4.1 Discrete and continuous spectrum

Figure 2 shows the numerical convergence results of the discrete spectrum with respect to the radial extent of the domain $r_{max}$ (figure 2 a) and the order of Chebyshev polynomial $N$ (figure 2 b). For the parametric space of figure 2, there are two discrete unstable modes ( $\unicode[STIX]{x1D714}_{i}>0$ ) with exponential growth, which quickly converge to fixed points in the complex- $\unicode[STIX]{x1D714}$ plane once the numerical tuning parameters of $r_{max}$ and $N$ are gradually increased. Note that the order of Chebyshev polynomials required here for such a convergence is larger than in Oberleithner et al. (Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011), who reported convergence around $N=300$ . This is not surprising since we do not use any additional stretching of the Gauss–Lobatto points, as discussed before in § 3.2, since it can potentially worsen the convergence of the continuous spectrum. Once any of the discrete modes cross the neutral axis ( $\unicode[STIX]{x1D714}_{i}=0$ ) toward the positive half-plane, they turn hydrodynamically unstable. The other discrete modes which are below this axis ( $\unicode[STIX]{x1D714}_{i}<0$ ) are stable modes with exponentially decaying eigenfunctions along the radial direction. Here, such stable exponential modes are seen to converge into branches of distinct shape, similar to e.g. the plane Couette and pipe flow spectra (see e.g. Schmid & Henningson Reference Schmid and Henningson2001).

The remaining spectrum is now referred to as the continuous spectrum, where we introduce classifications originally due to Obrist & Schmid (Reference Obrist and Schmid2010), Mao & Sherwin (Reference Mao and Sherwin2011). Within this spectrum, we classify as ‘potential modes’ those that have significant radial variations outside the jet core, while still being asymptotically stable, with a slower algebraic decay (rather than the exponential decay of typical stable discrete modes) at larger radial distances (see figure 3 for visualizations). In figure 2, these modes can be seen to be scattered over a large part of the spectrum, as their numbers depend upon  $N$ . Mathematically, such modes are also known to be highly non-orthogonal (Obrist & Schmid Reference Obrist and Schmid2003; Mao & Sherwin Reference Mao and Sherwin2011) and hence there is a greater chance for them to participate in the short time transient growth, which we investigate here. Although seemingly spurious in nature, we shall find in § 4.2 that these potential modes too can be bounded (i.e. numerically converged) within an area in the complex plane via computing a pseudospectrum. Further, potential modes, for which $\unicode[STIX]{x1D714}_{i}\rightarrow 0$ are called ‘free-stream modes’, following terminologies from boundary layer flows (see e.g. Mack Reference Mack1976; Zaki & Saha Reference Zaki and Saha2009), used for vortical flows by Mao & Sherwin (Reference Mao and Sherwin2011, Reference Mao and Sherwin2012), which at the limit have zero radial decay rate. As we shall see, these modes ensure a small but finite transient growth even at larger times. In this work, $r_{max}=10$ and $N=1000$ are used in all computations.

Figure 3. Radial decay of the potential mode (○) eigenfunctions $(|\hat{u} _{\unicode[STIX]{x1D703}}|)$ , shown in (b) for the modes highlighted in (a) at ●, $(1.479-0.198\text{i})$ and ▪, $(1.436-0.348\text{i})$ in the marked group A region and at ●, $(1.413-0.430\text{i})$ ; ▪, $(1.384-0.516\text{i})$ and ♦, $(1.352-0.605\text{i})$ in the group B region. The parameters are $m=1$ , $\unicode[STIX]{x1D6FC}=1$ at $z^{\ast }/D^{\ast }=1.0$ , while in (a) the solid lines indicate the pseudospectrum of $\unicode[STIX]{x1D716}=10^{-6}$ ; ♢ are discrete unstable and ▫ are discrete stable modes.

4.2 Pseudospectrum

The continuous modes which do not converge to fixed points in space may at first glance seem spurious. But on closer inspection, most of these modes in figure 2 are seen to roughly fill a rectangular area bounded approximately by the lines $\unicode[STIX]{x1D714}_{r}-1=0$ , $2\unicode[STIX]{x1D714}_{i}+\unicode[STIX]{x1D714}_{r}+1=0$ and the imaginary axis on its three sides, even as the numerical parameters are varied. These bounds do not appear to satisfy simple analytical relations, as they change significantly with the mean swirling flow developing downstream into the wake region.

Mathematically, the potential modes are not exact solutions of the modal system (3.4) and the corresponding boundary conditions, but satisfy them to some finite error. The $\unicode[STIX]{x1D716}$ -pseudospectrum $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D716}}$ of operator $\unicode[STIX]{x1D63E}$ in (3.4) is required to quantify such errors in estimating the continuous modes, defined as (e.g. Trefethen & Embree Reference Trefethen and Embree2005)

(4.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D716}}(\unicode[STIX]{x1D63E})=\{\unicode[STIX]{x1D706}\in \mathbb{C}:\Vert (\unicode[STIX]{x1D706}\unicode[STIX]{x1D644}-\unicode[STIX]{x1D63E})^{-1}\Vert _{2}<\unicode[STIX]{x1D716}\}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D706}$ is a complex eigenvalue and for $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D716}}$ the minimum singular value is used. For the discrete modes, $\unicode[STIX]{x1D716}=0$ , while for the continuous modes the finite magnitude of $\unicode[STIX]{x1D716}$ represents the error in estimating such modes.

In this work, the $\unicode[STIX]{x1D716}$ -pseudospectrum is computed after projection of (3.4) into a lower-dimensional subspace via a partial Schur decomposition and then an inverse Lanczos iteration is done to compute the smallest singular values, following procedures described in Trefethen (Reference Trefethen1999).

