3.1 Kelvin–Helmholtz wavepackets

In figure 3, the K–H wavepacket is identified as the most energetic coherent structure of the LES via POD for $m=0$ (a,c,e,g,i,k) and $m=4$ (b,d,f,h,j,l) over a range of frequencies. The wavepackets are characterized by a monotonic increase in wavenumber and a simultaneous decrease of their streamwise support with frequency. Their phase speed is $c_{ph}\approx 0.8U_{j}$ in all cases and was estimated as explained below in the context of figure 8.

Figure 3.

Spectral estimation of pressure POD modes (

, $-0.25\leqslant \unicode[STIX]{x1D6F9}_{p}/\Vert \unicode[STIX]{x1D6F9}_{p}\Vert _{\infty }\leqslant 0.25$ ) for different Strouhal numbers: (a,c,e,g,i,k)  $m=0$ ; (b,d,f,h,j,l)  $m=4$ .

The K–H instability amplifies to the end of the spatially unstable region of the mean flow and subsequently decays. For intermediate frequencies, the support of the resulting wavepacket extends from the nozzle up to approximately the end of the potential core, and PSE (Gudmundsson & Colonius Reference Gudmundsson and Colonius2011; Cavalieri et al. Reference Cavalieri, Rodríguez, Jordan, Colonius and Gervais2013; Baqui et al. Reference Baqui, Agarwal, Cavalieri and Sinayoko2015) and linearized Euler solutions (Baqui et al. Reference Baqui, Agarwal, Cavalieri and Sinayoko2015) were found to be in good agreement with measured or simulated structures. Recent studies based on the resolvent operator (Garnaud et al. Reference Garnaud, Lesshafft, Schmid and Huerre2013b ; Jeun, Nichols & Jovanović Reference Jeun, Nichols and Jovanović2016; Semeraro et al. Reference Semeraro, Lesshafft, Jaunet and Jordan2016; Qadri & Schmid Reference Qadri and Schmid2017; Tissot et al. Reference Tissot, Zhang, Lajús, Cavalieri and Jordan2017) furthermore showed that, due to the non-normal nature of the operator, the K–H instability can be forced efficiently through the Orr mechanism. In addition to the dominant K–H wavepacket, the trapped-wave component in the potential core is also clearly visible in figure 3(c).

3.2 Trapped acoustic waves

Trapped acoustic waves are extracted by use of CSD in figure 4 for the same Strouhal numbers considered in figure 3 above. The fluctuations inside the potential core are strongly correlated with the K–H wave in all cases. Inside the potential core, acoustic waves are observed for some $St$ – $m$ combinations, most prominently in figure 4(c,e,g,i,j). These trapped acoustic modes can be characterized by a radial order $n_{r}$ equal to the number of anti-nodes in the radial direction. By counting the anti-nodes, we find $n_{r}=1$ in figure 4(c,e), $n_{r}=2$ in figure 4(g,j) and $n_{r}=3$ in figure 4(i).

Figure 4.

CSD (

, $-0.25\leqslant \bar{C}_{pp}/\Vert \bar{C}_{pp}\Vert _{\infty }\leqslant 0.25$ ) correlating each point of the flow field with the location $(x_{0},r_{0})=(0.1,0.1)$ marked as ‘ $+$ ’ in the potential core for different Strouhal numbers and (a,c,e,g,i,k) $m=0$ and (b,d,f,h,j,l) $m=2$ .

4.1 Features of the continuous branch

Figure 6 shows three representative side-by-side comparisons of coherent structures distilled from the LES via CSD (a,c,e) and global modes of the continuous branch (b,d,f). The eigenvalue corresponding to the mode shown in figure 6(b) is located right below the least-damped mode $(0,1,1)$ in figure 5, i.e. at $St\approx 0.37$ and $\unicode[STIX]{x1D714}_{i}\approx -0.04$ . This mode is mainly of K–H type with only a weak acoustic component inside the potential core. The modal shape compares well with the CSD from the LES data shown in figure 6(a). Here, the correlation location of the CSD is chosen such that it extracts the shear-layer instability component. A correlation location inside the potential core brings to light the trapped acoustic wave, as discussed later in the context of figure 7(a,b).

