4.1 Velocity fluctuations studied with Taylor’s hypothesis

Three representative PIV planes ( $x/D=3,5,7$ ) are selected, and the time coordinate transformed using Taylor hypothesis, $x_{t}=tU_{ref}$ , where $U_{ref}$ is chosen to be the mean velocity at the radial station analysed ( $r/D=0.55$ ). A pseudo-spanwise coordinate $z$ is obtained by rescaling azimuth $\unicode[STIX]{x1D703}$ as $z=r\unicode[STIX]{x1D703}$ . Axial velocity fluctuations at the planes $(x_{t},z)$ can be seen in figure 2.

Figure 2.

Axial velocity fluctuations over a $(x_{t},z)$ plane for a $M=0.4$ jet for several axial stations ( $x/D=3$ , 5, 7, respectively) and $r/D=0.55$ (a–c). The data were normalised by the mean streamwise velocity at the given radial station. Colour ranges from $-0.3$ to 0.3. Axial velocity fluctuations of large-eddy simulation results of a turbulent boundary layer at $y^{+}=120$ and $Re_{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}}=4430$  (Eitel-Amor, Örlü & Schlatter Reference Eitel-Amor, Örlü and Schlatter2014) are also shown (d).

Figure 3.

Energy spectrum (colours, full contours) in the $(k_{x},k_{z})$ domain for a $M=0.4$ jet for several axial stations ( $x/D=3$ , 5, 7, respectively) and $r/D=0.42,0.55$ . A comparison with results from Tomkins & Adrian (Reference Tomkins and Adrian2005) is also shown (dotted contours). Spectrum axes are normalised by the momentum thickness $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ .

Looking closely at figure 2, one can identify elongated streaky structures, resembling those found in wall-bounded flows, such as boundary layers (Hutchins & Marusic Reference Hutchins and Marusic2007), pipes (Hellström, Sinha & Smits Reference Hellström, Sinha and Smits2011) and channel flow (Monty et al. Reference Monty, Stewart, Williams and Chong2007), usually associated with very large-scale motions or superstructures in wall-bounded turbulence studies (Smits et al. Reference Smits, McKeon and Marusic2011). As is the case in turbulent boundary layers superstructures (Hutchins & Marusic Reference Hutchins and Marusic2007), the streaks meander in the axial direction in this free-shear flow. Also, the pseudo-spanwise wavelength changes from small values at positions near the nozzle, to larger values as the PIV measurement station is moved downstream; the streak width for $x/D=3$ is smaller than for the two other positions considered. This behaviour can be expected since, moving downstream, the shear-layer thickness also increases; the scaling of streak sizes with the local thickness is explored next. To highlight the resemblance of these streaks with the large-scale structures from turbulent boundary layers, we show the results from a large-eddy simulation performed by Eitel-Amor et al. (Reference Eitel-Amor, Örlü and Schlatter2014) at $y^{+}=120$ and $Re_{\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}}=4430$ in figure 2(d). Visually, these structures, that scale with the boundary layer thickness ( $\unicode[STIX]{x1D6FF}_{99}$ ), show high similarity to the ones shown here for turbulent jets.

From the fields depicted in figure 2, a spatial Fourier transform is performed in both the pseudo-axial and spanwise directions for $r/D=0.42$ and $r/D=0.55$ , using the mean velocity at each radial station as the reference velocity for Taylor’s hypothesis. The wavenumber spectra are shown in figure 3, from which some characteristics of the flow can be educed.