Figure 2 also shows convergence of this $\unicode[STIX]{x1D716}$ -pseudospectrum where the $\unicode[STIX]{x1D716}=10^{-9}$ contour is plotted for different parametric combinations of $N$ and $r_{max}$ . In spite of an apparent lack of convergence of potential modes in the discrete sense, the $\unicode[STIX]{x1D716}$ -pseudospectrum contour clearly marks a region for such modes, whose top and right boundaries have converged for all parameters of the figure. This process also identifies a second distinct region for the potential modes at the junction of discrete stable mode branches, which are further classified in figure 3(a) for a different set of parameters. Such a splitting up of the potential modes region into two distinct sections is observed for the first time, which we label here as ‘group A’ and ‘group B’, respectively (see figure 3 a), where the former group lies at the junction of discrete stable mode branches. In figure 3(b), the absolute value of $\hat{u} _{\unicode[STIX]{x1D703}}$ is plotted for a selection of the potential modes, as marked in figure 3(a), gradually moving from the group A to group B regions. The nature of radial decay for all these modes in these twin regions follows $|\hat{u} _{\unicode[STIX]{x1D703}}|\sim r^{-\unicode[STIX]{x1D708}}$ (see e.g. Obrist & Schmid Reference Obrist and Schmid2010), where the exponent $\unicode[STIX]{x1D708}$ varies. The potential modes inside the group A region show characteristics resembling the discrete stable modes, with the first mode at $(1.479-0.198\text{i})$ in figure 3(a) being almost indistinguishable from a discrete stable mode with a peak near $r=0.5$ , as shown in figure 3(b). As we move toward the group B potential modes, a secondary peak appears outside the core vortical region, which becomes dominant for modes of the group B region, starting from the $(1.413-0.430\text{i})$ mode marked in figure 3(a), with the eventual disappearance of the peak inside the jet core (see figure 3 b). The group A potential modes show slower decay at $\unicode[STIX]{x1D708}\approx 0.1$ , while for the group B modes this is much faster at $\unicode[STIX]{x1D708}\approx 0.4$ . As we shall find out in § 5.2, such different decay rates play an important role in their respective transient gain calculations, where the group A modes show higher dominance.

The $\unicode[STIX]{x1D716}$ -pseudospectrum contours when plotted for the discrete modes appear as concentric circles (not shown here, but see e.g. Mao & Sherwin Reference Mao and Sherwin2011), which are possible to be resolved only at lower  $N$ , when some of these discrete modes are yet to be fully converged and thus have finite convergence errors.

The $\unicode[STIX]{x1D716}$ -pseudospectrum calculations as shown here, allow for a more definitive way of classifying the continuous spectrum discussed in § 4.1. We use this in the transient growth calculations of § 5, where we adopt the convention that potential modes belong to the part of the continuous spectrum bounded by a specific pseudospectrum contour level, chosen to be $\unicode[STIX]{x1D716}=10^{-6}$ in this work (see also figure 4), with the remaining spectrum automatically classified into free-stream modes.

Figure 4. Modal classification and local transient growths for (a,b)  $m=1$ , $\unicode[STIX]{x1D6FC}=1$ at $z^{\ast }/D^{\ast }=0.25$ ; (c,d)  $m=1$ , $\unicode[STIX]{x1D6FC}=1$ at $z^{\ast }/D^{\ast }=1.0$ and (e,f)  $m=2$ , $\unicode[STIX]{x1D6FC}=0.5$ at $z^{\ast }/D^{\ast }=2.6$ . The specific modes are ♦ discrete unstable; ▫ discrete stable; ○ potential and ▵ free-stream types, while the solid line in (a,c,e) show pseudospectrum of $\unicode[STIX]{x1D716}=10^{-6}$ . Transient gains in (b,d,f) are from –  –  – discrete stable; –   $\cdot$   –   $\cdot$ potential; –   $\cdot \cdot$   –   $\cdot \cdot$ potential $+$ discrete stable; —  —  — all stable (including free stream) and —— all unstable modes, while the thick grey line indicates net gain from all modes. In addition, in (d) gain curves for –   $\cdot$   –  ●  – group A potential; –   $\cdot$   –  ▪  – group B potential and –   $\cdot \cdot$   –  ●  – group A potential $+$ stable discrete modes are shown, where these potential mode groupings are labelled in (c). The insets in (a,c) show content of respective grey boxes.

5.1 Parametric space

The transient growths are computed at 10 streamwise locations as depicted in figure 1: $z^{\ast }/D^{\ast }=0.25$ , 0.5, 0.6, 1.0, 1.4, 1.5, 1.8, 2.2, 2.6 and 3.0, where the first five locations are approximately inside the recirculation bubble, while the others are inside its wake. We study perturbations composed of five azimuthal wavenumbers: $m=0$ , 1, 2, 3 and 4, where our choice of the higher wavenumbers $(m>2)$ reflects the expectation that higher-order helical modes should have significant role in high- $Re$ , high swirl number flows (see also Ruith et al. Reference Ruith, Chen, Meiburg and Maxworthy2003; Gallaire et al. Reference Gallaire, Ruith, Meiburg, Chomaz and Huerre2006), which point is as yet unexplored. The streamwise wavenumber, on the other hand, is known to play a key role in the selection of transient growth mechanism for swirling flows (e.g. Fontane et al. Reference Fontane, Brancher and Fabre2008), where completely different processes seem to dominate at the lower and higher end of the wavenumbers. Keeping this in mind, we have chosen 12 different values of  $\unicode[STIX]{x1D6FC}$ , with $\unicode[STIX]{x1D6FC}=0.05$ , 0.1 and 0.3 near the long-wavelength end, $\unicode[STIX]{x1D6FC}=0.5$ , 0.7 and 1.0 for intermediate values, and $\unicode[STIX]{x1D6FC}=1.5$ , 2.0, 2.5, 3.0, 3.5 and 4.0 to study the effect of higher wavenumber perturbations. In what follows, owing to the large size of this database, we attempt to highlight and summarize the major findings, while the detailed results for all the parameters are available in appendix A.

5.2 Role of potential modes in transient growths

The role of a continuous spectrum, if present, for the short-time transient growth has long been argued, especially for the boundary layer type flows (see e.g. discussions in Schmid & Henningson Reference Schmid and Henningson2001), and different techniques have often been required in order to include this continuous spectrum and to calculate its optimal transient growth (see Butler & Farrell Reference Butler and Farrell1992; Corbett & Bottaro Reference Corbett and Bottaro2000). Obrist & Schmid (Reference Obrist and Schmid2003) found important short-time transient growths in a class of boundary layer flows (swept Hiemenz flow) from algebraically decaying modes, which were later demonstrated to be part of the continuous spectrum of such a flow (see Obrist & Schmid Reference Obrist and Schmid2010). Mao & Sherwin (Reference Mao and Sherwin2011) took this forward and found the existence of such a continuous spectrum in the Batchelor vortex model, where the transient optimal growth from the full spectrum was found to be almost identical to that from just the continuous eigenmodes (see Mao & Sherwin Reference Mao and Sherwin2012), while the growth from only the discrete stable modes was one order of magnitude smaller.