Figure 6.

Comparison between (a,c,e) CSD of the LES with location $(x_{0},r_{0})$ marked by ‘ $+$ ’ and (b,d,f) global modes: (a,b)  $m=0$ , $St\approx 0.37$ ; (c,d)  $m=0$ , $St\approx 0.5$ ; (e,f)  $m=2$ , $St\approx 1$ . Modes are compared in terms of their normalized pressure components (

, $-0.25\leqslant \text{Re}(\tilde{\tilde{p}}_{m,\unicode[STIX]{x1D714}})/\Vert \text{Re}(\tilde{\tilde{p}}_{m,\unicode[STIX]{x1D714}})\Vert _{\infty }\leqslant 0.25$ , $-0.25\leqslant \text{Re}(\hat{p}_{m})/\Vert \text{Re}(\hat{p}_{m})\Vert _{\infty }\leqslant 0.25$ ).

A remarkable agreement is found for $m=0$ and $St=0.5$ as depicted in figure 6(c,d). Both the LES and the modal analyses identify a global structure that combines a K–H wave train with an acoustic core mode. The regions of support and the wavenumbers of both components are accurately predicted by the global mode. A similar modal structure can be seen in the $m=2$ , $St\approx 1$ case in figure 6(e,f). In this higher azimuthal wavenumber and frequency example, the CSD extracts the K–H wavepacket as the most energetic structure, whereas the trapped acoustic wave is dominant in the global mode.

In the spectrum, the continuous modes are unavoidably rendered discrete due to the numerical discretization. Changes in the numerical discretization scheme can freely shift their eigenvalues along the underlying continuous branch. Changing the computational domain size, as discussed in appendix C, is an example where such behaviour can be observed.

The physical aspect of interest for the present study is that the continuous modes consist of two coupled wave components: a downstream propagating K–H wavepacket, and a trapped acoustic wave in the potential core.

4.2 Features of the discrete modes

The leading discrete global modes of the first three branches for $m=0$ , and the first branch for $m=1$ are depicted in figures 7(b,d,f) and 7(h), respectively. Clearly, the discrete global modes represent the trapped acoustic mechanism. Unlike their continuous counterparts seen in figure 6(d,f) above, the discrete modes are confined within a small region $x\lesssim 2$ close to the nozzle. Their radial order $n_{r}$ becomes apparent by comparing modal shapes for different branches as in figure 7(b,d,f). For higher $m$ , the $n_{r}=1$ branch is shifted towards a higher frequency as can be deduced from figure 7(h) in comparison with the global spectra shown in figure 5. Note that all discrete global modes also possess a weaker K–H component, in contrast with the continuous modes which show K–H and trapped waves to be more strongly coupled. By comparison with the CSD (figure 7 a,c,e,g), it can be seen that the trapped acoustic modes are evidently present in the LES data. Their spatial extent and waveforms are accurately predicted by the linear global analysis. A similarly good agreement is found for all $m$ under consideration.

The pressure profile along the jet axis shown in figure 8(a) reveals the internal structure of the modes within a given branch. It can be seen that the absolute pressure signal possesses an increasing number of anti-nodes (now with respect to the $x$ axis) with increasing frequency. This suggests the definition of an integer streamwise wavenumber $n_{x}$ equal to the number of anti-nodes. Individual discrete modes can be identified by the triplet $(m,n_{r},n_{x})$ . The latter observation also has an important physical implication. The corresponding eigenvalues are reproduced in the detail of the spectrum in figure 8(b) for comparison. Figure 8(c) depicts the phase velocity of the same modes estimated as $\unicode[STIX]{x1D714}/(\unicode[STIX]{x2202}\unicode[STIX]{x1D703}_{p}/\unicode[STIX]{x2202}x)$ along the jet axis, where $\unicode[STIX]{x1D703}_{p}=\arg (\hat{p})$ is the local phase of the eigenvector pressure component. A jump discontinuity separates the pressure signal of each mode into part with a negative phase speed of $c_{p}\approx -0.2$ , and a part with a positive phase speed of $c_{p}\gtrsim 0.8U_{j}$ . The radial order $n_{r}$ previously addressed in the discussion of figure 7, is demonstrated in figure 8(d) in terms of radial velocity profiles of the leading modes of the first three branches for $m=0$ . In our parallel investigation, Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017) show that these waves behave like acoustic waves that experience the potential core as a cylindrical duct with a pressure release surface. Accordingly, they are effectively trapped within the potential core and their radial structure takes the form of Bessel functions. Therefore, we refer to them as duct modes.

