}$ , (b) $t=0.4~\text{m}\text{s}$ , (c) $t=0.6~\text{m}\text{s}$ , (d) $t=0.8~\text{m}\text{s}$ , (e) $t=1.0~\text{m}\text{s}$ and (f) $t=1.2~\text{m}\text{s}$ .

Figure 11 displays the axial vorticity $\unicode[STIX]{x1D714}_{z}$ overlapping with the planar velocity at $z/D=0.5$ . The axial vorticity of opposite signs appears in two symmetric circumferentially extended tadpole regions along both the inner and the outer swirling shear layers. Both regions are rotating clockwise, i.e. in the same direction as the mean tangential flow. It should be noted that the robust modes are displaying the dynamic behaviour since the mean flow has been subtracted out before the modal decomposition. Along the inner swirling shear layer, the compact region of peak axial vorticity is located closer to the centreline than to the tail region. Analysis of streamlines between the two compact regions reveals that the two skew-symmetric compact regions of peak axial vorticity cannot be interpreted as counter-rotating vortices. For counter-rotating vortices, the streamlines form closed orbits around each vortex centre. From figures 9 and 11, it is evident that the two skew-symmetric regions of peak planar velocity are fan-shaped, spanning the inner swirling shear layer, the swirling jet and the outer swirling shear layer. In the combustion literature, the helical shear-layer mode wraps around the inner swirling shear layer, rather than appearing as a fan-shaped structure (Roux et al. Reference Roux, Lartigue, Poinsot, Meier and Berat2005; Garcia-Villalba et al. Reference Garcia-Villalba, Frohlich and Rodi2006; Selle et al. Reference Selle, Benoit, Poinsot, Nicoud and Krebs2006; Wang et al. Reference Wang, Yang, Hsiao, Hsieh and Mongia2007; Huang & Yang Reference Huang and Yang2009; Valera-Medina et al. Reference Valera-Medina, Syred and Griffiths2009; Oberleithner et al. Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011; Stöhr et al. Reference Stöhr, Boxx, Carter and Meier2012). The reason for this is that in these investigations the coherent structure is identified from pressure, azimuthal vorticity, the swirling strength $\unicode[STIX]{x1D706}$ or the $\unicode[STIX]{x1D706}_{2}$ criterion rather than velocity. Comparisons of figures 9 and 11 clearly show that the phase difference between regions of peak axial vorticity and regions of peak axial velocity is approximately $90^{\circ }$ . This is expected since vorticity is the curl, i.e. a spatial derivative, of velocity.

Figure 12 displays the robust constituents of OH-PLIF (shown in colours) overlapping with the planar velocity (shown with arrows) at $z/D=0.5$ . It should be noted that the negative OH value in figure 12 is an artefact of DMD since the mean OH-PLIF intensity is subtracted before modal decomposition. As described above, ground-state OH exists both along the flame front and within the post-flame region; thus, the OH-PLIF intensity is not an indicator of the reaction zone. However, DMD works naturally with continuous variables instead of binary variables. The rationale of gaining qualitative information on the reaction zone and heat-release intensity from OH-PLIF imaging lies in the fact that the OH concentration is highest along and immediately downstream of the flame front. Combustion along both the inner and the outer swirling shear layers is affected by the shear-layer disturbance. The coherent-structure-affected reaction zone along the inner swirling shear layer is rather compact, commensurate with the compact mean reaction zone shown in figure 4(c). The vortex-affected reaction zone along the outer swirling shear layer extends to almost one-half of the perimeter. The vortex-affected heat-release zones are rotating clockwise, i.e. in the same direction as the mean tangential flow. Along the inner swirling shear layer, the OH-PLIF intensity is reduced when the reaction zone or the turbulent flame front is pushed towards the centreline, i.e. the product side, and vice versa; along the outer swirling shear layer, the OH-PLIF intensity is reduced when the turbulent flame front is pushed radially outwards towards the product side under the vortex motion, and vice versa. By correlating figure 12 with figure 11, one can see that along the outer swirling shear layer, the region of stronger OH-PLIF intensity overlaps with that of positive axial vorticity, and that along the inner swirling shear layer, the region of stronger OH-PLIF intensity overlaps with that of negative axial vorticity, suggesting the effects of turbulence and velocity gradients on the flame structure. The flame temperature and reaction rates decrease with smaller Damkohler number ( $Da$ ), i.e. the ratio of the characteristic flow time to the characteristic chemical time scale, because of an increasing rate of scalar dissipation (Giovangigli & Smooke Reference Giovangigli and Smooke1987; Law Reference Law2006). The OH-PLIF intensity becomes weaker when the flame is pushed towards the product side under vortex motion, similarly to the decreasing chemical reaction rate in counterflow flames under increasing strain rates.