In this section, we first investigate the source of transient gains in the swirling jet spectrum of figure 2 and whether potential modes contribute significantly to it, as has been found for the Batchelor vortex flow. For this, we pick three streamwise locations from figure 1, at $z^{\ast }/D^{\ast }=0.25$ , which is at the leading edge of the recirculation bubble (see figure 4 a,b), at $z^{\ast }/D^{\ast }=1.0$ , near the centre of this bubble (see figure 4 c,d) and at $z^{\ast }/D^{\ast }=2.6$ , well inside the downstream wake region (see figure 4 e,f). We first note that there appears to be significant broadening of the continuous spectrum at the downstream locations, where the leading pseudospectra move closer toward the unstable half of the complex plane. This points to increased expectations of strong transient growth at such downstream locations (e.g. Trefethen et al. Reference Trefethen, Trefethen, Reddy and Driscoll1993). This is especially true for profiles inside the wake region (see for $z^{\ast }/D^{\ast }=2.6$ in figure 4 e), which further indicates a greater probability for the wake region to support such non-modal growths. When compared to modal growths from the corresponding discrete unstable mode(s), such non-modal growths may dominate, as the former growth rates are also significantly weakened at the wake (consistent with the findings of Oberleithner et al. Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011). This is also reflected in the number of discrete unstable modes at $z^{\ast }/D^{\ast }=1.0$ (figure 4 c), which decreases drastically at the location $z^{\ast }/D^{\ast }=2.6$ (figure 4 e), confirming reduced modal dynamics at the latter location. Such a trend can now be visualized in figure 4(b,d,f), where at $z^{\ast }/D^{\ast }=1.0$ , the exponential growth from the modally unstable spectrum quickly exceeds the transient gain from continuous spectrum, at $\unicode[STIX]{x1D70F}<10$ , well before the latter reaches its peak (see figure 4 d). In contrast, at $z^{\ast }/D^{\ast }=2.6$ , for the parameters of figure 4, the transient gain from all stable modes peaks at $\unicode[STIX]{x1D70F}\approx 77$ , while the exponential growth from the sole unstable discrete mode exceeds this transient gain only at $\unicode[STIX]{x1D70F}>108$ , well after the latter starts to saturate. In figure 4(b), the nature of unstable energy gain is the result of exponential gains from two discrete unstable modes of almost equal strength (see figure 4 a), which at smaller $\unicode[STIX]{x1D70F}$ yields an algebraic behaviour with oscillations, whose amplitudes progressively diminish with time.

Figure 4 also indicates how different portions of the stable spectrum participate in transient amplification. Clearly, the discrete stable modes and potential modes by themselves, but especially the former group, yield relatively little transient gain, while when considered together the resulting increased non-orthogonality yields a growth higher by significant orders of magnitude (e.g. almost three orders higher than the transient growth due to only potential modes and five orders over discrete stable modes, as shown in figure 4 f), similar to observations for Batchelor vortex flows (Mao & Sherwin Reference Mao and Sherwin2012). Here, the role of free-stream modes seems to be rather minimal except at higher times by which point the transient gain from the rest of stable spectrum has almost disappeared, when these modes add a small but finite gain at all subsequent times, as evident in figure 4(b) but mostly invisible in the other two cases.

Figure 5. Local transient growths from the hydrodynamically stable modes inside the recirculation bubble at locations $z^{\ast }/D^{\ast }=0.6$ and 1.0 (as labelled) computed for (a,b)  $m=1$ , (c,d)  $m=2$ , (e,f)  $m=3$ and (g,h)  $m=4$ perturbations at the following streamwise wavenumbers $\unicode[STIX]{x1D6FC}$ , —  —  — 0.1; –   $\cdot$   –   $\cdot$  0.3; —— 0.5; —  —  ●  — 1.0; –   $\cdot$   –  ● 2.0 and ——● 3.0, shown as superimposed upon the corresponding exponential growths from all the unstable modes (in dark grey).

Next, focusing further on the details of potential spectrum, our particular choice of $\unicode[STIX]{x1D716}$ -pseudospectrum contour ( $\unicode[STIX]{x1D716}=10^{-6}$ ) in figure 4(c) has demarcated the entire potential spectrum into two groups, as classified before in figure 3(a). The group A modes, which show relatively slower algebraic decay in figure 3(b), are shown here to produce higher transient gains at higher times (see figure 4 d), while the group B spectrum does this at relatively shorter times, yielding the two-hump curve for the growth from the entire potential spectrum in figure 4(d). However, once the discrete stable spectrum is added to the group A potential modes, it is observed (in figure 4 d) to yield a growth almost identical to that from the entire stable spectrum, which clearly points to this part of the potential and discrete spectrum having the most significant non-modal behaviour, owing to their proximity to the unstable half-plane.

Figure 6. Same as in figure 5 but for locations in the wake region at $z^{\ast }/D^{\ast }=1.8$ and 2.6, as labelled.

5.3 Transient growth comparisons inside the bubble and wake

Once the potential for transient growths is apparent in § 5.2, in this section we demonstrate this short-time transient growth in detail at 4 selected locations: two locations inside the recirculation bubble at $z^{\ast }/D^{\ast }=0.6$ and 1.0 (near the middle of this bubble) and for two locations inside the wake at $z^{\ast }/D^{\ast }=1.8$ (‘near wake’) and 2.6 (‘far wake’). Figure 5 shows the results for locations inside the bubble for $m=1$ , 2, 3 and 4 at the following six selected streamwise wavenumbers: $\unicode[STIX]{x1D6FC}=0.1$ , 0.3, 0.5, 1.0, 2.0 and 3.0, while figure 6 shows the same for the wake locations. In all these figures, the exponential growth from the unstable exponential mode(s) are also plotted simultaneously to show the extent (in time) to which the transient growths dominate. Note here that in some of these figures, especially at the wake locations of figure 6, a missing exponential mode growth curve indicates an absence of any modal instability for those parameters.

Before proceeding further first note that, figure 5(a,b) clearly indicates the remarkable similarity of the $m=1$ transient gain curves for the exponential modes, across all the calculated axial wavenumbers (excepts perhaps at $\unicode[STIX]{x1D6FC}=0.1$ ), which moreover remains largely unchanged at locations around the centre of the recirculation bubble, unlike at the corresponding wake locations (see figure 6 a,b). Such observations provide important clues in support of the $m=1$ mode emerging as the dominant exponentially unstable global mode, as has been found by Oberleithner et al. (Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011) for this flow and Gallaire et al. (Reference Gallaire, Ruith, Meiburg, Chomaz and Huerre2006) for a different swirling jet. In fact, the apparent similarity of these gain profiles across the bubble locations in figure 5(a,b) also points to the possible origin of such a global mode being inside the recirculation bubble.