The first is that, for $r/D=0.42$ , the peak wavenumber is always located at $k_{x}\neq 0$ : it is already known that organised structures in the shape of axially extended wavepackets, with spatial evolution characterised by the Kelvin–Helmholtz instability, are present in this kind of flow (Cavalieri et al. Reference Cavalieri, Rodríguez, Jordan, Colonius and Gervais2013; Jordan & Colonius Reference Jordan and Colonius2013). Nevertheless, considerable energy is also located at $k_{z}\neq 0$ , which constitutes the signature of the streaky structures described above. As the streamwise position of the PIV plane is moved downstream, the energy spectrum becomes progressively concentrated around a specific spanwise wavenumber ( $k_{z}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}\approx 1$ ). The main cause of this change in the wavenumber peak is the growth and decay mechanism of wavepackets, the structures related to $k_{x}\neq 0$ and low $k_{z}$ : for regions near the nozzle, these structures have high amplitudes due to the high growth rate of the Kelvin–Helmholtz mode in regions of higher shear. These structures saturate and decay further downstream, causing their energy to decrease. On the other hand, the energy of the structures related to $k_{x}=0$ seems to grow as we move downstream. One important fact is that there is significant energy for $k_{x}\rightarrow 0$ and $k_{z}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}\approx 1$ for all positions considered. Figure 3(b,d,f) shows the same spectrum for $r/D=0.55$ , a region where wavepackets have lower amplitudes; one can see that the peak wavenumber in the spanwise direction remains around $k_{z}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}\approx 1$ , but now for $k_{x}=0$ . This is a sign that this region is not dominated by the wavepacket dynamics, but by the streamwise elongated structures seen in figure 2. Also, this result relates to what is visually perceived in figure 2: streaky structures have wavenumbers scaling with the local momentum thickness, and thus when plotted as a function of jet diameter one observes larger streaks at downstream stations.

To further highlight the relationship of these structures to those found in wall-bounded flows, we compare, in figure 3, the wavenumber spectrum of this flow with that of the turbulent boundary layer studied by Tomkins & Adrian (Reference Tomkins and Adrian2005). Even though the Reynolds number of the experiments differs ( $Re_{\unicode[STIX]{x1D703}}=92542$ at $x/D=7$ for the present experiment and $Re_{\unicode[STIX]{x1D703}}=1015$ for the boundary layer), the peak position in the low $k_{x}$ region is very similar for both cases (around $k_{z}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}=1$ ), confirming that the structures found experimentally in the present case are similar to those that are ubiquitous to boundary layers, having even a comparable energy spectrum. Previous work (Bogey, Marsden & Bailly Reference Bogey, Marsden and Bailly2011; Cavalieri et al. Reference Cavalieri, Rodríguez, Jordan, Colonius and Gervais2013) has highlighted this resemblance in the energy distribution via azimuthal Fourier-mode decomposition, showing similar peaks in the wavenumber spectrum with the proper scaling.

Finally, the premultiplied spectra for several axial positions of the PIV measurement, $k_{x}=0$ and $r/D=0.55$ can be seen in figure 4(a). The energy amplitudes of the streaks increase monotonically as we move the plane downstream, giving evidence that these structures are present with a higher amplitude in regions even further away from the nozzle, always peaking at the same value of $k_{z}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D703}}$ . The behaviour of the peak energy with the streamwise position of the PIV plane is shown in figure 4(b).

Figure 4.

Spectra for $k_{x}=0$ and $r/D=0.55$ .

4.2 Spectral proper orthogonal decomposition

Solving the eigenvalue problem defined by (3.3) using standard methods, we can obtain the energy of each SPOD mode $\unicode[STIX]{x1D70E}_{n}(m,\unicode[STIX]{x1D714})$ and the respective eigenfunction $\unicode[STIX]{x1D709}^{(n)}(x,r,m,\unicode[STIX]{x1D714})$ . Figure 5(a) shows the leading SPOD-mode energies for several Strouhal numbers $St=\unicode[STIX]{x1D714}D/(2\unicode[STIX]{x03C0}U_{jet})$ and azimuthal modes $m$ . The high-energy region for $m=0$ was shown to be related to a coherent wavepacket structure, as analysed by Semeraro et al. (Reference Semeraro, Jaunet, Jordan, Cavalieri and Lesshafft2016) and Lesshafft et al. (Reference Lesshafft, Semeraro, Jaunet, Cavalieri and Jordan2019), and is not the focus of the present work. Instead, we consider the $m>1$ region of the spectrum, where the reported streaky structures occur; in this region, the highest energies are found for $St=0$ (except for $m=1$ and 2), meaning that the structures are nearly steady in time, being related to a slow dynamic of the flow; here, SPOD modes for $St=0$ should be interpreted as the $St\rightarrow 0$ limit, with structures whose time scale is larger than the segment size used in Welch’s method. Even for $m=1$ and 2 the peak still occurs at low Strouhal numbers, supporting the hypothesis of a single mechanism. The energies of the first 4 SPOD modes for several azimuthal wavenumbers $m$ and $\unicode[STIX]{x1D714}=0$ can be seen in figure 5(b). As can be seen, a peak in $m=3$ arises for $n=1$ (which matches with the POD analysis performed by Hellström, Ganapathisubramani & Smits (Reference Hellström, Ganapathisubramani and Smits2015) for turbulent pipe flow) with a substantial separation between mode 1 and the suboptimals at the low $m$ region.