Figure 7.

Comparison between (a,c,e,g) CSD of the LES with location $(x_{0},r_{0})$ marked by ‘ $+$  (blue)’ and (b,d,f,h) global modes: (a,b)  $m=0$ , $St\approx 0.37$ , mode $(0,1,1)$ ; (c,d)  $m=0$ , $St\approx 0.87$ , mode $(0,2,1)$ ; (e,f)  $m=1$ , $St\approx 0.6$ , mode $(1,1,1)$ . Modes are compared in terms of their normalized pressure components (

, $-0.25\leqslant \text{Re}(\tilde{\tilde{p}}_{m,\unicode[STIX]{x1D714}})/\Vert \text{Re}(\tilde{\tilde{p}}_{m,\unicode[STIX]{x1D714}})\Vert _{\infty }\leqslant 0.25$ , $-0.25\leqslant \text{Re}(\hat{p}_{m})/\Vert \text{Re}(\hat{p}_{m})\Vert _{\infty }\leqslant 0.25$ ).

Figure 8.

Characterization of trapped acoustic modes in the potential core for $m=0$ : (a) pressure magnitude along the jet axis; (b) detail of the spectrum showing the $(0,1)$ -branch (▴ (red), ▪ (blue) and ♦ (green), modes from (a,c); ● (black), other discrete modes; ○, continuous modes); (c) local phase velocity of the pressure along the jet axis; (d) pressure magnitude profiles at $x=1$ . All quantities are normalized with respect to their maximum value. The discrete modes are labelled as ( $m$ , $n_{r}$ , $n_{x}$ ), corresponding to their azimuthal wavenumber $m$ , their radial order $n_{r}$ obtained as shown in (d) and their streamwise order obtained as shown in (a), respectively.

5.1 Comparison with LES

We next examine the structure of the discrete modes in more detail by comparing the modes of the $(0,1)$ branch with the simulation in terms of the estimated PSD defined by (2.9). In order to account for the small frequency spacing between individual modes, the spectral estimation is based on a larger number of 1024 snapshots in the following. This results in a smaller ensemble of $N_{b}=19$ realizations for the same overlap.

Figure 9.

Comparison between the pressure components of LES power spectral density estimate (

, $0\leqslant \bar{P}_{pp}/\Vert \bar{P}_{pp}\Vert _{\infty }\leqslant 1$ ) and global mode ( $\hat{p}_{m}\hat{p}_{m}^{\ast }$ , —— (blue)) for branch $(0,1)$ : (a)  $St\approx 0.374$ , mode $(0,1,1)$ ; (b)  $St\approx 0.382$ , mode $(0,1,2)$ ; (c)  $St\approx 0.39$ , mode $(0,1,3)$ ; (d)  $St\approx 0.397$ , mode $(0,1,4)$ ; (e)  $St\approx 0.405$ , mode $(0,1,5)$ ; (f)  $St\approx 0.41$ , mode $(0,1,6)$ . Contours of $\hat{p}_{m}\hat{p}_{m}^{\ast }$ correspond to 20 %, 40 %, 60 % and 80 % of its maximum value.

Figure 9 shows that we find a very good agreement between the global modes and the LES in terms of the location and structure of the trapped acoustic waves. The number of maximum nodes agrees precisely with our definition of the streamwise wavenumber $n_{x}$ , as introduced in the context of figure 8(a). The trapped acoustic waves are spatially confined in the streamwise direction. In the inlet plane, we find that the nozzle impedance associated with the LES simulation is well mimicked by the inlet sponge boundary condition. This favourable agreement was found robust with respect to changes in the sponge layer intensity, as detailed in appendix B. The length of the region over which the trapped waves have support is frequency dependent. For example, mode $(0,1,1)$ in figure 9(a) has a streamwise spatial support of $0\lesssim x\lesssim 2$ , whereas mode $(0,1,6)$ shown in figure 9(a) is confined within $0\lesssim x\lesssim 3.5$ . The observation is explained by the existence of a frequency-dependent end condition related to the streamwise contraction of the potential core. We identify this end condition using weakly non-parallel stability theory in Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017), and compare the prediction to the spatial support of the global modes in § 5.6.