Next, a quick glance through figure 5 reveals that at both locations inside the bubble, the exponential growth remains quite strong, but especially at $m=1$ and 2, which easily exceeds the corresponding transient gains at relatively short times $(\unicode[STIX]{x1D70F}<20)$ for all the streamwise wavenumbers explored here. Not until the higher helical modes of $m=3$ and 4, at the relatively lower streamwise wavenumbers of $\unicode[STIX]{x1D6FC}=0.1$ and 0.3, do we find such transient gains inside the recirculation bubble to dominate over the corresponding exponential modal growth, over longer time periods (see figure 5 e–h).

The fact that the wake region has weaker modal instabilities coupled with the increased shifting of the potential mode spectrum closer to the unstable half-plane (see figure 4 e) is illustrated in figure 6 by an increased importance of short-time transient gains at these locations. Especially, for the higher helical modes of $m>1$ , the transient gains are seen to reach their respective peaks before the corresponding exponential modal growths exceeding them, for all axial wavenumbers $\unicode[STIX]{x1D6FC}\leqslant 0.5$ , while for $m>2$ , this bound is higher at $\unicode[STIX]{x1D6FC}\leqslant 1.0$ .

In summary, short-time transient gains are indeed more dynamically important in the wake of the swirling flow, since the energy gains peak in time well before the exponential growths exceed them (if at all present), but especially for the higher-order helical $(m\geqslant 2)$ and moderate to low streamwise wavenumber $(\unicode[STIX]{x1D6FC}\lessapprox 1.0)$ perturbations.

The details of transient gains from the remaining azimuthal wavenumbers (including $m=0$ ) and axial wavenumbers, at other streamwise locations are detailed in appendix A. There, the axisymmetric $m=0$ perturbations can be seen to be quite ineffective (especially at the low to moderate $\unicode[STIX]{x1D6FC}$ of figures 16 and 17), which is not that surprising given the fact that the specific mean flow of figure 1 corresponds to an axisymmetric breakdown state.

5.4 Evolution of the maximum amplification $G_{max}$

Instead of dwelling on all the optimal transient growth data generated for the parameters of § 5.1, which can be overwhelming (see appendix A), in this section, we rather focus on the maximum amplification $G_{max}$ from the stable spectrum. $G_{max}$ is defined as the maximum gain $G(\unicode[STIX]{x1D70F})$ from all the stable modes over all times  $\unicode[STIX]{x1D70F}$ , computed in this section for several parametric combinations to make better physical sense of the database. With the other flow parameters fixed, we first vary perturbation wavenumbers $m$ and $\unicode[STIX]{x1D6FC}$ of (3.3), yielding the cases shown in figure 7, computed at each of the ten streamwise locations of figure 1. Figure 7(a) documents the integer azimuthal mode numbers $m_{max}$ at which the maximum amplification $G_{max}$ occurs, at each of the $z^{\ast }/D^{\ast }$ locations over all the $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D70F}$ investigated. Surprisingly, the first helical mode $m=1$ does not seem to peak at any of the locations studied, in contrast to the findings of e.g. Oberleithner et al. (Reference Oberleithner, Sieber, Nayeri, Paschereit, Petz, Hege, Noack and Wygnanski2011), for the same mean flow and Gallaire et al. (Reference Gallaire, Ruith, Meiburg, Chomaz and Huerre2006) for a lower $Re$ swirling flow, both of which have been found to be globally unstable for $m=1$ perturbations. Moreover, except at a couple of locations near the beginning of the recirculation bubble, at all other locations an even azimuthal mode ( $m=2$ or 4) yields the maximum optimal amplification. The fact that the higher (and perhaps even) azimuthal modes are more sensitive to transient growths possibly points to different mechanisms than at lower $Re$ flows, which we shall explore in § 5.5.

Figure 7. Maximum transient energy gain $G_{max}$ at each streamwise location of figure 1 shown as a function of (a) azimuthal wavenumber $(m)$ , for all $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70F})$ , (b) streamwise wavenumber  $\unicode[STIX]{x1D6FC}$ , for all $(m,\unicode[STIX]{x1D70F})$ , along with (c) the respective magnitudes of this gain $|G_{max}|$ and (d) the time $\unicode[STIX]{x1D70F}_{max}$ , at these locations. The shaded area represents the streamwise extent of the recirculation bubble, similar to figure 1.

Figure 8. Transient evolution of $G_{max}$ as a function of (a)  $m$ , for all $(\unicode[STIX]{x1D6FC},z^{\ast }/D^{\ast })$ , (b)  $\unicode[STIX]{x1D6FC}$ , for all $(m,z^{\ast }/D^{\ast })$ and (c)  $z^{\ast }/D^{\ast }$ , for all $(m,\unicode[STIX]{x1D6FC})$ . In (d), spatial evolution of $G_{max}$ is shown at ——  $\unicode[STIX]{x1D70F}=10$ , –  –  –  $\unicode[STIX]{x1D70F}=30$ , –   $\cdot$   –   $\cdot$   $\unicode[STIX]{x1D70F}=75$ and —  —  —  $\unicode[STIX]{x1D70F}=125$ , with the three distinct regions differently shaded (see text). In (a–c), there are 25 $G_{max}$ contour levels in between 1 (blue) and $2\times 10^{7}$ (red), distributed exponentially. The dashed straight line in (c,d) indicates the approximate collapsing location of the recirculation bubble.

Figure 9. Same evolution of $G_{max}$ as in figure 8, but in the following planes: (a)  $(\unicode[STIX]{x1D6FC}-m)$ , for all $(z^{\ast }/D^{\ast },\unicode[STIX]{x1D70F})$ , (b)  $(z^{\ast }/D^{\ast }-m)$ , for all $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70F})$ and (c)  $(z^{\ast }/D^{\ast }-\unicode[STIX]{x1D6FC})$ , for all $(m,\unicode[STIX]{x1D70F})$ values.

On the other hand, the maximum axial wavenumbers $\unicode[STIX]{x1D6FC}_{max}$ as computed in figure 7(b) at first glance may seem to be scattered all over, but at the highest growth locations (on comparing with figure 7 c) their numbers are always within a range of $0.5\lessapprox \unicode[STIX]{x1D6FC}\lessapprox 2.5$ , except for the result at $z^{\ast }/D^{\ast }=1.4$ . Recall that at the locations $z^{\ast }/D^{\ast }=0.25$ , 1.4 and 1.5, which are near the ends of the recirculation bubble, the computed $\unicode[STIX]{x1D6FC}_{max}$ is clearly dubious due to the strong non-parallel effects present in the mean flow at these locations.