Figure 13.

Robust constituents for stable swirling combustion at $z/D=1.0$ . (a) Velocity ( $\text{m}~\text{s}^{-1}$ ), (b) axial vorticity ( $\text{s}^{-1}$ ), (c) OH-PLIF intensity (a.u.).

4.2.2 $z/D=1.0$

To delineate the spatial development of the helical mode downstream of the air swirler, DMD analyses were conducted farther downstream of the air swirler at $z/D=1.0$ . A single conjugate pair of robust modes with the same frequency as those at $z/D=0.5$ , i.e. around 800 Hz, can be identified on the $R{-}\unicode[STIX]{x1D703}$ plane at $z/D=1.0$ . The robust constituents rotate clockwise, i.e. in the same direction as the mean tangential flow. For simplicity, in figure 13, only one snapshot of the robust constituents for velocity, axial vorticity and OH-PLIF is shown. From comparison of figure 13(a) with figure 9, it is evident that the two skew-symmetric regions of peak axial velocity have extended circumferentially from kidney-shaped at $z/D=0.5$ to banana-shaped at $z/D=1.0$ and are located at a larger radius, which is commensurate with radial expansion of the swirling jet. The peak axial velocity has decreased by approximately a factor of 2 from $z/D=0.5$ to $z/D=1.0$ , suggesting strong decay of the PVC downstream of the air swirler. Unlike at $z/D=0.5$ , at $z/D=1.0$ , no discernible streamlines between the two skew-symmetric regions of peak axial vorticity are observed, indicating that the PVC is no longer able to penetrate deep into the central recirculation zone. By comparing figure 13(b) and figure 11, it is evident that the peak axial vorticity of the vortex ‘annulus’ has decayed more than three times from $z/D=0.5$ to $z/D=1.0$ . In addition, the axial vorticity along the outer swirling shear layer has become much stronger than that of the vortex annulus. Figure 13(c) shows that the OH-PLIF intensity is stronger when the flame front is pushed towards the reactant under PVC motion, and vice versa, similarly to what is shown in figure 12. The OH-PLIF intensity is higher along the outer swirling shear layer than along the inner swirling shear layer.

Figure 14.

Robust constituents for stable swirling combustion at $z/D=1.5$ . (a) Velocity ( $\text{m}~\text{s}^{-1}$ ), (b) axial vorticity ( $\text{s}^{-1}$ ), (c) OH-PLIF intensity (a.u.).

4.2.3 $z/D=1.5$ and $z/D=2.0$

At $z/D=1.5$ , no robust modes are found until $\unicode[STIX]{x1D6E5}$ is increased to ${\geqslant}$ $0.50$ , and the pair of robust modes has the same frequency as that at $z/D=0.5$ and $z/D=1.0$ , i.e. around 800 Hz. Figure 14 displays a snapshot of robust constituents for velocity, axial vorticity, and OH-PLIF intensity, all of which are much weaker than those at $z/D=0.5$ and $z/D=1.0$ . Robust constituents are located at larger radius, which is commensurate with radial expansion of the swirling jet and swirling shear layers. From figure 14(b), it is difficult to differentiate the fragmented vortex annulus from vortices along the swirling shear layers, which are mainly generated by combustion and hydrodynamic instabilities. Figure 14(c) shows that the turbulent flame brush along the inner swirling shear layer has merged with that of the outer swirling shear.