5.2 Experimental observations

Figure 10 compares discrete mode frequencies to spectra obtained from two sets of experimental data. In a recent experimental campaign dedicated to the validation of the present LES, the near-field pressure of a $M=0.9$ turbulent jet was measured in the ‘Bruit and Vent’ jet noise facility of the Pprime Institute, Poitiers. In a similar experiment by Suzuki & Colonius (Reference Suzuki and Colonius2006, appendix B) for $M_{j}=0.98$ , $T_{j}/T_{\infty }=0.84$ and $\text{Re}=16\times 10^{5}$ , the measured tones were attributed to an acoustic duct phenomenon upstream of the nozzle. Both experiments used a six microphone ring array located in the near-nozzle region. Spectra for the axisymmetric and first helical Fourier components are reproduced in panels 10(a) and 10(b), respectively. It can be seen that the peaks in the spectra agree well with the frequencies of the leading discrete modes found by the global analysis.

Figure 10.

Near-nozzle exit spectrum comparing Strouhal numbers of the least-damped global modes ( $\cdots \cdots$ ) to the sound pressure level (SPL) of recent ‘Bruit and Vent’ measurements (——), and the data of Suzuki & Colonius (Reference Suzuki and Colonius2006) (– – –) for $M_{j}=0.98$ , $T_{j}/T_{\infty }=0.84$ : (a)  $m=0$ ; (b)  $m=1$ .

5.3 Frequency–wavenumber analysis

Our next goal is to describe the trapped acoustic instabilities in the frequency–wavenumber domain. For a fixed frequency $\unicode[STIX]{x1D714}$ and azimuthal wavenumber $m$ , the streamwise wavenumber content of the trapped modes can readily be obtained from a Fourier decomposition of pressure signal on or parallel to the jet axis. Here, we chose $\tilde{\tilde{p}}_{m,\unicode[STIX]{x1D714}}(0\leqslant x\leqslant 5,r_{0})$ , and denote by $k$ the streamwise wavenumber. The exact length and radial location of the line segment used for the decomposition has no significant effect on the results, as long as it remains located inside the potential core. Windowing or padding of the segment is not necessary for the same reasons.

Figure 11.

Frequency–wavenumber diagram (

, $\bar{P}_{pp}(0\leqslant x\leqslant 5,r_{0})$ ) of the LES data in the potential core region for $m=0$ and $r_{0}=0.1$ . Lines of constant propagation velocity $c_{p}=\unicode[STIX]{x1D714}/k=c_{g}=\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}k$ and their corresponding values are shown in colour. $c_{p}=0.8U_{j}$ corresponds to the propagation speed of K–H instabilities.

The resulting frequency–wavenumber diagram for $m=0$ is shown in figure 11. The PSD is estimated by means of equation (2.9). In the positive $k$ -plane, energy is concentrated along the line of constant slope $\unicode[STIX]{x1D714}/k\approx 0.8U_{j}$ (—— (pink)), corresponding to the typical downstream propagation velocity of the initial jet shear-layer instability. The curved bands of energy in the angular sector of negative phase velocities $-a_{j}\leqslant c_{p}\leqslant U_{j}-a_{j}$ are the signatures of the trapped acoustic modes. These bands can be interpreted as empirical dispersion relations of the K–H and trapped acoustic modes, as they relate their frequency to their wavenumber. The lowest band originates from trapped acoustic modes of radial order $n_{r}=1$ , the second of $n_{r}=2$ and so forth. Note that the shape of the energy bands implies that multiple waves with negative phase velocity can coexist at certain frequencies. Furthermore, it can be observed that the two ends of each band asymptotically approach the lines $U_{j}-a_{j}$ and $-a_{j}$ , respectively. These two limits correspond to acoustic duct behaviour (– – – (green)) with waves propagating against the jet velocity inside the potential core and acoustic free-stream behaviour (– – – (blue)), with waves propagating upstream in the ambient flow.