Figure 7(c) shows the location $z^{\ast }/D^{\ast }=1.0$ yielding the maximum $G_{max}$ among all the streamwise locations studied here. However, this amplification, almost an order higher than the next highest gain location at $z^{\ast }/D^{\ast }=1.8$ in the wake, occurs at a relatively higher time $\unicode[STIX]{x1D70F}_{max}\approx 90$ (see figure 7 d), by which point the corresponding exponential growth easily exceeds it, rendering the algebraic growth insignificant. In contrast, in the near wake region, the significant optimal amplifications ( $10^{5}<G_{max}<10^{6}$ ) occur within a much shorter duration $\unicode[STIX]{x1D70F}_{max}<50$ , when the corresponding exponential growths may not be significant, if at all present. For example, at $z^{\ast }/D^{\ast }=1.8$ , the peak transient amplification occurs before exponential modal growth exceeds it, as seen in figure 6(g) (at $m_{max}=4$ and $\unicode[STIX]{x1D6FC}_{max}=2.0$ ), which is thus dynamically more important for short-time transients than a location inside the bubble at $z^{\ast }/D^{\ast }=1.0$ .

In figures 8 and 9, we plot the contours of $G_{max}$ in several parametric planes to further visualize its significance. The temporal evolution seen in figure 8 repeats some of the observations just made in figure 7, including the fact that higher $G_{max}$ contours tend to be clustered at $m=2$ at larger times, while at even higher helical modes $(m=3,4)$ this appears at much smaller times $\unicode[STIX]{x1D70F}<50$ (see figure 8 a). As for $\unicode[STIX]{x1D6FC}$ in figure 8(b), several such ‘hot spots’ do seem to exist, especially at $\unicode[STIX]{x1D6FC}<2$ that is spread over a longer $\unicode[STIX]{x1D70F}$ and at $2<\unicode[STIX]{x1D6FC}<3$ , active over much shorter times. In figure 8(c), it is interesting to note that at $z^{\ast }/D^{\ast }=1$ , which has been found in figure 7 to yield the highest amplification at relatively longer times, in fact, also supports high optimal growth at relatively smaller $\unicode[STIX]{x1D70F}<50$ . This particular figure also clearly identifies the two other locations in the wake that are important for transient dynamics, at $z^{\ast }/D^{\ast }=1.8$ , where this is relevant at shorter times $(\unicode[STIX]{x1D70F}<50)$ , while at the far wake location of $z^{\ast }/D^{\ast }=2.6$ this occurs at two distinct time instants of $\unicode[STIX]{x1D70F}\approx 75$ , and then later at $\unicode[STIX]{x1D70F}\approx 175$ , with the former being more important.

In this context, it may be noted that the $G_{max}$ contours in figure 8(a–c) undergo a relatively slow decay, thereby stretching the time axis to relatively larger values of  $\unicode[STIX]{x1D70F}$ , which may not be physically realizable (see discussions in Pradeep & Hussain Reference Pradeep and Hussain2006; Mao & Sherwin Reference Mao and Sherwin2012). In contrast, the growth phase of $G_{max}$ happens quite fast, so that at $\unicode[STIX]{x1D70F}<30$ , the maximum transient gain exceeds $G_{max}>5\times 10^{6}$ , at $z^{\ast }/D^{\ast }=1.0$ , while by $\unicode[STIX]{x1D70F}<75$ this is at $G_{max}>7\times 10^{5}$ , located at the far wake $z^{\ast }/D^{\ast }=2.6$ (see figure 8 d). Such times taken to reach peak energy growth compare favourably with the peak nonlinear growth of a Batchelor vortex, observed at $\unicode[STIX]{x1D70F}\approx 60$ , via direct simulations (see Mao & Sherwin Reference Mao and Sherwin2012). Further, figure 8(d) shows that as $\unicode[STIX]{x1D70F}$ rises, the $G_{max}$ curves appear more discontinuous over the streamwise extent of the flow domain. Now this is not surprising, since the inherent non-parallelism of the underlying mean flow is expected to be more severe at higher  $\unicode[STIX]{x1D70F}$ . At the same time, this procedure confirms the existence of three distinct regions inside the flow (see figure 8 d), within which the maximum transient growth curves appear to be smooth, each of which are perhaps subjected to distinct transient dynamics, as we explore in the next section.

Figure 10. Absolute values of the optimal input perturbations (grey curves) and outcomes (coloured curves) computed for (a)  $m=1$ , $\unicode[STIX]{x1D6FC}=0.1$ at $z^{\ast }/D^{\ast }=0.6$ and (b)  $m=1$ , $\unicode[STIX]{x1D6FC}=2.0$ at $z^{\ast }/D^{\ast }=2.6$ , showing ——  $\hat{u} _{\unicode[STIX]{x1D703}}$ , –   $\cdot$   –   $\cdot$   $\hat{u} _{z}$ and –  –  –  $\hat{u} _{r}$ .

When plotted on the $m{-}\unicode[STIX]{x1D6FC}$ plane as in figure 9(a), this procedure shows higher azimuthal wavenumbers to yield higher $G_{max}$ levels at increasingly higher $\unicode[STIX]{x1D6FC}$ levels. For example, $m=2$ yields the highest amplification at $\unicode[STIX]{x1D6FC}\approx 1.5$ , $m=3$ at $\unicode[STIX]{x1D6FC}\approx 2.5$ and $m=4$ at $\unicode[STIX]{x1D6FC}\approx 3$ . Along the streamwise locations of figure 9(b,c), the expected locations $z^{\ast }/D^{\ast }=1.0$ , 1.8 and 2.6 get highlighted yet again, where at $z^{\ast }/D^{\ast }=1.0$ and 2.6, the azimuthal mode $m=2$ produces the largest optimal amplification, while at $z^{\ast }/D^{\ast }=1.8$ both $m=2$ and 4 seem to have the highest sensitivities, consistent with the findings of figure 7(a). As for  $\unicode[STIX]{x1D6FC}$ , figure 9(c) shows that as we move toward the wake region the highest transient amplifications tend to increasingly appear at the lower wavenumbers that yield longer-wavelength waves, which are therefore more likely to penetrate into these wake regions, further downstream.