Figure 5.

First SPOD energy for several Strouhal numbers and azimuthal modes (a) and SPOD energies for each azimuthal mode for $\unicode[STIX]{x1D714}=0$ (b).

Following the analysis performed by Hellström, Marusic & Smits (Reference Hellström, Marusic and Smits2016), we define the azimuthal wavelength as $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}=2\unicode[STIX]{x03C0}/m$ , which leads to a characteristic length of $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}R$ , where $R=D/2$ . To obtain the characteristic streamwise development of the first SPOD mode, the eigenfunctions were radially integrated ( $\unicode[STIX]{x1D6EF}(x)=\int |\unicode[STIX]{x1D709}^{(1)}|r\,\text{d}r$ ) and then the axial coordinate was normalised using this new characteristic length. Normalising also the amplitude of each mode by its maximum, we find a collapse for these radially integrated first SPOD modes for azimuthal wavenumbers $m\geqslant 3$ . This underlines the fact that $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ is the important length for the most energetic structures in this flow, highlighting the similarity with what is found for turbulent pipe flow in Hellström et al. (Reference Hellström, Marusic and Smits2016). However, the present analysis differs from the cited work mainly in the direction where this similarity was found; Hellström et al. (Reference Hellström, Marusic and Smits2016) found the disturbances in pipe flow to be self-similar in the radial direction, whereas we found a similarity and a characteristic length in the streamwise direction.

Figure 6.

Shape of the first SPOD mode for several azimuthal modes.

Figure 7.

Streamwise velocity fluctuations of the first SPOD mode for several azimuthal modes.

Figure 8.

Streamwise velocity fluctuations of the second SPOD mode for several azimuthal modes.

Figure 9.

Isosurfaces of 40 % of the maximum amplitude of the first SPOD mode (positive values in red and negative ones in blue) and $m=10$ ; contours rendered for $0<\unicode[STIX]{x1D719}<\unicode[STIX]{x03C0}$ for clarity. An instantaneous field from the experiments at $x/D=2$ is also depicted.

A striking feature of figure 6(a) is that the leading modes are very large-scale motions, extending several jet diameters in $x$ . For instance, $m=10$ structures have a length of approximately 5 jet diameters, and lower- $m$ modes are even larger than that. The aspect ratio of structures can be estimated through the collapse found in figure 6(b), and corresponds, for $m\geqslant 3$ , to a ratio of axial over azimuthal length equal to approximately 15, highlighting again the streaky shape of these modes.

The shapes of the first and second SPOD modes for $St=0$ and several azimuthal wavenumbers can be seen in figures 7 and 8. The main behaviour is similar for all modes: the structures start to grow from the near-nozzle region, at radial locations close to the shear layer, where linear mechanisms are expected to be relevant, and experience a substantial growth in space, with high amplitudes in different regions of the flow. As a consequence, we can identify the regions where each azimuthal mode is acting more effectively. For instance, $m=1$ and 2 are related to disturbances more concentrated at regions with $r/D<0.5$ and higher $x/D$ . The oscillations in the streamwise direction of the second SPOD mode for higher azimuthal wavenumbers, shown in figure 8, also support the idea that these modes are related to streaky structures, representing their meandering motion observed in figure 2 by the phase opposition between upstream and downstream positions. One can see that the first and second SPOD modes do not decay within the region covered by the measurements for $m=1$ ; 2. Figure 7 also shows that all modes are located close to the nozzle as we increase the azimuthal wavenumber; this information, combined with the fact that streaks have higher amplitudes at downstream positions, which was inferred from figure 4(a), explains what is seen in the energy distribution of the SPOD modes: since lower azimuthal modes are related to more elongated structures, modes related to lower $m$ should be more energetic than the others at downstream positions.