Figure 12.

LES power spectral density in form of a frequency–wavenumber diagram obtained from the pressure signal along the jet axis in the potential core $x\in [0\;5]$ : (a)  $m=0$ ; (b)  $m=1$ ; (c)  $m=2$ ; (d)  $m=3$ ; (e)  $m=4$ . The Strouhal numbers of the leading global modes of each discrete branch ( $\cdots \cdots$  (red); marked by ‘ $\rightarrow$  (blue)’ in figure 5) are shown for comparison.

Details focusing on the trapped acoustic instability mechanism for higher azimuthal wavenumbers are presented in figure 12. It can be seen that the onset of the acoustic instability is shifted towards higher frequencies with increasing azimuthal wavenumber $m$ . A comparison with the global stability spectra shown in figure 5 confirms that the the local minima (valleys) occur precisely at the frequencies of the leading global modes of each discrete branch. For $m=0$ and $m=1$ , the PSD peaks within the $n_{r}=1$ energy bands, whereas higher radial order modes are most energetic for $m>1$ .

5.4 Reconstruction of the dispersion relation from global modes

We now focus on the wavenumber content of the global modes for a direct comparison with the LES. A frequency–wavenumber diagram analogous to figure 11 can be constructed from the global modes using exactly the same method as described above. For each mode, a Fourier decomposition of the pressure $\hat{p}_{m}(0\leqslant x\leqslant 5,r_{0})$ yields a wavenumber spectrum as depicted in figure 13(c). In the particular case of mode $(0,1,2)$ in this example, the spectrum exhibits three distinct peaks denoted by ‘ $k^{-}$ ’, ‘ $k^{+}$ ’ and ‘K–H’.

Figure 13.

Reconstruction of the dispersion relations of the trapped acoustic and K–H instabilities from a spectral analysis of the pressure signal along the jet axis in the potential core $x\in [0\;5]$ : (a) frequency–wavenumber diagram comparing LES (

, $\bar{P}_{pp}$ ) as in figure 12(a), and global modes (● (red), mode $(0,1,1)$ ; —— (red), – – – (red), trapped-mode branches; – – – (green), – – – (blue), continuous modes; —— (pink), K–H component); (b) LST spectrum around branch $(0,1)$ (● (red), mode $(0,1,1)$ ; $+$  (red)branch intersection mode; ○ (red) other $(0,1)$ modes; ▿ (blue), ▫ (green) continuous modes); (c) wavenumber spectrum of mode $(0,1,2)$ . Colours and symbols in (a) and (b) correspond. Since all modes comprise a K–H component, it is not indicated in (b).

The ‘ $k^{-}$ ’, ‘ $k^{+}$ ’ and ‘K–H’ peaks are identified for all discrete modes. Connecting them in the $St$ – $k$ plane yields the red lines in figure 13(a). Analogously, the leading (– – – (blue), free-stream behaviour) and trailing (– – – (green), duct behaviour) sections are found by connecting the dominant peaks in the spectrum of the continuous modes in negative $k$ -plane. Continuous segments can now readily be interpreted as reconstructed dispersion relations for the trapped waves and the K–H waves.

A detail of the spectrum around branch $(0,1)$ is reproduced in figure 13(b) using the same colours and symbols as in the frequency–wavenumber diagram in figure 13(a) for comparison. The concave part (red) of the lower band is directly associated with the discrete branch $(0,1)$ modes. Analogously, the upper band at $St\gtrsim 0.8$ is associated with the $(0,2)$ branch.

The mode located at the discrete-continuous branch intersection ( $+$  (red)) in figure 13(a,b) separates continuous from discrete, and also continuous duct modes from continuous free-stream modes. The local slope along each segment of the dispersion relation determines the group velocity of the instability wave at the corresponding wavenumber and frequency. This motivates the distinction between a $k^{+}$ (—— (red)) segment of positive, and a $k^{-}$ (– – – (red)) segment of negative group velocity, respectively. A remarkable agreement between the reconstructed and the empirical dispersion relations is found. The discrete global modes contain both the $k^{+}$ and the $k^{-}$ wave. This observation shows that these two waves are intimately linked, and suggests a resonance phenomenon, as we will confirm in the following.