In summary, the above analysis clearly establishes the locations of $z^{\ast }/D^{\ast }=1.0$ , inside the bubble and $z^{\ast }/D^{\ast }=1.8$ and 2.6, at the near and far wake, respectively, as most sensitive for transient amplifications, with the latter two locations far more dynamically important when considering the nature of corresponding exponential modal growth.

5.5 Transient growth mechanisms from optimal perturbations

After establishing the presence of strong short-time transient gains in the swirling jet of figure 1 in §§ 5.3 and 5.4, where the observed growths at the downstream wake region are shown to be more dynamically relevant to those occurring in the recirculation bubble, in this section, we focus on growth mechanisms to potentially explain the observed differences. Prior investigation of stability mechanisms in vortical flows have focused on several mathematical versions of such flows, starting from the classical Rankine vortex model to the Lamb–Oseen and Batchelor vortices. Among these, the Batchelor vortex model, that can represent both jet and wake-like vortices, is perhaps the best suited to our understanding of growth mechanisms in the swirling jet considered here. Mao & Sherwin (Reference Mao and Sherwin2011, Reference Mao and Sherwin2012), in fact, have demonstrated the presence of strong transient growths in Batchelor vortex flows, but they did not investigate the corresponding growth mechanisms in much detail. Instead, their analysis simply corroborates similar observations with the Lamb–Oseen vortex, for which extensive knowledge of growth mechanisms exists via the works of Antkowiak & Brancher (Reference Antkowiak and Brancher2004, Reference Antkowiak and Brancher2007), Pradeep & Hussain (Reference Pradeep and Hussain2006), Fontane et al. (Reference Fontane, Brancher and Fabre2008), among others. Although, the evolution of the measured swirling jet considered in this work is unlikely to be accurately modelled by any of the model vortex flows, including the Batchelor vortex (see e.g. Ruith et al. Reference Ruith, Chen, Meiburg and Maxworthy2003), it is still worthwhile exploring connections (if any) of the established transient evolution mechanisms in such model vortex flows, including the anti-lift-up, Orr and core induction mechanisms (see, especially Fontane et al. Reference Fontane, Brancher and Fabre2008; Mao & Sherwin Reference Mao and Sherwin2012), with the transient evolution of the present swirling jet, which we attempt in this section.

Toward that objective, figure 10(a) shows the radial structures of the optimal initial perturbations and their respective outcomes (at the corresponding $t=\unicode[STIX]{x1D70F}_{max}$ times) at a streamwise location inside the recirculation bubble ( $z^{\ast }/D^{\ast }=0.6$ ), for the parameters listed in the figure caption. Clearly, the swirl velocity $\hat{u} _{\unicode[STIX]{x1D703}}$ dominates over other velocity perturbations at the initial time. Such an optimal $\hat{u} _{\unicode[STIX]{x1D703}}$ perturbation, computed here over the continuous and stable discrete spectra (as discussed in § 5.2), peaks outside the vortex core, reconfirming its dominance in that region. The low-amplitude oscillations that can be seen at regions of $r<1$ (also in figure 10 b) for $\hat{u} _{\unicode[STIX]{x1D703}}$ (but curiously not for $\hat{u} _{z}$ and  $\hat{u} _{r}$ ) are likely due to the extra sensitivity that optimal initial azimuthal perturbations have when not including the unstable discrete modes in the optimal growth calculations. As these unstable modes dominate inside the swirling jet core at all times, especially during the input perturbation times when the response from stable modes is predominantly outside, their non-inclusion therefore fails to form a complete basis for an arbitrary perturbation inside the core region. As for the optimal output perturbations at time $t=\unicode[STIX]{x1D70F}_{max}$ , shown in the same figure, it is easy to see the decay of this initially large $\hat{u} _{\unicode[STIX]{x1D703}}$ component. The other velocity components, initially negligible, grow significantly, particularly  $\hat{u} _{z}$ , which dominates at $t=\unicode[STIX]{x1D70F}_{max}$ . Further, optimal outputs for $\hat{u} _{\unicode[STIX]{x1D703}}$ and $\hat{u} _{z}$ exhibit two peaks, one inside the core vortex region while the second one is immediately outside. At the low wavenumber of figure 10(a), this reflects observations of Mao (Reference Mao2010) in the context of Batchelor vortices, which in our case appears to point to a mechanism in which there is communication between regions inside the vortex core with those located immediately outside.

In figure 10(b), when the same components are plotted (at a higher $\unicode[STIX]{x1D6FC}=2.0$ ) at a location in the far wake ( $z^{\ast }/D^{\ast }=2.6$ ), the presence of an equally strong $\hat{u} _{z}$ component (along with $\hat{u} _{\unicode[STIX]{x1D703}}$ ) can also be seen in the initial perturbations, essentially reflecting the dominant $\bar{u}_{z}$ component at these wake locations, associated with the gradual reduction of the velocity deficit (see figure 1). The output structures at $t=\unicode[STIX]{x1D70F}_{max}$ indicate a slight reduction in $\hat{u} _{z}$ but a steep rise in the $\hat{u} _{\unicode[STIX]{x1D703}}$ and $\hat{u} _{r}$ components, especially at the jet core, where the twin-peak structures seen inside the bubble (figure 10 a) are not much apparent. At such higher wavenumbers in the wake, this is due to a strong energy transfer toward the core vortical region, with a near depletion of fluctuating energy outside.

The observations of figure 10 reveal that similar optimal initial perturbations can yield radically different output structures, thereby pointing to separate transient growth mechanisms active in such swirling flows that likely depend upon the streamwise location and the perturbation wavenumbers. This is now explored via a detailed parametric analysis through figures 11–15, where optimal spatial structures of the input (initial) $\tilde{\boldsymbol{q}}_{0}$ and the corresponding output perturbations $\tilde{\boldsymbol{q}}_{\unicode[STIX]{x1D70F}}$ at time $t=\unicode[STIX]{x1D70F}_{max}$ (corresponding to figures 5 and 6), at the two streamwise locations ( $z^{\ast }/D^{\ast }=1.0$ and 2.6) are presented for several combinations of $\unicode[STIX]{x1D6FC}$ and  $m$ . In each case, the visual evolution of optimal structures between the input and output times is achieved via isocontours of azimuthal velocity $\hat{u} _{\unicode[STIX]{x1D703}}$ on the streamwise ( $r-z$ ) planes and axial vorticity $\unicode[STIX]{x1D714}_{z}$ on the cross-stream planes, with a dashed line at $r=1$ locating the local maximum mean streamwise velocity $\bar{u}_{z}^{\ast }|_{max}$ (see § 3.1). Our primary aim is to investigate connection between these optimal structures and the corresponding nature of transient growth, particularly if certain spatial structures can be associated with the strongest gains. The transient gain $G(\unicode[STIX]{x1D70F})$ curves at these parameters are included within the larger dataset of figures 16–18 of appendix A, where for quick identification the specific two streamwise locations ( $z^{\ast }/D^{\ast }=1.0$ and 2.6) are separately highlighted.