At $z/D=2.0$ , no robust modes are found, even when $\unicode[STIX]{x1D6E5}$ is increased to ${\geqslant}$ 0.75. As can be observed from the robust constituents at $z/D=0.5$ , 1.0 and 1.5, the helical mode undergoes rapid decay downstream of the air swirler. At $z/D=2.0$ , the helical disturbance has probably decayed to a point where it is of minimal practical significance. This result is consistent with prior computations and experimental observations that show relatively rapid decay of these helical shear-layer modes (Roux et al. Reference Roux, Lartigue, Poinsot, Meier and Berat2005; Garcia-Villalba et al. Reference Garcia-Villalba, Frohlich and Rodi2006; Selle et al. Reference Selle, Benoit, Poinsot, Nicoud and Krebs2006; Wang et al. Reference Wang, Yang, Hsiao, Hsieh and Mongia2007; Huang & Yang Reference Huang and Yang2009; Valera-Medina et al. Reference Valera-Medina, Syred and Griffiths2009; Oberleithner et al. Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011; Stöhr et al. Reference Stöhr, Boxx, Carter and Meier2012).

5.2.1 $z/D=0.5$

Figures 18(a) and 18(b) show the energy content and eigenvalues respectively of the first 100 most energetic DMD modes for the first-half data segment during unstable combustion at $z/D=0.5$ . Only DMD modes with a positive imaginary part are shown in figure 18(b). Eigenvalues are widely scattered up to 2500 Hz, and most DMD modes are not reproducible. Figure 18(c) shows that the number of robust modes reaches a plateau of 2 with increasing thresholds for eigenvalues ( $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{R}=\unicode[STIX]{x1D6FF}_{I}$ ) at $\unicode[STIX]{x1D6E5}=0.25$ . The conjugate pair of robust DMD modes around 95 Hz has the smallest damping ratio (i.e. least damped) among all of the DMD modes. The mean value and the standard deviation of the real part of the eigenvalues for the robust modes are $-39.4~\text{s}^{-1}$ and $4.46~\text{s}^{-1}$ respectively. The two robust modes account for ${\sim}20.7\,\%$ of the total turbulent kinetic energy, which is almost two times that of robust modes for stable combustion. The phases of the two robust modes vary almost linearly with time, and the sines of the two modes (i.e. the Lissajous figure) completely overlap.

Figure 19.

Robust constituents of velocity at $z/D=0.5$ during combustion instabilities across one acoustic cycle. Panels (a), (b), (c), (d), (e), (f), (g), (h) and (i) correspond to $t=0$ , 1.2, 2.4, 3.6, 4.8, 6.0, 7.2, 8.4 and 9.6 ms respectively.

Figure 19 displays the robust constituents of velocity for unstable combustion at $z/D=0.5$ , which are essentially axisymmetric. The helical shear-layer mode visualized in figures 9, 11 and 12 for stable swirling combustion has been significantly suppressed and overwhelmed by the axisymmetric flow response to planar acoustic excitations during combustion instabilities. Azimuthal decomposition (figure 20) shows that the amplitude of the axisymmetric mode (i.e. $m=0$ ) is almost 10 times higher than that of helical or azimuthal modes (i.e. $m\neq 0$ ). The dump plane, $z=0$ , can be treated as a pressure anti-node, because of the large blocking ratio ( ${\sim}90\,\%$ ). Pressure oscillations within the combustor generate nearly periodic unsteady flows across the air swirler and along the swirling jet, resulting in nearly periodic variations of the flame surface area and heat release. As shown in figure 19, fluctuations in the axial velocity reach $70~\text{m}~\text{s}^{-1}$ , which is comparable to the mean axial velocity. In previous investigations, it has been found that pressure oscillations along the combustion chamber are essentially acoustical; however, unsteady flows along the inlet pipe to the combustor cannot be fully described by linear acoustics – particularly during low-frequency combustion instabilities (Yi & Santavicca Reference Yi and Santavicca2012). Figure 19 shows that flow reversal around the centreline occurs when the planar velocity is in the same direction as the mean tangential flow, i.e. clockwise, and that no flow reversal occurs when the planar velocity is in the direction opposite to the mean tangential flow. Flow reversal around the centreline is caused by increasing pressure gradients downstream of the air swirler. When the tangential velocity of the robust constituents is in the same direction as the mean tangential flow, i.e. clockwise, the pressure gradient is higher, thus favouring formation of VBD and flow reversal along the centreline.

Figure 20.

Normalized amplitude of azimuthal modes for axial velocity at $r/D=0.25$ and $z/D=0.5$ during combustion instabilities. Here, $V_{0}$ refers to the average velocity at the exit of the air swirler at standard atmospheric conditions, $50~\text{m}~\text{s}^{-1}$ .