Of particular interest are the two end points of the $k^{+}$ segment as they mark zeros of the slope of the dispersion relation, and therefore points of zero group velocity. These points are of particular interest as they determine the long-time behaviour of a system in a fixed laboratory reference frame (Briggs Reference Briggs1964). The lower point, denoted as S1, is associated with the leading mode and the upper one, denoted by S2, with the mode that marks the intersection of the discrete branch with the continuous spectrum. These points describe the transition from a mechanism that is predominantly active close to or inside the shear layer, i.e. continuous free-stream modes, to a duct-like behaviour inside the core, i.e. to continuous core modes. The anti-nodes used to determine the integer streamwise wavenumber $n_{x}$ , and as seen in figures 8(a) and 9, can now be interpreted in the light of the above discussion as a beating

(5.1) $$\begin{eqnarray}\text{e}^{\text{i}k^{+}x}+\text{e}^{\text{i}k^{-}x}=2\cos \left(\frac{\unicode[STIX]{x0394}k^{\pm }}{2}x\right)\exp \left(\text{i}\frac{k^{+}+k^{-}}{2}x\right)\end{eqnarray}$$

resulting from the superposition of the $k^{+}$ and the $k^{-}$ waves. As an example, consider mode $(0,1,3)$ with $k^{+}=-7.1$ , $k^{-}=-13.0$ and a resulting difference of $\unicode[STIX]{x0394}k^{\pm }=5.9$ . The corresponding wavelength of the beating, $\unicode[STIX]{x1D706}^{\pm }=4\unicode[STIX]{x03C0}/\unicode[STIX]{x0394}k^{\pm }=2.1$ , can be seen to accurately match twice the distance of the maximum nodes of mode $(0,1,3)$ in figure 8(a). The average wavenumber $(k^{+}+k^{-})/2$ on the other hand determines the wavelength seen in instantaneous, or visualizations of the real part as in figure 7. Together with the observation that the acoustic modes are confined within a short region in the streamwise direction by end conditions, it can be concluded at this point that a resonance between the $k^{+}$ and $k^{-}$ segments occurs. The resonance phenomenon in turn explains the high energy content along the frequency band in which $k^{+}$ and $k^{-}$ coexist. For the same reason, the corresponding modes appear as discrete eigenvalues in the spectrum.

For frequencies below the lowest radial order discrete branch, e.g. $St\lesssim 0.37$ for $m=0$ , the potential core does not support duct modes. The absence of the duct mechanism appears to be directly related to the sudden drop of the eigenvalues at low $St$ in the global stability spectra, as seen in figure 5 above. This becomes particularly apparent at higher azimuthal wavenumbers. For $m=3$ in figure 5(d), for example, all modes are heavily damped up to $St\approx 1$ . As soon as duct modes are supported, the discrete mode branch $(3,1)$ appears above a branch of K–H modes which now support acoustic core waves.

A caveat on the comparison between the reconstructed and the empirical dispersion relations is that the discrete nature of the trapped acoustic modes is not evident from the LES data. In the LES frequency–wavenumber diagram, the energy appears to be smoothly distributed along the individual bands. Presumably, a much longer time series would be needed to converge the statistics of individual discrete modes.

5.5 Isolation of the trapped acoustic waves

In classical local stability theory, each spatial mode is associated with a complex wavenumber given by its eigenvalue. By contrast, a single global mode of a specific complex frequency can encapsulate a variety of waves. We are interested in isolating the trapped acoustic wave component from the global modes to determine their spatial support, and for comparison with local theory.

Figure 14.

Same as figure 9 but comparing LES power spectral density estimate (

, $0\leqslant \bar{P}_{pp}/\Vert \bar{P}_{pp}\Vert _{\infty }\leqslant 1$ ) and the isolated trapped acoustic modes ( $\Vert \hat{q}_{m}(x_{0},r_{0})\Vert \cdot \Vert \hat{q}_{m}^{\dagger }(x_{0},r_{0})\Vert$ , —— (blue)). Contours of the product map correspond to 20 %, 40 %, 60 % and 80 % of its maximum value.