On the other hand, for $m\geqslant 3$ , the modes have higher amplitudes close to the lipline, highlighting their action in the region of maximum shear. The axial support of the modes decreases with increasing $m$ , following the decrease of $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D703}}$ , as seen in figure 6, approaching progressively the nozzle. The fact that the peak position of these high azimuthal wavenumber streaks is in the upstream region indicates that these modes may well be captured by linear analysis. Particularly, the first SPOD mode for $m=10$ has a high amplitude at $x/D=2$ ; figure 9 shows a sketch allowing this mode to be visualised with an instantaneous field at that position. It can be seen that the flow also possesses the oscillatory behaviour in azimuth, displaying an instantaneous cross-section of the streaky structures at that position with the correct radial support. Looking at lower radial positions, the signature of an $m=1$ streaky structure can also be identified, albeit with smaller amplitudes, which again agrees with the analysis of figure 7, since low- $m$ modes are more relevant for higher $x/D$ . The similarities between the structure of this mode and the instantaneous velocity field are clear, showing that these streaks are indeed present in snapshots of the turbulent jet.

In the present manuscript, we attempt to evaluate the correlation (or coherence) length of streaky structures using SPOD. It is also possible to use the cross-spectral density to evaluate length scales, similar to what is done by Jaunet et al. (Reference Jaunet, Jordan and Cavalieri2017); however, this requires taking specific choices for streamwise and radial positions, and a complete analysis for all possible positions becomes cumbersome. The use of SPOD circumvents these problems, as the dominant spatial structure in the whole cross-spectral density field is obtained in the leading eigenfunction. This allows a global estimate of length scales of dominant turbulent structures.

4.3 Comparison with linear models

We now focus on analysis of the radial shapes of the most energetic disturbances in the flow for $St=0$ and azimuthal wavenumbers $m=3$ , $m=7$ and $m=10$ at $x/D=2$ , a region where linear mechanisms are expected to be more relevant due to the higher shear. Cross-sections of the leading SPOD modes are shown in figure 10(a,c,e) and, for these azimuthal wavenumbers, it can be seen that most of mode dynamics is concentrated in the sheared region, especially around the lipline, with low amplitudes in the central region where the mean velocity is approximately uniform. As previously mentioned, the shapes of the SPOD modes are similar to what is found for turbulent pipe flow (Hellström et al. Reference Hellström, Ganapathisubramani and Smits2015, Reference Hellström, Marusic and Smits2016); the main difference in the present flow is the absence of the wall, showing that this is not a sine qua non condition for the appearance of streaky structures in a flow and that the sheared mean flow of the jet alone supports their emergence, consistent with the analysis of Ellingsen & Palm (Reference Ellingsen and Palm1975). Another conclusion is that the radial decay of the structures is also fast. However, one should be careful in the analysis of the radial decay of the structures; since the PIV window covered the interval $0<r/D<0.8$ , structures with a radial support larger than that would be suppressed. Further analysis of the SPOD modes will be performed in the following sections, where a comparison between the most energetic structures with results from other tools (mostly used in wall-bounded turbulence analysis) will be performed. All analyses were performed using the experimental Reynolds number, considering the molecular viscosity.

Figure 10.

First SPOD and resolvent modes for azimuthal wavenumbers $m=3$ , $m=7$ and $m=10$ (real part). Optimal responses from resolvent analysis are shown for the same radial interval of the experiments for comparison ( $r/D<0.8$ ). The dashed line indicates the position of the lipline.

4.3.1 Resolvent analysis

Using the formulation described in § 3.2, we obtain resolvent modes for the same frequency $\unicode[STIX]{x1D714}=0$ and wavenumbers $m=3;7;10$ for the $M=0.4$ jet at $x/D=2$ . The structures of the first mode can be seen in figure 10(b,d,f). We note that these plots retain the key characteristics of the SPOD modes, suggesting that these structures can be obtained via resolvent analysis of the turbulent mean. Comparing figures 10(b,d,f) and 10(a,c,e), we see a good agreement in terms of both the position of the peak for $m=7$ and 10, using the lipline as reference point, and the radial support of these modes. For $m=3$ , however, only the position of peak amplitude is captured by the resolvent analysis, which resulted in modes with radial shape that differ in width from the SPOD results. One of the reasons for that is the axial structure of the mode: as seen in figure 7, amplitudes of the leading $m=3$ SPOD mode at $x/D=2$ are rather low, and become more relevant only further downstream. The local resolvent analysis is expected to better match the SPOD results if these do not vary considerably in $x$ , since in the local resolvent analysis it is assumed that $k_{x}=0$ , which leads to constant amplitudes in $x$ . Still, even with this assumption, the local resolvent analysis is, on the whole, able to predict the shapes of the leading SPOD modes, particularly for higher azimuthal wavenumbers.