Figure 21.

Transfer functions for axial velocity at $z/D=0.5$ during combustion instabilities. (a) Amplitude and (b) phases (degree).

To illustrate the spatial variations in flow response to acoustic oscillations, the transfer function of axial velocity is shown in figure 21,

where $\hat{V_{z}}$ denotes the Laplace transform of axial velocity. The peak axial velocity along the swirling jet is located around ( $x=0$ , $y=0.6$ ). Figure 21(a) shows that axial-velocity fluctuations are most intense along the swirling jet and that they decay radially outward across the outer swirling shear layer and radially inward across the inner swirling shear layer. Figure 21(b) shows that axial flows along the inner swirling shear layer and within the central recirculation zone lag behind those of the annular swirling jet up to $135^{\circ }$ . The fluctuating swirling jet is the driving force behind the unsteady flows within the central recirculation zone and along the swirling shear layers.

Figure 22.

Robust constituents of axial vorticity and planar velocity across one acoustic cycle during combustion instabilities at $z/D=0.5$ . Panels (a), (b), (c), (d), (e), (f), (g), (h) and (i) correspond to $t=0$ , 1.2, 2.4, 3.6, 4.8, 6.0, 7.2, 8.4 and 9.6 ms respectively.

Figure 22 displays time traces of axial vorticity across one acoustic cycle. The distribution of axial vorticity along the outer swirling shear layer, the inner swirling shear layer and the central region around the centreline is nearly axisymmetric. When the planar velocity of the robust constituents is in the same direction as the mean tangential flow, i.e. clockwise, the axial vorticity is in the same direction as the mean flow, and vice versa. It should be noted that the axial vorticity along the outer swirling shear layer is always in the direction opposite to that along the inner swirling shear layer and around the centreline. The magnitude of the axial vorticity increases with the planar velocity since the axial vorticity is a spatial derivative of the planar velocity.

Figure 23.

Robust DMD modes for velocity during combustion instabilities: (a) $z/D=0.5$ , (b) $z/D=1.0$ and (c) $z/D=1.5$ .

Figure 24.

Robust DMD modes for axial vorticity during combustion instabilities: (a) $z/D=0.5$ , (b) $z/D=1.0$ and (c) $z/D=1.5$ .

5.2.2 $z/D=1.0$ and $z/D=1.5$

Dynamic-mode decomposition analyses were conducted farther downstream of the air swirler at $z/D=1.0$ and $z/D=1.5$ . At $z/D=1.0$ and $z/D=1.5$ , a conjugate pair of robust DMD modes was identified with the same cutoff thresholds as those at $z/D=0.5$ . The pair of robust modes oscillated at 95 Hz with the smallest damping ratio among all of the DMD modes, and their phases varied linearly with time. Similarly to what occurred at $z/D=0.5$ , the flow response to acoustic oscillations during combustion instabilities was axisymmetric. Without loss of information, only the eigenmode for velocity is shown in figure 23 and the axial vorticity in figure 24. Reverse axial flows around the centreline appear when the planar velocity is in the same direction as the mean tangential flow, i.e. clockwise, and vice versa. The axial-velocity oscillations along the swirling jet decrease sharply downstream of the air swirler – $70~\text{m}~\text{s}^{-1}$ at $z/D=0.5$ , $60~\text{m}~\text{s}^{-1}$ at $z/D=1.0$ and $42~\text{m}~\text{s}^{-1}$ at $z/D=1.5$ – because of radial expansion of the swirling jet. Figure 24 shows that the vortex annulus expands radially outwards downstream of the air swirler with sharp reduction in strength. The axial vorticity of the vortex annulus is negative when the robust modes are rotating clockwise, i.e. in the same direction as the mean tangential flow, and vice versa. The intact vortex core upstream of VBD (and the vortex annulus downstream of VBD) is formed from vortex sheets that are shed from the boundary layer around the swirler hub (Leibovich Reference Leibovich1984). Obviously, the axial vorticity is stronger when the velocity across the air swirler increases and weaker when the velocity decreases.

At $z/D=2.0$ , robust DMD modes cannot be identified until $\unicode[STIX]{x1D6E5}$ is set to ${\geqslant}0.5$ . The swirling jet and the outer swirling shear layer are out of the imaging domain; therefore, results for $z/D=2.0$ are not discussed.