We have observed that adjoint trapped modes have a negligible K–H component. Examples and a brief discussion of the adjoint modes used here can be found in appendix E. Hence, the map of the local product

(5.2) $$\begin{eqnarray}\Vert \hat{q}_{m}(x_{0},r_{0})\Vert \cdot \Vert \hat{q}_{m}^{\dagger }(x_{0},r_{0})\Vert\end{eqnarray}$$

is a way of isolating the trapped-wave mechanism from the K–H wavepackets. In a different context, such maps have been used by Chomaz (Reference Chomaz2005) and Giannetti & Luchini (Reference Giannetti and Luchini2007) to determine an upper bound for the sensitivity of eigenvalues to spatially localized feedback. Figure 14 compares the map of the direct-adjoint product with the estimated PSD from the simulation.

The analysis clearly identifies the maximum nodes of the pressure PSD as the regions of highest sensitivity, and therefore closely resembles the PSD distribution of the global modes as seen in figure 9. Comparing figures 9 and 14, it becomes evident that multiplying the direct solution by the adjoint largely removes the shear-layer wavepackets. We can thus make use of the direct-adjoint product as a filter. The K–H instability constitutes a convective non-normality characterized by an upstream–downstream separation of the spatial support of the adjoint and direct modes (Sipp et al. Reference Sipp, Marquet, Meliga and Barbagallo2010; Nichols & Lele Reference Nichols and Lele2011). It is therefore widely eliminated from the product and the trapped waves are clearly isolated. Furthermore, the match between the PSD and the isolated trapped modes suggests that the trapped instability mechanism is nearly self-adjoint.

5.6 Local linear theory

The study of the resonant trapped acoustic modes in terms of local and weakly non-parallel spatio-temporal linear theory is at the core of the parallel investigation by Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017). Here, we revisit one of the main findings of the weakly non-parallel theory regarding the end condition of the resonant modes, as it allows for a direct comparison with the global modes. In particular, we are interested in finding saddle points in the complex $k$ -plane, i.e. the points $k_{0}$ at which $\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}k=0$ . In such saddle points, $k_{0}$ is a double root of the dispersion relationship. These points are relevant to provide end conditions if they are formed between a $k^{+}$ and a $k^{-}$ mode (Huerre & Monkewitz Reference Huerre and Monkewitz1990). Defining $\unicode[STIX]{x1D714}_{0}=\unicode[STIX]{x1D714}(k_{0})$ (i.e. $\unicode[STIX]{x1D714}_{0}$ and $k_{0}$ satisfy the locally parallel dispersion relationship), if $\unicode[STIX]{x1D714}_{0}$ is real, a saddle point presumably corresponds to the points S1 and S2 in the reconstructed dispersion relation shown in figure 13. The reader is referred to Towne et al. (Reference Towne, Cavalieri, Jordan, Colonius, Schmidt, Jaunet and Brès2017) for a detailed discussion. Under the assumption of a locally parallel flow, i.e. homogeneity in the streamwise direction, the normal-mode ansatz (2.2) takes the form

(5.3) $$\begin{eqnarray}q^{\prime }(r,\unicode[STIX]{x1D703},t)=\hat{q}_{m}(r)\text{e}^{\text{i}(kx+m\unicode[STIX]{x1D703}-\unicode[STIX]{x1D714}t)},\end{eqnarray}$$

where the streamwise wavenumber $k\in \mathbb{C}$ is a free parameter. The resulting eigenvalue problem for $\unicode[STIX]{x1D714}$ at some streamwise location $x_{0}$ takes the same form as in 2.3, but with ${\mathcal{L}}_{m}={\mathcal{L}}_{m}(\bar{q}(x_{0},r),k)$ now as the local linear stability operator. Saddle points are identified in the complex $k$ -plane by solving the local eigenvalue problem over a range of complex wavenumbers $k$ and subsequently finding the zeros of the complex derivative of $\unicode[STIX]{x1D714}(k)$ for a specific mode. In figure 15, the acoustic core modes of radial order $n_{r}=1$ are traced in the complex plane. Two saddles S1 and S2 are found for $x$ locations close to the nozzle. Both saddles reside on the real axis, and move closer together with increasing downstream distance, as seen in figure 15(a–c). At a certain streamwise location, the two saddles collide. For locations further downstream, no saddle points with real valued $k_{0}$ are found. Beyond this point, no $k^{+}$ -mechanism exists, and resonance cannot occur. This case is depicted in figure 15(d).