Another conclusion can be drawn from this analysis. It was shown by Semeraro et al. (Reference Semeraro, Jaunet, Jordan, Cavalieri and Lesshafft2016) and Towne et al. (Reference Towne, Schmidt and Colonius2018) that resolvent and SPOD modes should be identical if the nonlinear terms in the Navier–Stokes system are white in space. With that in mind, the resemblance between SPOD and resolvent modes in the present work suggests that the nonlinear terms at that position (considered as forcing herein), may be related to uncorrelated structures, that behave in the manner of white noise, at least for high azimuthal wavenumbers.

Information about the optimal forcing shape is also of interest. One should recall that, for wall-bounded flows, the growth of streaks via the lift-up effect plays an important role (Schmid & Henningson Reference Schmid and Henningson2012; Brandt Reference Brandt2014), and is one of the dominant phenomena in the flow dynamics. These streaks are forced by streamwise vortices (or rolls) in a particular manner: positive streaks are related to forcing that advects flow from faster to slower regions, with the opposite happening for negative streaks. This roll–streak dynamics is also suggested to be one of the building blocks that allows turbulence to self-sustain, as shown in Hamilton et al. (Reference Hamilton, Kim and Waleffe1995), Waleffe (Reference Waleffe1995) and Hwang & Bengana (Reference Hwang and Bengana2016). The optimal forcing from the resolvent analysis, and that leads to the streaky responses in figure 10(b,d,f), is shown in figure 11. The optimal forcing is seen to consist mainly of radial and azimuthal fluctuations; the streamwise component is negligible for the azimuthal wavenumbers studied here. We can see that the same lift-up mechanism occurs for the present case: whenever a positive streak occurs in the response mode, it is always related to the movement of fluid from fast to slow regions (or from the inner part of the jet to the outer region). The presence of these vortices in the forcing modes supports the hypothesis that the lift-up mechanism is also relevant for turbulent jets.

Figure 11.

Radial and azimuthal components of the optimal forcing (arrows) and normalised streamwise component of optimal response (grey scale) from resolvent analysis for $m=3$ , $m=7$ and $m=10$ .

Figure 12.

Comparison between SPOD modes ( $\unicode[STIX]{x1D714}=0$ ), response modes from resolvent analysis ( $\unicode[STIX]{x1D714}=0$ , $k_{x}=0$ ) and response at the time of optimal growth from transient growth analysis ( $k_{x}=0$ ). The optimal is shown for $m=3$ , $m=7$ and $m=10$ .

Figure 13.

Growth curves from transient growth analysis ( $k_{x}=0$ ).

4.3.2 Transient growth

A transient growth study of Michalke’s velocity profiles for jets and slowly diffusing jet flows was performed by Jiménez-González & Brancher (Reference Jiménez-González and Brancher2017). In the present work, we extend the analysis to the turbulent jet, considering as base flow the experimental mean flow at $x/D=2$ , using an implementation similar to the resolvent analysis described in the previous section. We are thus not looking for initial disturbances in a laminar flow that will generate the largest growth, triggering transition, as in the cited work; we focus, rather, on turbulent structures with the largest transient growth in time, and which would then naturally emerge as large-scale structures. This amounts to an identification of the responses that predominate in the flow and the initial disturbances that lead to them. Also, using transient growth analysis, we obtain an idea of the time development of these disturbances, giving relevant information about the time scales over which this dynamics takes place.

The response at time of optimal growth for the transient growth analysis is similar to what was obtained by the resolvent analysis, with streaky structures concentrated close to the lipline. Figure 12 shows a comparison between the modes with optimal transient growth, those obtained by resolvent analysis and a cross-section of the global SPOD at $x/D=2$ . These plots emphasise the conclusions drawn in the previous section: the transient growth analysis captures well the peak position of the streaks for the cases studied and even the radial support of the modes is well represented for higher azimuthal modes ( $m=7;10$ ). It can also be seen that the structure of $m=3$ SPOD mode at this position has a much more experimental noise than in the other ones, which is probably due to its low relative amplitude at $x/D=2$ . Still, the optimal responses from both resolvent and transient growth analyses match the SPOD modes for the azimuthal wavenumbers studied, with improving results as $m$ increases.