Figure 11. Real contours of $\hat{u} _{\unicode[STIX]{x1D703}}$ shown in $\unicode[STIX]{x1D703}$ -planes and $\hat{\unicode[STIX]{x1D714}}_{z}$ in $z$ -planes for the first asymmetric mode $m=1$ at $z^{\ast }/D^{\ast }=1.0$ for the wavenumbers (a–d)  $\unicode[STIX]{x1D6FC}=0.5$ , (e–h)  $\unicode[STIX]{x1D6FC}=1.0$ and (i–l)  $\unicode[STIX]{x1D6FC}=2.0$ ; and at $z^{\ast }/D^{\ast }=2.6$ for (m–p)  $\unicode[STIX]{x1D6FC}=0.5$ and (q–t)  $\unicode[STIX]{x1D6FC}=2.0$ . In each case, the first two columns represent the input structure for optimal perturbations, while the last two columns are the corresponding outcome. The dashed lines in $\unicode[STIX]{x1D703}$ -planes refer to $r=1$ , while the dashed circles in $z$ -planes have a radius of 1. There are 20 uniformly spaced $\hat{u} _{\unicode[STIX]{x1D703}}$ and $\hat{\unicode[STIX]{x1D714}}_{z}$ contours in between $\pm 1$ and $\pm 75$ , respectively.

Here, we note that because of the procedures employed in § 3.3 in calculating the transient amplification, our method only yields the optimal initial and final conditions via (3.17). This method does not yield the optimal structures at intermediate times which may provide important clues to their transient evolution and hence the associated mechanisms. An algorithm involving sequential time integrations of the forward and adjoint evolution operators can yield such intermediate optimal states (see e.g. Corbett & Bottaro Reference Corbett and Bottaro2001; Barkley, Blackburn & Sherwin Reference Barkley, Blackburn and Sherwin2008), which is not pursued here. However, in the following we find several of the optimal input and output pairs in the present swirling jet to be remarkably similar to the documented optimal structures in a Lamb–Oseen vortex (see Fontane et al. Reference Fontane, Brancher and Fabre2008), while the one Batchelor vortex evolution case shown in Mao & Sherwin (Reference Mao and Sherwin2012) also looks identical. Such extensive similarity clearly suggests the associated mechanisms in the present cases to be of similar origin, in spite of the unavailable intermediate states of evolution.

As shown in § 5.4, the even azimuthal modes $m=2,4$ yield the highest transient amplifications, which will therefore be our primary focus. However in the following, we first use results from the lowest helical mode $m=1$ perturbations, shown in figure 11, to classify the different optimal structures observed in the present swirling jet by comparing them with similar observations from the Lamb–Oseen vortex. Here, the optimal initial perturbations appear as stacks of azimuthal velocity streaks in the streamwise planes or axial vorticity rings in the cross-stream planes (similar to observations elsewhere, e.g. Antkowiak & Brancher Reference Antkowiak and Brancher2004, Reference Antkowiak and Brancher2007; Fontane et al. Reference Fontane, Brancher and Fabre2008; Mao & Sherwin Reference Mao and Sherwin2012), which are seen to roll up differently depending upon their location with respect to the core vortical region (marked approximately by dashed lines in figures 11–15). If the streamwise $\hat{u} _{\unicode[STIX]{x1D703}}$ streaks are located almost exclusively inside the vortex core, as in $\unicode[STIX]{x1D6FC}=1.0$ at $z^{\ast }/D^{\ast }=1.0$ (see figure 11 e) and for $\unicode[STIX]{x1D6FC}=0.5$ at $z^{\ast }/D^{\ast }=2.6$ (see figure 11 m), these structures are uncoiled via the classical Orr mechanism, wherein the initial $\hat{\unicode[STIX]{x1D714}}_{z}$ spirals condense into the output structures of figure 11(h) and (p), respectively, which are now concentrated over smaller areas of the core vortical region. There is also a distinct reorientation of these streaks in the respective outputs (better visible in figure 11 g) toward the direction of maximum shear, again a characteristic of the Orr mechanism. In other situations, the input optimal streamwise $\hat{u} _{\unicode[STIX]{x1D703}}$ streaks are either predominantly located outside the vortex core (see figure 11 a,q), or these span across the $r=1$ line (see figure 11 i). Such initial structures also evolve over time via the Orr mechanism, induced by the mean flow differential rotation, but also undergo core contamination via a velocity induction, eventually leading to the dominant optimal structures of figure 11(d,l) and (t), respectively. Each of these has an inner core structure flanked by thin vortex spiral arms just outside of $r=1$ . In the meridian plane, these output optimal perturbations resemble wavy (sometimes breaded) structures in $\hat{u} _{\unicode[STIX]{x1D703}}$ (see figure 11 c,k,s).