Figure 15.

Saddle point location of the $(0,1)$ branch in the complex wavenumber plane obtained from local spatio-temporal stability theory at different streamwise positions: (a)  $x=0$ ; (b)  $x=1.5$ ; (c)  $x=2.5$ ; (d)  $x=3.8$ . Isolines of the real and imaginary part of the complex frequency (—— (blue), $\unicode[STIX]{x1D714}_{r}$ ; —— (red), $\unicode[STIX]{x1D714}_{i}$ ) are shown, and the locations of the two saddle points (● (red), saddle S1; ○ (red), saddle S2) are obtained as the zeros of the complex derivative $\text{Re}(\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}ks)=\text{Im}(\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}ks)=0$ (– – – (blue), $\text{Re}(\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}k)$ ; – – – (red), $\text{Im}(\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}k)$ . Note that in (d), the saddles S1 and S1 cannot be distinguished, and the assignment is therefore arbitrary.

Figure 16.

Comparison between local spatio-temporal stability theory (—— (red), first saddle point location; – – – (red), second saddle point location), isolated trapped modes (blue areas), and mode at the branch intersection (green areas), denoted by ‘ $+$  (red)’ in figure 13(a,b) for branch $(0,1)$ ). The normalized direct-adjoint product is shown along the jet axis for $m=0$ , and along $r_{0}=0.25$ for $m=1$ in (b). The trapped modes clearly confined between the inlet and the saddle point locations.

In figure 16, the frequency of the two saddle points is plotted over the streamwise location (—— (red), – – – (red)) for three representative branches. The $(0,1)$ branch is considered in figure 16(a) and higher azimuthal wavenumber and radial order examples in panels 16(b) and 16(c), respectively. The isolated resonant acoustic modes (▪ (blue)) and the branch intersection mode (▪ (green)) are shown in the same plots for comparison. A good agreement between the spatial support of the resonant waves, as identified by the global stability analysis and the location of the S1 saddle, is found in all cases. It becomes apparent that the saddle point serves as an end condition for the trapped acoustic modes. With increasing $St$ , resonance first occurs for the leading discrete mode of axial order $n_{x}=1$ at the lowest possible frequency that permits the interaction between two waves of opposite group velocity with $k^{+}\approx k^{-}$ . At higher frequencies, the beating between the two waves $k+>k^{-}$ causes the observed modulation of the trapped modes of streamwise order $n_{x}>1$ . The frequency of the highest streamwise wavenumber mode approximately coincides with the minimum frequency of the S2 saddle. Beyond this point, no resonant core modes are to be found. Apparently, the S2 saddle provides an alternative end condition. In contrast to the S1 saddle, S2 is associated with a $k^{-}$ wave which apparently has a lower reflection coefficient at the nozzle than the duct modes. This becomes apparent from an inspection of the branch intersection modes (▪ (green)). The signature of the S1 saddle is clearly visible, but the fluctuation levels between the nozzle and S1 are much lower than for the discrete modes. From the detail views depicted in panels 16(d) and 16(e), respectively, the influence of the S2 saddle can be inferred in the case of the $(0,1)$ and $(1,1)$ branches, respectively. A clear beating signature is visible only between the nozzle exit and S2, whereas the fluctuation has a richer wavenumber content between S1 and S2.

The side-by-side comparison of local and global stability results in figure 16 highlights the complementary merits of both approaches. The global mode analysis reveals the connection between the different stability phenomena, identifies internal feedback and predicts the acoustic resonance. On the other hand, the local approach provides further physical insight about the physical identity of the waves and the nature of the end conditions.