One should recall that resolvent and SPOD modes were calculated for $\unicode[STIX]{x1D714}=0$ , focusing on structures that have a very slow evolution (not to say invariant) in time. On the other hand, transient growth involves a different approach: the nonlinear terms are set aside (there is no forcing) and we investigate initial disturbances that would have the highest growth for a given wavenumber. The fact that the present analysis shows a good agreement with resolvent modes is an indication that the dominant dynamics in this flow occurs in a low-frequency, low wavenumber range, and is thus a long-time behaviour. This suggests the possibility of using non-modal stability methods for turbulent jets not only for the low-energy wavepackets, but also for the high-energy structures, similar to the approaches used for wall-bounded flows.

The slow dynamics of the phenomenon is supported by the growth curves for each case, depicted in figure 13(a). The large values of time for peak transient growth are also in accordance with the behaviour found by Jiménez-González & Brancher (Reference Jiménez-González and Brancher2017), who showed that the time of maximum growth scales with the Reynolds number; their results highlight a monotonic reduction of the optimal time of growth with increasing $m$ , which also agrees with the trend identified in figure 13(a). One should nonetheless bear in mind that, since this analysis is restricted to a slice of the jet, this time scale is not related to a streamwise vortex generated close to the nozzle, which would evolve in time and space and become the resulting streak. In the locally parallel framework, the time is related to a vortex initially generated at the same position as the resulting streak, clouding the link between this result and the real time evolution of the structure.

Figure 13(b) shows how the growth curve changes with the axial station of the base flow chosen for the calculation for $m=7$ , an azimuthal wavenumber whose first SPOD mode extends through the entire domain. This result shows that this linear mechanism is more relevant in regions closer to the nozzle, supporting our previous hypothesis. This behaviour is also similar to what was found by Jiménez-González & Brancher (Reference Jiménez-González and Brancher2017), who found smoother growth curves with lower optimal gains and higher optimal times for increasing shear layer momentum thickness; in our case, the momentum thickness of each axial section increases as we move downstream, leading to growth curves that have the same behaviour.

The initial disturbance that generates the optimally growing streaks can be seen in figure 14. The similarities with results from Jiménez-González & Brancher (Reference Jiménez-González and Brancher2017) and with the optimal forcing from the resolvent analysis confirms that the streaks are generated via the lift-up mechanism. Initial perturbations in the shape of streamwise vortices are those that will suffer the largest growth, evolving into streamwise elongated streaks, resembling those found by Cossu et al. (Reference Cossu, Pujals and Depardon2009). These vortices are also related to those found by Tinney et al. (Reference Tinney, Glauser and Ukeiley2008); the main novelty in the present study is their connection with the dominant streaks. Once again, the presence of these structures concentrated in the shear region, close to the lipline, is a suggestion that this, the lift-up mechanism, is related to any shear flow, not necessarily with the presence of a wall.

Figure 14.

Radial and azimuthal components of the optimal initial disturbance (arrows) and streamwise component of optimal response (grey scale) from transient growth analysis for $m=3$ , $m=7$ and $m=10$ .

4.3.3 Streaks downstream of the potential core

In the previous sections we have chosen to perform the numerical analysis in regions closer to the nozzle, since the linear mechanisms we consider are expected to be stronger in this region due to the higher mean shear. Still, as shown in figures 6 and 7, these structures reach positions downstream of the potential core ( $x/D>5.5$ for this jet), especially for low azimuthal wavenumbers; we here evaluate if linear models are able to predict streak shapes in downstream locations. Figure 15 shows the comparison between the radial dependency of the SPOD modes at $x/D=7$ for $m=3$ ; 4; 5 (wavenumbers for which the streaks have relevant amplitudes in this position) and the linear models using the base flow at this axial station, keeping the experimental Reynolds number in the analysis. At this position, the SPOD results show that the radial support of the most energetic structure considerably exceeds the measurement window, not only showing that streaks are larger in this region, but also making the analysis problematic, since about half of the radial information is lost. However, the models were able to qualitatively reproduce the sharp radial growth of the streak amplitude from the jet axis, with better results for transient growth analysis, especially for higher azimuthal wavenumbers.

Figure 15.

Comparison between SPOD modes ( $\unicode[STIX]{x1D714}=0$ ), response modes from resolvent analysis ( $\unicode[STIX]{x1D714}=0$ , $k_{x}=0$ ) and response at the time of optimal growth from transient growth analysis ( $k_{x}=0$ ) for $x/D=7$ . The optimal is shown for $m=3$ , $m=4$ and $m=5$ .