Such optimal structures of the present swirling jet as shown in figure 11 and the following figures can now be explained with reference to the classification of Kelvin waves in the Lamb–Oseen vortex, extensively documented in Fabre, Sipp & Jacquin (Reference Fabre, Sipp and Jacquin2006). Strictly speaking, Kelvin waves (or simply vortex waves) are a family of damped and oscillatory modes, originally identified in the Rankine vortex. Here, this definition of Kelvin waves is extended to include modes in any such generic flows that include shear and rotation. Such waves appear via a transient resonance phenomenon, whereas the vortex spirals of the optimal initial condition uncoil and advect via the classical Orr mechanism, they concurrently yield a local reorganization of the vortical perturbations which supports velocity induction toward the core, leading to its eventual contamination (see discussions in Antkowiak & Brancher Reference Antkowiak and Brancher2004; Fabre et al. Reference Fabre, Sipp and Jacquin2006; Pradeep & Hussain Reference Pradeep and Hussain2006; Fontane et al. Reference Fontane, Brancher and Fabre2008). In the present case, two broad types of Kelvin wave structures are apparent, as discussed via observations in figure 11. The output perturbations of figure 11(d,l) and (t) represent an inner dipolar wave structure complemented by spiralling vorticity sheets located beyond the outer periphery of the vortex core. Indeed, via stochastic forcing of the Lamb–Oseen vortex similar structures can emerge (see Fontane et al. Reference Fontane, Brancher and Fabre2008), which are referred to as ‘critical-layer waves’ by Fabre et al. (Reference Fabre, Sipp and Jacquin2006). At moderate to high wavenumbers such two-component wave structures are expected in vortex flows, which can be ideal candidates for transferring the energy from an external perturbation field to the inner core. Antkowiak & Brancher (Reference Antkowiak and Brancher2004) have verified this, who observed the largest transient growths in the Lamb–Oseen vortex when these waves appear. In contrast, appearance of such optimal output structures only yield moderate maximum transient amplifications here (see in figures 17 d and 18 d). The second type of Kelvin wave appears when the perturbations starting from time $t=0$ (see figure 11 f,n) are mostly contained inside the core vortical region (instead of spreading into the quasi-potential region) even at $t=\unicode[STIX]{x1D70F}_{max}$ (see figure 11 h,p). This yields optimal output spiral structures via the Orr uncoiling, which are more compact than the respective initial optimal perturbations. Although not immediately apparent, these outputs are similar to the ‘core waves’ or perhaps the ‘purely viscous modes’, yet to be identified in practical vortex flows (see discussions in Fabre et al. Reference Fabre, Sipp and Jacquin2006), whose structures are fully contained inside the vortex core. In some cases (e.g. figure 11 p), such spiral structures tend to fill the vortex core with a larger number of spiral turns. Appearance of these Kelvin core waves, by far, yields the strongest transient amplifications if the corresponding output spiral structures are more concentrated inside the vortex core (see figure 11 h), in this case for the $(m=1,\unicode[STIX]{x1D6FC}=1.0)$ perturbations, as can be confirmed via figure 17(f) of appendix A. In contrast, the input $\unicode[STIX]{x1D6FC}=0.5$ perturbation case of figure 11(n) has an increased damping rate via its output core wave structure (see figure 11 p), thus yielding only moderate levels of transient growth (see figure 17 d). Therefore, appearance of structures akin to Kelvin core waves with a lower damping rate seems sufficient for strong transient growth for the present swirling jet, which will be further confirmed next with the higher- $m$ perturbations.

Figure 12. Same as in figure 11, but for $m=2$ at $z^{\ast }/D^{\ast }=1.0$ , shown for the wavenumbers (a–d)  $\unicode[STIX]{x1D6FC}=0.5$ ; (e–h)  $\unicode[STIX]{x1D6FC}=1.0$ ; (i–l)  $\unicode[STIX]{x1D6FC}=2.0$ and (m–p)  $\unicode[STIX]{x1D6FC}=3.0$ .

Figure 13. Same as in figure 12, but at $z^{\ast }/D^{\ast }=2.6$ .

Figure 14. Same as in figure 12, but for $m=4$ .

Figure 15. Same as in figure 13, but for $m=4$ .

For the higher helical modes, we show results for the even azimuthal modes $m=2$ (figures 12, 13) and $m=4$ (figures 14, 15), which are previously shown to cause the largest transient amplifications in figures 7(a), 8(a) and 9(b). The nature of optimal perturbations in the four cases of figure 12 is quickly apparent, where at all times the streamwise streaks of azimuthal velocity are mostly confined inside the vortex core (in contrast to, say, figure 11) thus resembling Kelvin core modes, with the potential for higher transient amplifications. The $\unicode[STIX]{x1D6FC}=2.0$ case (figure 12 i–l) clearly supports this with condensed spirals at the core region yielding the highest growth, which can be further confirmed from the corresponding transient amplification curve in figure 18(g). The other wavenumber cases pick output structures that fill the vortical core, except the higher wavenumber $\unicode[STIX]{x1D6FC}=3.0$ case (figure 12 m–p) that yields braided structures in the streamwise plane and a higher-order flat-top Kelvin wave structure in the cross-plane, again similar to observations by Fabre et al. (Reference Fabre, Sipp and Jacquin2006), Fontane et al. (Reference Fontane, Brancher and Fabre2008) at such wavenumbers. This particular structure is also seen in figure 18(h) to cause maximal amplification for the parameters. Figure 13 shows the situation at the wake location of $z^{\ast }/D^{\ast }=2.6$ . The input perturbation structures here are more radially spread out (see especially figure 13 f,j,n) than those observed inside the bubble in figure 12 yielding typical optimal output $\hat{\unicode[STIX]{x1D714}}_{z}$ structures of figure 13(h,l,p), each of which is composed of a compact inner structure bounded by four uncoiled initial vorticity sheets. In contrast to the observations of Fontane et al. (Reference Fontane, Brancher and Fabre2008), such output structures that have undergone core contamination, yield moderate to low transient amplifications here (see figures 17 i, 18 g,h). On the other hand, as the azimuthal streaks are simply reoriented to resemble the output $(\unicode[STIX]{x1D6FC}=0.5)$ core wave, shown in figure 13(a–d), it easily yields one of the highest transient amplifications among all cases studied, as can be confirmed from figure 17(g).

At even higher helical orders, the appearance of critical-layer Kelvin wave patterns diminishes over the wavenumbers studied, due to an apparent lack of the transient resonance mechanisms which creates such patterns. Instead, at the lower wavenumbers, the Orr mechanism which uncoils the initially spiralling structures yields the most amplification via the appearance of core wave-like modes. This is shown in figures 14 and 15 for the $m=4$ perturbations. The higher wavenumber $\unicode[STIX]{x1D6FC}=2.0$ case at $z^{\ast }/D^{\ast }=1.0$ (figure 14 i–l) and $\unicode[STIX]{x1D6FC}=3.0$ case at $z^{\ast }/D^{\ast }=2.6$ (figure 15 m–p) cause patterns that are primarily at the periphery of the vortical core region, apparently due to the maximum base-flow shear at these locations. Such instances also yield low transient amplifications, but are rare at these higher azimuthal wavenumbers. Instead, optimal output core modes of figure 14(a–d), figure 14(m–p) at $z^{\ast }/D^{\ast }=1.0$ and figure 15(e–h) at $z^{\ast }/D^{\ast }=2.6$ that undergo minimal damping with compact spiral structures yield the most significant transient gains (see figures 17 m, 18 n and 17 o, respectively).